Discovering
GeometryAn Investigative Approach
Discovering
GeometryAn Investigative Approach
Practice Your Skills
with Answers
Teacher’s Materials Project Editor: Elizabeth DeCarli
Project Administrator: Brady Golden
Coordinating Writer: Jennifer North Morris
Contributors: David Rasmussen, Ralph Bothe, Judy Hicks, Michael Serra
Accuracy Checker: Dudley Brooks
Production Editor: Holly Rudelitsch
Copyeditor: Jill Pellarin
Editorial Production Manager: Christine Osborne
Production Supervisor: Ann Rothenbuhler
Production Coordinator: Jennifer Young
Text Designers: Jenny Somerville, Garry Harman
Composition, Technical Art, Prepress: ICC Macmillan Inc.
Cover Designer: Jill Kongabel
Printer: Data Reproductions
Textbook Product Manager: James Ryan
Executive Editor: Casey FitzSimons
Publisher: Steven Rasmussen
©2008 by Key Curriculum Press. All rights reserved.
Cover Photo Credits: Background image: Doug Wilson/Westlight/Corbis.
Construction site image: Sonda Dawes/The Image Works. All other images:
Ken Karp Photography.
Limited Reproduction Permission
The publisher grants the teacher whose school has adopted Discovering Geometry,
and who has received Discovering Geometry: An Investigative Approach, Practice
Your Skills with Answers as part of the Teaching Resources package for the book,
the right to reproduce material for use in his or her own classroom. Unauthorized
copying of Discovering Geometry: An Investigative Approach, Practice Your Skills
with Answers constitutes copyright infringement and is a violation of federal law.
®Key Curriculum Press is a registered trademark of Key Curriculum Press.
All registered trademarks and trademarks in this book are the property of
their respective holders.
Key Curriculum Press
1150 65th Street
Emeryville, CA 94608
510-595-7000
editorial@keypress.com
www.keypress.com
Printed in the United States of America
1098765432 131211100908 ISBN 978-1-55953-894-7
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Chapter 1
Lesson 1.1:Building Blocks of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Lesson 1.2: Poolroom Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Lesson 1.3: What’s a Widget? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Lesson 1.4: Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Lesson 1.5: Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lesson 1.6:Special Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Lesson 1.7:Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Lesson 1.8:Space Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Lesson 1.9: A Picture Is Worth a Thousand Words . . . . . . . . . . . . . . . . . . . 9
Chapter 2
Lesson 2.1:Inductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lesson 2.2: Finding the nth Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Lesson 2.3:Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Lesson 2.4:Deductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Lesson 2.5:Angle Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Lesson 2.6:Special Angles on Parallel Lines . . . . . . . . . . . . . . . . . . . . . . .
Chapter 3
Lesson 3.1: Duplicating Segments and Angles . . . . . . . . . . . . . . . . . . . . 16
Lesson 3.2: Constructing Perpendicular Bisectors . . . . . . . . . . . . . . . . . . 17
Lesson 3.3: Constructing Perpendiculars to a Line . . . . . . . . . . . . . . . . . 18
Lesson 3.4: Constructing Angle Bisectors . . . . . . . . . . . . . . . . . . . . . . . . . 19
Lesson 3.5: Constructing Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lesson 3.6: Construction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Lesson 3.7: Constructing Points of Concurrency . . . . . . . . . . . . . . . . . . . 22
Lesson 3.8: The Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter 4
Lesson 4.1: Triangle Sum Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Lesson 4.2: Properties of Isosceles Triangles . . . . . . . . . . . . . . . . . . . . . .
Lesson 4.3: Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Lesson 4.4: Are There Congruence Shortcuts? . . . . . . . . . . . . . . . . . . . . . 27
Lesson 4.5: Are There Other Congruence Shortcuts? . . . . . . . . . . . . . . . 28
Lesson 4.6: Corresponding Parts of Congruent Triangles . . . . . . . . . . . 29
Lesson 4.7: Flowchart Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lesson 4.8: Proving Special Triangle Conjectures . . . . . . . . . . . . . . . . . . 31
iii
Chapter 5
Lesson 5.1: Polygon Sum Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Lesson 5.2:Exterior Angles of a Polygon . . . . . . . . . . . . . . . . . . . . . . . . . 33
Lesson 5.3:Kite and Trapezoid Properties . . . . . . . . . . . . . . . . . . . . . . . . 34
Lesson 5.4: Properties of Midsegments . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Lesson 5.5: Properties of Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Lesson 5.6: Properties of Special Parallelograms . . . . . . . . . . . . . . . . . . 37
Lesson 5.7: Proving Quadrilateral Properties . . . . . . . . . . . . . . . . . . . . . . 38
Chapter 6
Lesson 6.1: Tangent Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Lesson 6.2:Chord Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Lesson 6.3:Arcs and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Lesson 6.4: Proving Circle Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Lesson 6.5: The Circumference/Diameter Ratio . . . . . . . . . . . . . . . . . . . . 43
Lesson 6.6:Around the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Lesson 6.7:Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Exploration: Intersecting Secants, Tangents, and Chords . . . . . . . . . . . 46
Chapter 7
Lesson 7.1: Transformations and Symmetry . . . . . . . . . . . . . . . . . . . . . . . 47
Lesson 7.2: Properties of Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Lesson 7.3: Compositions of Transformations . . . . . . . . . . . . . . . . . . . . . 49
Lesson 7.4: Tessellations with Regular Polygons . . . . . . . . . . . . . . . . . . . 50
Lessons 7.5–7.8: Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter 8
Lesson 8.1: Areas of Rectangles and Parallelograms . . . . . . . . . . . . . . . 52
Lesson 8.2: Areas of Triangles, Trapezoids, and Kites . . . . . . . . . . . . . . . 53
Lesson 8.3:Area Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Lesson 8.4:Areas of Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Lesson 8.5:Areas of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Lesson 8.6:Any Way You Slice It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Lesson 8.7:Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 9
Lesson 9.1: The Theorem of Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Lesson 9.2: The Converse of the Pythagorean Theorem . . . . . . . . . . . . 60
Lesson 9.3: Two Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Lesson 9.4:Story Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Lesson 9.5:Distance in Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . 63
Lesson 9.6: Circles and the Pythagorean Theorem . . . . . . . . . . . . . . . . . 64
Chapter 10
Lesson 10.1: The Geometry of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Lesson 10.2: Volume of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . 66
Lesson 10.3: Volume of Pyramids and Cones . . . . . . . . . . . . . . . . . . . . . . 67
Lesson 10.4: Volume Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Lesson 10.5:Displacement and Density . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Lesson 10.6: Volume of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Lesson 10.7:Surface Area of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter 11
Lesson 11.1:Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Lesson 11.2:Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Lesson 11.3: Indirect Measurement with Similar Triangles . . . . . . . . . . 74
Lesson 11.4: Corresponding Parts of Similar Triangles . . . . . . . . . . . . . 75
Lesson 11.5: Proportions with Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Lesson 11.6: Proportions with Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Lesson 11.7: Proportional Segments Between Parallel Lines . . . . . . . . 78
Chapter 12
Lesson 12.1: Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Lesson 12.2: Problem Solving with Right Triangles . . . . . . . . . . . . . . . . 80
Lesson 12.3: The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Lesson 12.4: The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Lesson 12.5: Problem Solving with Trigonometry . . . . . . . . . . . . . . . . . . 83
Chapter 13
Lesson 13.1: The Premises of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Lesson 13.2:Planning a Geometry Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Lesson 13.3: Triangle Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Lesson 13.4:Quadrilateral Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Lesson 13.5:Indirect Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Lesson 13.6:Circle Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Lesson 13.7:Similarity Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
v
Introduction
The author and editors of Discovering Geometry: An Investigative Approach are
aware of the importance of students developing geometry skills along with
acquiring concepts through investigation. The student book includes many
skill-based exercises. These Practice Your Skills worksheets provide problems
similar to the introductory exercises in each lesson of Discovering Geometry. Like
those exercises, these worksheets allow students to practice and reinforce the
important procedures and skills developed in the lessons. Some of these problems
provide non contextual skills practice. Others give students an opportunity to
apply geometry concepts in fairly simple, straightforward contexts. Some are more
complex problems that are broken down into small steps.
You might assign the Practice Your Skills worksheet for every lesson, or only for
those lessons your students find particularly difficult. Or, you may wish to
assign the worksheets on an individual basis, only to those students who need
extra help. One worksheet has been provided for nearly every lesson. There are
no worksheets for Chapter 0, and the optional tessellation lessons have been
combined into two worksheets. To save you the time and expense of copying
pages, you can give students the inexpensive Practice Your Skills Student Workbook,
which does not have answers. Though the copyright allows you to copy pages
from Practice Your Skills with Answers for use with your students, the consumable
Practice Your Skills Student Workbook should not be copied. Students, parents, and
mentors can also download the student worksheets from www.keymath.com.
Lesson 1.1 • Building Blocks of Geometry
Name
Period Date
For Exercises 1–7, complete each statement. PS 3 cm.
NS
1. The midpoint of PQis ________________.
2. NQ ________________.
3. Another name for NSis ________________.
4. S is the ________________ of SQ.
5. P is the midpoint of ________________.
6. NS ________________.
7. Another name for SN is ________________.
8.
Name all pairs of congruent segments in KLMN. Use the
congruence symbol to write your answer.
K
9. M(4, 8) is the midpoint of DE. D has coordinates (6, 1). Find the
L
coordinates of E.
For Exercises 10 and 11, use a ruler to draw each figure. Label the
figure and mark the congruent parts.
P Q
N
M
8 cm
8 cm
O
10. ABand CD
with M as the midpoint
of both ABand CD
. AB 6.4 cm
and CD 4.0 cm. A, B, and C are
not collinear.
12.
Sketch six points A, B, C, D, E, and F, no
three of which are collinear. Name the lines
defined by these points. How many lines
are there?
11. AB. C is the midpoint of AB and CD, with
AC 1.5 cm. D, not on AB
, is the
midpoint of AE, with AD 2BC.
13.
In the figure below, {B, C, H, E} is a set of
four coplanar points. Name two other sets
of four coplanar points. How many sets of
four coplanar points are there?
B
F
D
C
A
G
E
H
Cube
Discovering Geometry Practice Your Skills CHAPTER 1
©2008 Key Curriculum Press
Lesson 1.2 • Poolroom Math
Name Period Date
For Exercises 1–5, use the figure at right to complete
each statement.
1. A is the ________________ of BAE.
2. AD is the ________________ of BAE.
3. AD is a ________________ of DAE.
4. If mBAC 42°, then mCAE ________________.
5. DAB ________________.
For Exercises 6–9, use your protractor to find the measure of
each angle to the nearest degree.
P
6. mPRO 7. mORT
R
8. mO 9. mRTO
B
C
D
E
A
O
TA
For Exercises 10–12, use your protractor to draw and then label
each angle with the given measure.
10. mMNO 15° 11. mRIG 90° 12. mz 160°
For Exercises 13–15, find the measure of the angle formed by the
hands at each time.
13. 3:00 14. 4:00 15. 3:30
For Exercises 16 and 17, mark each figure with all the given information.
16. mADB 90°, AD BD, DAB DBA
12
1
5
2
4
11
7
10
8
6
39
ADCB
17. mRPQ 90°, QR TZ, RT QZ, Q T
QPZTR
CHAPTER 1 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 1.3 • What’s a Widget?
Name
Period Date
For Exercises 1–9, match each term with one of the items (a to i) below.
a. 2
b. ? c.
?
90°
1
d.
e.
f.
?
?
g.
h.
i.
P
?
Q
R
1. _____ Vertical angles
2. _____ Obtuse angle
3. _____ Right angle
4. _____ Complementary angles
5. _____ Congruent angles
6. _____ Linear pair of angles
7. _____ Bisected angle
8. _____ Perpendicular lines
9. _____ Congruent segments
10.
If mP 13°, mQ 77°, and Q and R are complementary, what
can you conclude about P and R? Explain your reasoning.
For Exercises 11–13, sketch, label, and mark a figure showing
each property.
11. 1 2, 2 3 12. PQ PR.
13. BAC XAY, CX BC
Discovering Geometry Practice Your Skills CHAPTER 1
©2008 Key Curriculum Press
Lesson 1.4 • Polygons
Name
Period Date
For Exercises 1–8, complete the table.
Polygon name Number of sides Number of diagonals
1. Triangle
2. 2
3. 5
4. Hexagon
5. Heptagon
6. 8
7. 35
8. 12
For Exercises 9 and 10, sketch and label each figure. Mark the congruences.
9.
Concave pentagon PENTA, with external 10. Equilateral quadrilateral QUAD, with
diagonal ET, and TA PE. Q U.
For Exercises 11–14, sketch and use hexagon ABCDEF.
11. Name the diagonals from A.
12. Name a pair of consecutive sides.
13. Name a pair of consecutive angles.
14. Name a pair of non-intersecting diagonals.
For Exercises 15–18, use the figures at right.
MNOPQ RSTUV
15. mN _____
16.
VR _____
P
17. mP _____
18. ON _____
19.
The perimeter of a regular pentagon is 31 cm. Find the length of
each side.
16.1
8.4
7.2 S R
U
V
T
O
N
M
Q
82°58°
61°
CHAPTER 1 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 1.5 • Triangles
Name
Period Date
For Exercises 1–5, use the figure at right. Name a pair of
1. Parallel segments
EA B
2. Perpendicular segments
CD
F
3. Congruent segments
4. Supplementary angles
GH I
5. Linear angles
For Exercises 6 and 7, sketch, label, and mark each figure.
6. Isosceles obtuse triangle TRI with vertex angle T.
7. Scalene right triangle SCA with midpoints L, M, and N on SC,CA,
and SA, respectively.
For Exercises 8 and 9, use your geometry tools to draw each figure.
8.
Acute isosceles triangle ACD with vertex 9. Scalene right triangle RGH.
angle A measuring 40°.
For Exercises 10–12, use the graph at right.
10. Locate F so that ABF is a right triangle.
11. Locate D so that ABD is an isosceles triangle.
12. Locate G so that ABG is scalene and not a right triangle.
y
x
B (8, 0) A (0, 0)
C (8, 3)
Discovering Geometry Practice Your Skills CHAPTER 1
©2008 Key Curriculum Press
Lesson 1.6 • Special Quadrilaterals
Name Period Date
For Exercises 1–6, sketch, label, and mark each figure.
1. Parallelogram PGRA 2. Square SQRE
3. Rhombus RHOM with acute H. 4. Trapezoid TRAP with TR AP,RE PA,
and P, E, and A collinear.
5. Kite KITE with EK KI and obtuse K. 6. Rectangle RANG with perimeter 2a 4b
For Exercises 7–10, name each polygon in the figure. Assume that
the grid is square.
7. Square 8. Parallelogram
9. Rhombus 10. Kite
For Exercises 11–13, use the graph at right.
11. Locate D so that ABCD is a rectangle.
12. Locate E so that ABCE is a trapezoid.
13. Locate G so that points A, B, C, and G
determine a parallelogram that is not
a rectangle.
y
J
G
D F
E
A B C
H
I
x
B (8, 0) A (0, 0)
C (8, 3)
CHAPTER 1 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 1.7 • Circles
Name
Period Date
For Exercises 1–4, use the figure at right.
O48°
P
1. mQR _____ 2. mPR _____
3. mPQR _____ 4. mQPR _____
Q
R
5.
Sketch a circle with an inscribed pentagon. 6. Sketch a circle with a circumscribed
quadrilateral.
7.
A circle with center (3, 2) goes through 8. Use a compass, protractor, and straightedge
(2, 2). Give the coordinates of three to draw circle O with diameter AB; radius
other points on the circle. OC AB;OD
with OC, the angle
bisector of AOC, with D on the circle;
y
chords ACand BC; and a tangent at D.
(2, 2)
(3, 2)
x
9.
Use a compass to construct a circle. Label 10. Use your compass and protractor to make
the center P. Sketch two parallel tangents. an arc with measure 50°, an arc with
Connect the points of tangency. What do measure 180°, and an arc with measure
you notice about the chord? 290°. Label each arc with its measure.
11.
Use your compass to construct two circles with different radii that
intersect in two points. Label the centers P and Q and the points
of intersection A and B. Construct quadrilateral PAQB. What type
of quadrilateral is it?
Discovering Geometry Practice Your Skills CHAPTER 1
©2008 Key Curriculum Press
Lesson 1.8 • Space Geometry
Name Period Date
For Exercises 1 and 2, draw each figure.
1. A prism with a rectangular base. 2. A cylinder with
than height.
base diameter greater
For Exercises 3 and 4, sketch the three-dimensional figure formed by
folding each net into a solid. Name the solid.
3.
4.
For Exercises 5 and 6, sketch the section formed when each solid is sliced
by the plane as shown.
5.
6.
7.
The prism below is built with 1-cm 8. Find the lengths of x and y.
cubes. How many cubes are completely
hidden from sight, as seen from this angle?
4
y
x
2
3
4
CHAPTER 1 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 1.9 • A Picture Is Worth a Thousand Words
Name
Period Date
Read and reread each problem carefully, determining what information you
are given and what it is that you trying to find.
1.
A pair of parallel interstate gas and power lines run 10 meters apart
and are equally distant from relay station A. The power company
needs to locate a gas-monitoring point on one of the lines exactly
12 meters from relay station A. Draw a diagram showing the locus of
possible locations.
2.
The six members of the Senica High School math club are having a
group photo taken for the yearbook. The photographer has asked the
club to submit the height of each member so that he can quickly
arrange them in order. The math club sent him the following
information. Anica is 4 inches taller than Bruce. Charles is the same
height as Ellen but an inch taller than Anica. Fred is midway between
Bruce and Dora. Dora is 2 inches taller than Anica. Help out the
photographer by arranging the club members in order from tallest
to shortest.
3.
Create a Venn diagram showing the relationships among triangles,
acute triangles, isosceles triangles, and scalene triangles.
4. Sketch a possible net for each solid.
a.
b.
c.
Discovering Geometry Practice Your Skills CHAPTER 1
©2008 Key Curriculum Press
Lesson 2.1 • Inductive Reasoning
Name
Period Date
For Exercises 1–7, use inductive reasoning to find the next two terms in
each sequence.
1. 4, 8, 12, 16, _____, _____
2.
400, 200, 100, 50, 25, _____, _____
1214
3. , , , , _____, _____
8
725
4. 5, 3, 2, 1, 1, 0, _____, _____
5. 360, 180, 120, 90, _____, _____
6. 1, 3, 9, 27, 81, _____, _____
7. 1, 5, 14, 30, 55, _____, _____
For Exercises 8–10, use inductive reasoning to draw the next two shapes in
each picture pattern.
8.
9.
10. y
x
(3, 1)
y
x
(–1, 3)
y
x
(–3, –1)
For Exercises 11–13, use inductive reasoning to test each conjecture.
Decide if the conjecture seems true or false. If it seems false, give
a counterexample.
11. The square of a number is larger than the number.
12.
Every multiple of 11 is a “palindrome,” that is, a number that reads the
same forward and backward.
13. The difference of two consecutive square numbers is an odd number.
10 CHAPTER 2 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 2.2 • Finding the nth Term
Name Period Date
For Exercises 1–4, tell whether the rule is a linear function.
1.
2.
n 1 2 3 4 5
f(n) 8 15 22 29 36
n 1 2 3 4 5
g(n) 14 11 8 5 2
3.
4.
n 1 2 3 4 5
h(n) 9 6 2 3 9
n 1 2 3 4 5
j(n) 3
2 1 1
2 0 1
2
For Exercises 5 and 6, complete each table.
5.
6.
n 1 2 3 4 5
f(n) 7n 12
n 1 2 3 4 5
g(n) 8n 2
For Exercises 7–9, find the function rule for each sequence. Then find the
50th term in the sequence.
7.
n 1 2 3 4 5 6 . . . n ... 50
f(n) 9 13 17 21 25 29 . . . . . .
8.
n 1 2 3 4 5 6 . . . n ... 50
g(n) 6 1 4 9 14 19 ... ...
9.
n 1 2 3 4 5 6 . . . n ... 50
h(n) 6.5 7 7.5 8 8.5 9 . . . . . .
10. Use the figures to complete the table.
n 12345 . . . n ... 50
Number
of triangles 159 ... ...
11. Use the figures above to complete the table. Assume that the area of the
first figure is 1 square unit.
n 12 34 5 . . . n ... 50
Area of
figure 14 16 ... ...
Discovering Geometry Practice Your Skills CHAPTER 2 11
©2008 Key Curriculum Press
Lesson 2.3 • Mathematical Modeling
Name
Period Date
1. Draw the next figure in this pattern.
a.
How many small squares will there be in the
10th figure?
b. How many in the 25th figure?
c. What is the general function rule for this pattern?
2.
If you toss a coin, you will get a head or a tail. Copy and H
complete the geometric model to show all possible results of
three consecutive tosses.
a. How many sequences of results are possible?
b. How many sequences have exactly one tail?
c.
Assuming a head or a tail is equally likely, what is the
probability of getting exactly one tail in three tosses?
3.
If there are 12 people sitting at a round table, how many different pairs
of people can have conversations during dinner, assuming they can
all talk to each other? What geometric figure can you use to model
this situation?
4.
Tournament games and results are often displayed using a geometric
model. Two examples are shown below. Sketch a geometric model for
a tournament involving 5 teams and a tournament involving 6 teams.
Each team must have the same chance to win. Try to have as few
games as possible in each tournament. Show the total number
of games in each tournament. Name the teams a, b, c . . . and number
the games 1,2,3 ....
a
a
1
b
3
13
c
c2
b
d
2
H
H
HHH
T
T
HHT
3 teams, 3 games 4 teams, 3 games
(round robin) (single elimination)
12 CHAPTER 2 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 2.4 • Deductive Reasoning
Name
Period Date
1.
ABC is equilateral. Is ABD equilateral? Explain your answer.
What type of reasoning, inductive or deductive, do you use when
solving this problem?
A
2.
A and D are complementary. A and E are supplementary.
What can you conclude about D and E? Explain your answer.
What type of reasoning, inductive or deductive, do you use when
solving this problem?
3.
Which figures in the last group are whatnots? What type of reasoning,
inductive or deductive, do you use when solving this problem?
C
B
D
d. e. f.
a. b. c.
Whatnots Not whatnots
Which are whatnots?
4. Solve each equation for x. Give a reason for each step in the process.
What type of reasoning, inductive or deductive, do you use when
solving these problems?
a. 4x 3(2 x) 8 2x b.
19 2(
5
3x 1)
x 2
5. A sequence begins 4, 1, 6, 11 . . .
a.
Give the next two terms in the sequence. What type of reasoning,
inductive or deductive, do you use when solving this problem?
b.
Find a rule that generates the sequence. Then give the 50th term in
the sequence. What type of reasoning, inductive or deductive, do
you use when solving this problem?
Discovering Geometry Practice Your Skills CHAPTER 2 13
©2008 Key Curriculum Press
Lesson 2.5 • Angle Relationships
Name
Period Date
For Exercises 1–6, find each lettered angle measure without using
aprotractor.
1.
2.
3.
c
40°
a
15°
a 38°
112°
bc
d
a
70°
b
4.
5.
6.
dce 132°
a
c
b
b
100°
70°
c
a
110°
e
d
138°
25°
ba
For Exercises 7–10, tell whether each statement is always (A),
sometimes (S), or never (N) true.
7.
_____ The sum of the measures of two acute angles equals the
measure of an obtuse angle.
8.
_____ If XAY and PAQ are vertical angles, then either X, A, and P
or X, A, and Q are collinear.
9. _____ If two angles form a linear pair, then they are complementary.
10. _____ If a statement is true, then its converse is true.
For Exercises 11–15, fill in each blank to make a true statement.
11. If one angle of a linear pair is obtuse, then the other is ____________.
12. If A B and the supplement of B has measure 22°, then
mA ________________.
13.
If P is a right angle and P and Q form a linear pair, then
mQ is ________________.
14.
If S and T are complementary and T and U are supplementary,
then U is a(n) ________________ angle.
15.
Switching the “if” and “then” parts of a statement changes the
statement to its ________________.
14 CHAPTER 2 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 2.6 • Special Angles on Parallel Lines
Name Period Date
For Exercises 1–3, use your conjectures to find each angle measure.
1.
a
b
c
54°
2.
a
b
cd
65°
3.
54°
b
a
For Exercises 4–6, use your conjectures to determine whether 1 2,
and explain why. If not enough information is given, write “cannot
be determined.”
4.
5.
6.
118°
1
1
95°
2
62°
2
25°
2
48°
48°
1
7. Find each angle measure.
44°
78°
64°
f
e
d
b
c
a
8. Find x. 9. Find x and y.
4y
2.
x
3x
160°
182 4x
5x
2.
Discovering Geometry Practice Your Skills CHAPTER 2 15
©2008 Key Curriculum Press
Lesson 3.1 • Duplicating Segments and Angles
Name
Period Date
In Exercises 1–3, use the segments and angles below. Complete the
constructions on a separate piece of paper.
A
B
P
Q R S
1.
Using only a compass and straightedge, duplicate each segment and
angle. There is an arc in each angle to help you.
2. Construct a line segment with length 3PQ 2RS.
3.
Duplicate the two angles so that the angles have the same vertex and
share a common side, and the nonshared side of one angle falls inside
the other angle. Then use a protractor to measure the three angles you
created. Write an equation relating their measures.
4.
Use a compass and straightedge to construct an isosceles triangle with
two sides congruent to ABand base congruent to CD
.
A
B
5. Repeat Exercise 4 with patty paper and a straightedge.
.
6. Construct an equilateral triangle with sides congruent to CD
C
D
C D
16 CHAPTER 3 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 3.2 • Constructing Perpendicular Bisectors
Name
Period Date
For Exercises 1–6, construct the figures on a separate sheet of paper using
only a compass and a straightedge.
1. Draw a segment and construct its perpendicular bisector.
2.
Construct two congruent segments that are the perpendicular bisectors
of each other. Form a quadrilateral by connecting the four endpoints.
What type of quadrilateral does this seem to be?
5
3. Duplicate AB. Then construct a segment with length AB.
4
A
B
. CM
ABC.Is ABC unique? If not, construct a different triangle,
ABC, also having CM
4. Draw a segment; label it CM is a median of ABC.Construct
as a median.
5. Draw a segment; label it PQ. PQis a midsegment of ABC.Construct
ABC.Is ABC unique? If not, construct a different triangle,
ABC, also having PQas a midsegment.
6.
Construct a right triangle. Label it ABC with right angle B.Construct
median BD. Compare BD, AD, and CD.
7. Complete each statement as fully as possible.
a.
L is equidistant from ________________.
b.
M is equidistant from ________________.
c.
N is equidistant from ________________.
A
d.
O is equidistant from ________________.
D
C
B
E
L
M N
O
Discovering Geometry Practice Your Skills CHAPTER 3 17
©2008 Key Curriculum Press
Lesson 3.3 • Constructing Perpendiculars to a Line
Name
Period Date
For Exercises 1–5, decide whether each statement is true or false. If the
statement is false, explain why or give a counterexample.
1.
In a triangle, an altitude is shorter than either side from the
same vertex.
2.
In a triangle, an altitude is shorter than the median from the
same vertex.
3.
In a triangle, if a perpendicular bisector of a side and an altitude
coincide, then the triangle is isosceles.
4. Exactly one altitude lies outside a triangle.
5.
The intersection of the perpendicular bisectors of the sides lies inside
the triangle.
For Exercises 6 and 7, use patty paper. Attach your patty paper to your
worksheet.
6.
Construct a right triangle. Construct the altitude from the right angle
to the opposite side.
7. Mark two points, P and Q. Fold the paper to construct square PQRS.
Use your compass and straightedge and the definition of distance to
complete Exercises 8 and 9 on a separate sheet of paper.
.
8. Construct a rectangle with sides equal in length to ABand CD
A
BCD
9.
Construct a large equilateral triangle. Let P be any point inside the
triangle. Construct WX
equal in length to the sum of the distances
from P to each of the sides. Let Q be any other point inside the
triangle. Construct YZequal in length to the sum of the distances
from Q to each side. Compare WX and YZ.
18 CHAPTER 3 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 3.4 • Constructing Angle Bisectors
Name
Period Date
1. Complete each statement as fully as possible.
a.
M is equidistant from _________________.
b.
P is equidistant from _________________. 3
c.
Q is equidistant from _________________.
d.
R is equidistant from _________________.
2.
If the converse of the Angle Bisector Conjecture is true,
what can you conclude about this figure?
M
P
R
Q
1
4
5
2
A
B
P
C
3. If BE bisects ABD, find x and mABE.
BA
E D
C
2x 10°3x 20°
4.
Draw an obtuse angle. Use a compass and straightedge to construct the
angle bisector. Draw another obtuse angle and fold to construct the
angle bisector.
5.
Draw a large triangle on patty paper. Fold to construct the three angle
bisectors. What do you notice?
For Exercises 6 and 7, construct a figure with the given specifications using
a straightedge and compass or patty paper. Use additional sheets of paper
to show your work.
6.
Using only your compass and straightedge, construct an isosceles
right triangle.
7.
Construct right triangle RGH with right angle R. Construct median
RM from M to RG from
, perpendicular MN, and perpendicular MOM to RH
. Compare RN and GN, and compare RO and HO.
Discovering Geometry Practice Your Skills CHAPTER 3 19
©2008 Key Curriculum Press
Lesson 3.5 • Constructing Parallel Lines
Name
Period Date
For Exercises 1–6, construct a figure with the given specifications using a
straightedge and compass or patty paper. Use additional sheets of paper to
show your work.
1.
Draw a line and a point not on the line. Use a compass and
straightedge to construct a line through the given point parallel to the
given line.
2.
Repeat Exercise 1, but draw the line and point on patty paper and fold
to construct the parallel line.
3. Use a compass and straightedge to construct a parallelogram.
4. Use patty paper and a straightedge to construct an isosceles trapezoid.
5. Construct a rhombus with sides equal in length to ABand having an
angle congruent to P.
A
B
P
6. Construct trapezoid ZOID with ZOand IDas nonparallel sides and
AB as the distance between the parallel sides.
Z
O
I D A B
20 CHAPTER 3 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 3.6 • Construction Problems
Name Period Date
For Exercises 1–5, construct a figure with the given specifications using
either a compass and straightedge or patty paper. Use additional sheets of
paper to show your work.
1. Construct kite KITE using these parts.
I T
I
K I
2. Construct a rectangle with perimeter the length of this segment.
3. Construct a rectangle with this segment as its diagonal.
4. Draw obtuse OBT. Construct and label the three altitudes OU,
,BS
and TE.
5. Construct a triangle congruent to ABC. Describe your steps.
A
B
C
In Exercises 6–8, construct a triangle using the given parts. Then, if
possible, construct a different (noncongruent) triangle using the
same parts.
S1
S2
6.
7. S1 8. S1
S2
A
B
A
S3
Discovering Geometry Practice Your Skills CHAPTER 3 21
©2008 Key Curriculum Press
Lesson 3.7 • Constructing Points of Concurrency
Name
Period Date
For Exercises 1 and 2, make a sketch and explain how to find the answer.
1.
A circular revolving sprinkler needs to be set up to water every part of
a triangular garden. Where should the sprinkler be located so that it
reaches all of the garden, but doesn’t spray farther than necessary?
2.
You need to supply electric power to three transformers, one on
each of three roads enclosing a large triangular tract of land. Each
transformer should be the same distance from the power-generation
plant and as close to the plant as possible. Where should you build the
power plant, and where should you locate each transformer?
For Exercises 3–5, construct a figure with the given specifications using
a compass and straightedge. Use additional sheets of paper to show
your work.
3.
Draw an obtuse triangle. Construct the inscribed and the
circumscribed circles.
4.
Construct an equilateral triangle. Construct the inscribed and the
circumscribed circles. How does this construction differ from
Exercise 3?
5.
Construct two obtuse, two acute, and two right triangles. Locate
the circumcenter of each triangle. Make a conjecture about the
relationship between the location of the circumcenter and the
measure of the angles.
22 CHAPTER 3 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 3.8 • The Centroid
Name
Period Date
For Exercises 1–3, use additional sheets of paper to show your work.
1. Draw a large acute triangle. Construct the centroid.
2. Construct a regular hexagon and locate its center of gravity.
3.
Use a ruler and compass to find the center of gravity of a sheet-metal
triangle with sides measuring 6 cm, 8 cm, and 10 cm. How far is the
center from each vertex, to the nearest tenth of a centimeter?
4.
ABC has vertices A(9, 12), B(3, 2), and C(3, 2). Find the
coordinates of the centroid.
C (3, 2)
A (9, 12)
y
x
B (3, 2)
P
Q
L
R
M
C
K
5. PL 24, QC 10, and KC 7. Find PC, CL, QM, and CR.
6.
Identify each statement as describing the incenter, circumcenter,
orthocenter, or centroid.
a.
________________ The point equally distant from the three sides
of a triangle.
b.
________________ The center of gravity of a thin metal triangle.
c.
________________ The point equidistant from the three vertices.
d.
________________ The intersection of the perpendicular bisectors
of the sides of a triangle.
e.
________________ The intersection of the altitudes of a triangle.
f.
________________ The intersection of the angle bisectors of
a triangle.
g.
________________ The intersection of the medians of a triangle.
Discovering Geometry Practice Your Skills CHAPTER 3 23
©2008 Key Curriculum Press
Lesson 4.1 • Triangle Sum Conjecture
Name Period Date
In Exercises 1–9, determine the angle measures.
1. p ______, q ______ 2. x ______, y ______ 3. a ______, b ______
79
50
23a b
28
17
53x
y
82
98
qp
31
4.
r ______, s ______, 5. x ______, y ______ 6. y ______
t ______
85x
31
y
100s
tr
100 .
x
y
30 4x7x
7. s ______
8. m ______ 9. mP ______
s
76
m
35
P
b
c
c
a
a
10.
Find the measure of QPT. 11. Find the sum of the measures of
the marked angles.
P
S
R
T
Q
12.
Use the diagram to explain why 13. Use the diagram to explain why
A and B are complementary. mA mB mC mD.
A
D
B
E
B
A
24 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.2 • Properties of Isosceles Triangles
Name
Period Date
In Exercises 1–3, find the angle measures.
1. mT ______ 2. mG ______
3. x ______
T
A
110.
x
58
R
I
G
N
In Exercises 4–6, find the measures.
4. mA ______, perimeter 5. The perimeter of LMO 6. The perimeter of QRS is
of ABC ______ is 536 m. LM ______, 344 cm. mQ ______,
mM ______ QR ______
A
13 cm
39
102
x 30x
Q
M
210 m
a 7 cm
B
68
S
a
L
O
163 m
C
7. a. Name the angle(s) congruent to DAB.
R
y
b. Name the angle(s) congruent to ADB.
A
c. What can you conclude about AD? Why?
and BC
8.
x _____, y _____ 9. PR QR and QS RS. 10. Use the diagram to explain
If mRSQ 120°, what is why PQR is isosceles.
B C
D
4y
2x y 79x
mQPR?
Q
Q
T
R
55.
70
P
RS
P
S
Discovering Geometry Practice Your Skills CHAPTER 4 25
©2008 Key Curriculum Press
Lesson 4.3 • Triangle Inequalities
Name
Period Date
In Exercises 1 and 2, determine whether it is possible to draw a triangle
with sides of the given measures. If it is possible, write yes. If it is not
possible, write no and make a sketch demonstrating why it is not possible.
1. 16 cm,30 cm,45 cm
2. 9 km, 17 km, 28 km
3.
If 17 and 36 are the lengths of two sides of a triangle, what is the range
of possible values for the length of the third side?
In Exercises 4–6, arrange the unknown measures in order from greatest
to least.
a
bc
c
4.
5.
61
32
6.
2840
71
dc
18
13
a
b
b
20
a
7. x _____ 8. x _____
9. What’s wrong with
66
142.
10. Explain why PQS is isosceles.
P
R
this picture?
x 2x
Q
120
160
C
A
B
x
158
S
In Exercises 11 and 12, use a compass and straightedge to construct a
triangle with the given sides. If it is not possible, explain why not.
Q
11. A
B
12. P
C
Q
R
B
A
CPR
26 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.4 • Are There Congruence Shortcuts?
Name Period Date
In Exercises 1–3, name the conjecture that leads to each congruence.
1. PAT IMT 2. SID JAN bisects MA AT,
3. TS,MTand MST AST
AI S
A 6
8
M
6 N
TI
9
9
DS
T
8
J
PM
A
In Exercises 4–9, name a triangle congruent to the given triangle and
state the congruence conjecture. If you cannot show any triangles to be
congruent from the information given, write “cannot be determined” and
redraw the triangles so that they are clearly not congruent.
Z
I
4. M is the midpoint of AB 5. KITE is a kite with KI TI. 6. ABC _____
and PQ.
KIE _____
C
AY
APM _____
T
PB
E
M
BX
K
A
Q
7. MON _____ 8. SQR _____ 9. TOP _____
y
NQ
T
U
R
10
S
M
O
T
2 4 6 810
2
4
6
8
G
T
D
O
P
In Exercises 10–12, use a compass and a straightedge or patty paper and a
straightedge to construct a triangle with the given parts. Then, if possible,
construct a different (noncongruent) triangle with the same parts. If it is
not possible, explain why not.
10. S
T
11.
12. XY
C
B
C
T
U
XZ
A
US
B
X
Discovering Geometry Practice Your Skills CHAPTER 4 27
©2008 Key Curriculum Press
Lesson 4.5 • Are There Other Congruence Shortcuts?
Name
Period Date
In Exercises 1–6, name a triangle congruent to the given triangle and state
the congruence conjecture. If you cannot show any triangles to be congruent
from the information given, write “cannot be determined” and explain why.
1. PIT _____ 2. XVW _____ 3. ECD _____
BE
PO
VW
C
X
T
YZ
AD
4. PSis the angle bisector 5. ACN _____ 6. EFGH is a parallelogram.
of QPR. P GQ EQ.
PQS _____
EQL _____
RC
Q
Q
H
G
K
F
P
S
A
R
NL
E
7.
The perimeter of QRS is 350 cm. 8. The perimeter of TUV is 95 cm.
Is QRS MOL? Explain. Is TUV WXV? Explain.
In Exercises 9 and 10, construct a triangle with the given parts. Then, if
possible, construct a different (noncongruent) triangle with the same parts.
If it is not possible, explain why not.
x
40
x 25
2x 10
T U
V
X W
L Q R
x
125
70
x 55
2x15
OS
M
9.
P
Q
P Q
10.
C
A
AB
28 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.6 • Corresponding Parts of Congruent Triangles
Name
Period Date
1.
Give the shorthand name for each of the four triangle
congruence conjectures.
In Exercises 2–5, use the figure at right to explain why
each congruence is true. WXYZ is a parallelogram.
2. WXZ YZX 3. WZX YXZ
WX
4. WZX YXZ 5. W Y
For Exercises 6 and 7, mark the figures with the given information. To
demonstrate whether the segments or the angles indicated are congruent,
determine that two triangles are congruent. Then state which conjecture
proves them congruent.
6. M is the midpoint of WX and
7. ABC is isosceles and CD
is the bisector
ZX BD
YZ.Is YW? Why?
of the vertex angle. Is AD? Why?
X
C
Z Y
BADYWZM
In Exercises 8 and 9, use the figure at right to
write a paragraph proof for each statement.
8. DE CF 9. EC FD
A
D C
BE F
10. TRAP is an isosceles trapezoid with TP RA and PTR ART.
Write a paragraph proof explaining why TA RP.
TRAP
Discovering Geometry Practice Your Skills CHAPTER 4 29
©2008 Key Curriculum Press
Lesson 4.7 • Flowchart Thinking
Name
Period Date
Complete the flowchart for each proof.
1. Given:
PQ SRand PQ SR.
Show: SP QR
P
Flowchart Proof
RS
Q
Given
PQS ______PQ SR SP QR
QS ______
2. Given:
Kite KITE with KE KI I
Show: KTbisects EKI and ETI K
Flowchart Proof
E
T
KE KI
ETK ITK
KITE is a kite
KET ______
Definition
of bisect
3. Given:
ABCD is a parallelogram DC
Show: A C
A
Flowchart Proof
B
ABCD is a parallelogram
_________________________
AB CD
______________
Definition of
___________
Same segment
_____________
____________ _____________
30 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.8 • Proving Special Triangle Conjectures
Name
Period Date
In Exercises 1–3, use the figure at right.
1.
CD is a median, perimeter ABC 60, and AC 22. AD _____
2.
CD is an angle bisector, and mA 54°. mACD _____
3.
CD is an altitude, perimeter ABC 42, mACD 38°, and AD 8.
mB _____, CB _____
A
D
4. EQU is equilateral.
5. ANG is equiangular
mE _____ and perimeter ANG 51.
AN _____
6. ABC is equilateral, ACD is isosceles with base AC,
C
perimeter ABC 66, and perimeter ACD 82.
B
Perimeter ABCD _____
7. Complete a flowchart proof for this conjecture: In an isosceles triangle,
C
B
C
D
A
the altitude from the vertex angle is the median to the base.
Given: Isosceles ABC with AC BCand altitude CD
Show: CD
is a median
A
Flowchart Proof
D
B
• • •
A ________
ADC BDC
ADC and BDC
are right angles
____________________
Definition of altitude
__________________
__________________
CD is an altitude
AC BC
Given __________________
C
8. Write a flowchart proof for this conjecture: In an isosceles triangle, the
median to the base is also the angle bisector of the vertex angle.
Given: Isosceles ABC with AC BCand median CD
Show: CD
bisects ACB
A
B
D
Discovering Geometry Practice Your Skills CHAPTER 4 31
©2008 Key Curriculum Press
Lesson 5.1 • Polygon Sum Conjecture
Name
Period Date
In Exercises 1 and 2, find each lettered angle measure.
1.
a _____, b _____, c _____, 2. a _____, b _____, c _____,
d _____, e _____ d _____, e _____, f _____
a
e d
cb26
97.
85
44
a
c
b
d
f
e
3.
One exterior angle of a regular polygon measures 10°. What is
the measure of each interior angle? How many sides does the
polygon have?
4. The sum of the measures of the interior angles of a regular polygon is
2340°. How many sides does the polygon have?
5. ABCD is a square. ABE is an equilateral 6. ABCDE is a regular pentagon. ABFG
triangle. is a square.
x _____ x _____
D C D
E
C
x
E
FxG
A
B
A
B
7.
Use a protractor to draw pentagon ABCDE with mA 85°,
mB 125°, mC 110°, and mD 70°. What is mE?
Measure it, and check your work by calculating.
32 CHAPTER 5 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 5.2 • Exterior Angles of a Polygon
Name
Period Date
1. How many sides does a regular polygon
2. How many sides does a polygon have if the
have if each exterior angle measures 30°?
sum of the measures of the interior angles
is 3960°?
3.
If the sum of the measures of the interior 4. If the sum of the measures of the interior
angles of a polygon equals the sum of the angles of a polygon is twice the sum of its
measures of its exterior angles, how many exterior angles, how many sides does it
sides does it have? have?
In Exercises 5–7, find each lettered angle measure.
5.
a _____, b _____ 6. a _____, b _____ 7. a _____, b _____,
c _____
116
82
a
b
a
b
82
134
72
a
b
c
3x
2x
x
8. Find each lettered angle measure.
a _____
b _____
c _____
d _____
a
d
b
c
150
95
9. Construct an equiangular quadrilateral that is not regular.
Discovering Geometry Practice Your Skills CHAPTER 5 33
©2008 Key Curriculum Press
Lesson 5.3 • Kite and Trapezoid Properties
Name
Period Date
In Exercises 1–4, find each lettered measure.
1. Perimeter 116. x _____
2. x _____, y _____
28 x
x
y
56
3. x _____, y _____
4. x _____, y _____
78
41x
y
137
22
x
y
5. Perimeter PQRS 220. PS _____ 6. b 2a 1. a _____
S 4x 1 R
M
34
b
a
N
L
2x 3
Q
P
T 4
K
In Exercises 7 and 8, use the properties of kites and trapezoids to construct
each figure. Use patty paper or a compass and a straightedge.
7. Construct an isosceles trapezoid given base AB, B, and distance
between bases XY.
BXYA B
8. Construct kite ABCD with AB,BC, and BD.
9.
Write a paragraph or flowchart proof of the Converse of the Isosceles
Trapezoid Conjecture. Hint: Draw AEparallel to TPwith E on TR.
Given: Trapezoid TRAP with T R
Show: TP RA T
A B BC DB
P A
R
34 CHAPTER 5 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 5.4 • Properties of Midsegments
Name Period Date
In Exercises 1–3, each figure shows a midsegment.
1. a _____, b _____, 2. x _____, y _____,
c _____ z _____
3. x _____, y _____,
z _____
b
a c 37
54
y
16
41
14 zy
13
11
x
x
21
z
29
4. X, Y, and Z are midpoints. Perimeter PQR 132, RQ 55, and PZ 20.
Perimeter XYZ
_____
PQ _____
ZX _____ P
5. MN is the midsegment. Find the 6. Explain how to find the width of the lake from
coordinates of M and N. Find the A to B using a tape measure, but without
slopes of ABand MN. using a boat or getting your feet wet.
Z Y
X Q
R
y
C (20, 10)
A
Lake
B
M
A (4, 2)
N
x
B (9, –6)
7.
M, N, and O are midpoints. What type of quadrilateral
is AMNO? How do you know? Give a flowchart proof
showing that ONC MBN.
A
M
8.
Give a paragraph or flowchart proof.
Given: PQR with PD DF FH HR
and QE EG GI IR
Show: HI FG DE PQ.
P
B
O N
C
D E
F G
H I
R
Q
Discovering Geometry Practice Your Skills CHAPTER 5 35
©2008 Key Curriculum Press
Lesson 5.5 • Properties of Parallelograms
Name
Period Date
In Exercises 1–7, ABCD is a parallelogram.
15 cm
1.
Perimeter ABCD _____ 2. AO 11, and BO 7.
26 cm C AC _____, BD _____
D
DC
B
A
O
AB
4. a _____, b _____, 5. Perimeter ABCD 119, and
c _____ BC 24. AB _____
bc
a19
62
110
DC DC
A
B
AB
3. Perimeter ABCD 46.
AB _____, BC _____
C
D
2x 7
B
Ax 9
6.
a _____, b _____,
c _____
D
C
A
acb
27
68
26.
B
7.
Perimeter ABCD 16x 12. AD _____ 8. Ball B is struck at the same instant by two
forces, F
1 and F
2. Show the resultant force
DC
on the ball.
4x 3
A 63 B
F2
F1
B
9.
Find each lettered angle measure.
a _____ g _____
b _____ h _____
c _____ i _____
d _____ j _____ D
e _____ k _____
A
B
f
_____
F
h
e d i
k
j
ba
g
Ef
G
C
81
38
38
26
c
10. Construct a parallelogram with diagonals ACand BD.
Is your parallelogram unique? If not, construct a different
(noncongruent) parallelogram.
A
CB
D
36 CHAPTER 5 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 5.6 • Properties of Special Parallelograms
Name Period Date
1. PQRS is a rectangle and 2. KLMN is a square and 3. ABCD is a rhombus,
OS 16. NM 8. AD 11, and DO 6.
OQ _____ mOKL _____ OB _____
mQRS _____
mMOL _____ BC _____
PR _____
Perimeter KLMN _____ mAOD _____
S
RNM D
O
O
PQ
A
B
O
K
L
In Exercises 4–11, match each description with all the terms that fit it.
a. Trapezoid b. Isosceles triangle c. Parallelogram d. Rhombus
e. Kite
f. Rectangle g. Square h. All quadrilaterals
4. _____ Diagonals bisect each other. 5. _____ Diagonals are perpendicular.
6.
_____ Diagonals are congruent. 7. _____ Measures of interior angles sum
to 360°.
8. _____ Opposite sides are congruent. 9. _____ Opposite angles are congruent.
10.
_____ Both diagonals bisect angles. 11. _____ Diagonals are perpendicular
bisectors of each other.
In Exercises 12 and 13, graph the points and determine whether ABCD is a
trapezoid, parallelogram, rectangle, or none of these.
12. A(4, 1), B(0, 3), C(4, 0), D(1, 5) 13. A(0, 3), B(1, 2), C(3, 4), D(2, 1)
5
5
–5
5
–5
5
–5
–5
14. Construct rectangle ABCD with diagonal ACand CAB.
AC
CAB
A
Discovering Geometry Practice Your Skills CHAPTER 5 37
©2008 Key Curriculum Press
Lesson 5.7 • Proving Quadrilateral Properties
Name
Period Date
Write or complete each flowchart proof.
DQ
C
1. Given: ABCD is a parallelogram and AP QC
Show: ACand PQbisect each other
A
Flowchart Proof
BP
R
APR _____
___________________
_____________Given Given
AIA Conjecture
DC AB
APR CQR
AR _____
AC and QP
bisect each other
Definition of bisect
2. Given: Dart ABCD with ABand CD BC ADShow: A C
A
3.
Show that the diagonals of a rhombus divide the rhombus into DC
four congruent triangles.
Given: Rhombus ABCD
Show: ABO CBO CDO ADO A
C
B
D
B
O
4. Given: Parallelogram ABCD, BY
AC,DX AC D C
Show: DX BY
A
B
Y
X
38 CHAPTER 5 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 6.1 • Tangent Properties
Name
Period Date
1. Rays r and s are tangents. w _____ 2. AB
is tangent to both circles and
w 54
r
mAMC 295°. mBQX _____
M
PA
BQX
C
s
3. PQ
is tangent to two externally tangent noncongruent
circles, M and N.
a. mNQP _____, mMPQ _____
b.
What kind of quadrilateral is MNQP? Explain
your reasoning.
M N
P
Q
4. AT5. PA,PC are tangents. is tangent to circle P. Find the ,PB, and PDequation of ATExplain why PA PD. .
y
A (3, 9)
T
P
x
A
B
C
D
P
6. Circle A has diameter 16.4 cm. Circle B has diameter 6.7 cm.
a. If A and B are internally tangent, what is the distance between
their centers?
b. If A and B are externally tangent, what is the distance between
their centers?
7.
Construct a circle, P. Pick a point, A, on the circle. Construct a tangent
through A. Pick a point, T, on the tangent. Construct a second tangent
to the circle through T.
Discovering Geometry Practice Your Skills CHAPTER 6 39
©2008 Key Curriculum Press
Lesson 6.2 • Chord Properties
Name
Period Date
In Exercises 1–6, find each unknown or write “cannot be determined.”
1.
a _____, b _____, 2. w _____, v _____ 3. z _____
c _____
a
b
c
95
v
w
66
z
4. w _____, x _____, 5. w _____, x _____, 6. x _____, y _____
y _____ y _____
w
y
35
66
x
y
x
y
50
100
8 cm
100
w
x
7. AB AC. AMON is a 8. What’s wrong with 9. Find the coordinates of
________________. this picture? P and M.
Justify your answer. y
B
N
M
O
2
6
A (3, 6)
6
C
A
B (5, –2)
P
M
10. mAB _____
11. Trace part of a circle onto patty paper. Fold
CO
49
107
to find the center. Explain your method.
mABC _____
mBAC _____
mACB _____
B
A
40 CHAPTER 6 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 6.3 • Arcs and Angles
Name
Period Date
1. mXM 80°
X
80.
mXNM _____
Y M
mXN _____
mMN _____
O
N
3.
a _____
b _____
c _____
70b
c
a
5. AB and AC are tangents.
40
x
x _____
7.
mAD _____
mD _____
mAB _____
mDAB _____
B
A
C
A
B
D
80
70
2. AB
is a tangent.
x _____
y _____
z _____
4. a _____
120.
A
60
B
x
z
y
140.
b _____
c _____
a bc
100.
6. AD is a tangent. ACis a diameter.
mA _____
54
BOA D
mAB _____
mC _____
mCB _____
C
8.
p _____
87
29 rqp
s
q _____
r _____
s _____
9. Find the lettered angle and arc measures. AT and AZ
are tangents.
a _____ b _____ c _____
d _____ e _____ f _____
g _____ h _____ j _____
k _____ m _____ n _____
25
50
a
g
h
j
k
b
f
n
m
cd
e1
2
T
Z
A
Discovering Geometry Practice Your Skills CHAPTER 6 41
©2008 Key Curriculum Press
Lesson 6.4 • Proving Circle Conjectures
Name
Period Date
In Exercises 1–4, complete each proof with a paragraph or a flowchart.
1. Given:
Circles O and P are externally tangent, with common
tangents CD.
and AB
Show: AB at X
bisects CD
A P
D
X
C
O
B
. OE.
2. Given: Circle O with diameter ABand chord AD AD
Show: DE
BE
3. Given: PQ are tangent to both circles.
and RS
Show: PQ RS.
M X
N
P
S
R
Q
B
4. Prove the converse of the Chord Arcs Conjecture: If two arcs in a circle
OC
are congruent, then their chords are congruent. Hint: Draw radii.
D
Given: AB
CD
Show: AB CD.
A
O
A
B
D
E
42 CHAPTER 6 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 6.5 • The Circumference/Diameter Ratio
Name Period Date
In Exercises 1–4, leave your answers in terms of .
1. If r 10.5 cm, find C. 2. If C 25 cm, find r.
3. What is the circumference of a circle whose
radius is 30 cm?
4. What is the diameter of a circle whose
circumference is 24 cm?
In Exercises 5–9, round your answer to the nearest 0.1 unit. Use the
symbol to show that your answer is an approximation.
5. If d 9.6 cm, find C.
6. If C 132 cm, find d and r.
7.
A dinner plate fits snugly in a square box with perimeter 48 inches.
What is the circumference of the plate?
8.
Four saucers are part of the same set as the dinner plate in Exercise 7.
Each has a circumference of 15.7 inches. Will they fit, side by side, in
the same square box? If so, how many inches will there be between the
saucers for padding?
9. AT are tangents. AT 12 cm.
and AS10. How can you use a large carpenter’s
What is the circumference of circle O? square to find the circumference of a tree?
T A
SO
11. In order to increase the circumference of a circle from 16 cm to
20 cm, by how much must the diameter increase?
Discovering Geometry Practice Your Skills CHAPTER 6 43
©2008 Key Curriculum Press
Box
10 ft
7.5 in.
Box
10 ft
7.5 in.
Lesson 6.6 • Around the World
Name
Period Date
1.
Alfonzo’s Pizzeria bakes olive pieces in the outer crust of its 20-inch
(diameter) pizza. There is at least one olive piece per inch of crust.
How many olive pieces will you get in one slice of pizza? Assume the
pizza is cut into eight slices.
2.
To use the machine at right, you turn the crank, which turns the pulley
wheel, which winds the rope and lifts the box. Through how many
rotations must you turn the crank to lift the box 10 feet?
3.
A satellite in geostationary orbit stays over the same spot on Earth.
The satellite completes one orbit in the same time that Earth rotates
once about its axis (23.93 hours). If the satellite’s orbit has radius
4.23 107 m, calculate the satellite’s orbital speed (tangential velocity)
in meters per second.
4.
You want to decorate the side of a cylindrical can by coloring a
rectangular piece of paper and wrapping it around the can. The paper
is 19 cm by 29 cm. Find the two possible diameters of the can to the
nearest 0.01 cm. Assume the paper fits exactly.
5.
As you sit in your chair, you are whirling through space with Earth as
it moves around the sun. If the average distance from Earth to the
sun is 1.4957 1011 m and Earth completes one revolution every
364.25 days, what is your “sitting” speed in space relative to the sun?
Give your answer in km/h, rounded to the nearest 100 km/h.
44 CHAPTER 6 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 6.7 • Arc Length
Name
Period Date
In Exercises 1–10, leave your answers in terms of .
1. Length of AB _____ 2. The circumference is 24.
is 5.
3. The length of EF
and mCD 60°. Length Radius _____
B
of CD _____
1206
.
30
E
D
F
AC
4. Length of XY _____ 5. The radius is 20. Length 6. The circumference is 25.
of AB _____
Length of AB _____
A
Y
B
70
40
AB
7. The diameter is 40. Length 8. The length of XY9. Length of AB _____ is 14.
of AC _____ Diameter _____ B
10
X
50
80
36
110
A
X
A
T
Y
C
10.
A circle has an arc with measure 80° and length 88. What is the
diameter of the circle?
Discovering Geometry Practice Your Skills CHAPTER 6 45
©2008 Key Curriculum Press
Exploration • Intersecting Secants, Tangents, and Chords
Name Period Date
1. x _____ 2. FC is tangent to circle A at point C.
x44
mDC ______, mED ______
F
C
86.
140.
E
A
D
35
3. ED and EC are tangents. is a tangent, mBC
4. CE 150°
mDC ______, mDEC ______ mBCE ______, mBAC ______
B
A
246B
DEC
C
E
5. x ______, y ______, z ______ 6. x ______, y ______, z ______
79 60.
72zx
39
44.
60
y
70
x
zy
7. AB and AC8. AB are tangents. is a tangent, mABC 75°
x ______, y ______, z ______ x ______, y ______
A
z
xy
B
C
A
O
127
34
C
97
AB
x
85
y
46 CHAPTER 6 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 7.1 • Transformations and Symmetry
Name
Period Date
In Exercises 1–3, perform each transformation.
1. Reflect TRI across line . 2. Rotate PARL 270° clockwise 3. Translate PENTA by
about Q. the given vector.
I
LT
T
R
NP
R
A
QA
PE
4.
ABCDE and its reflected image, ABCDE, are shown below.
Use construction tools to locate the line of reflection, . Explain
your method.
D
D
E.
E
BA.
In Exercises 5–8, identify the type(s) of symmetry in each figure.
5. Equilateral triangle 6. Rectangle 7. Isosceles triangle 8. Square
C
B
C
A
In Exercises 9–12, draw each polygon and identify the type(s) of symmetry
in each. Draw all lines of reflection and mark centers of rotation.
9. Rhombus 10. Parallelogram 11. Isosceles trapezoid 12. Square
Discovering Geometry Practice Your Skills CHAPTER 7 47
©2008 Key Curriculum Press
Lesson 7.2 • Properties of Isometries
Name Period Date
In Exercises 1–3, draw the image according to the rule and identify the
type of transformation.
1. (x, y) . (x, y) 2. (x, y) . (x 4, y 6) 3. (x, y) . (4 x, y)
y
yy
2
–4
–2
4
6
–2–4 642
–4
4
8
x–8 –4 4
–2
x–4
–6
–6 –4 –2 2 64
2
4
6
In Exercises 4 and 5, the Harbour High Geometry Class is holding a Fence
Race. Contestants must touch each fence at some point as they run from
S to F. Use your geometry tools to draw the shortest possible race path.
Fence 1
4.
5.
Fence
S
S
F
Fence 2
F
In Exercises 6–8, complete the ordered pair rule that transforms
each triangle to its image. Identify the transformation. Find all
missing coordinates.
6. (x, y) . (_____, _____) 7. (x, y) . (_____, _____) 8. (x, y) . (_____, _____)
y
y
y
B.
R(4, 5) RPP(–7, –3)
Q(–2, 1) Q(2, 1)
T(0, 7)
(5, 2)
B(–5, 2)
A(8, 2)
C.
S(3, 3)
R
x
x
S
x
C
R(3, 0) T
A(–5, –4)
48 CHAPTER 7 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 7.3 • Compositions of Transformations
Name
Period Date
In Exercises 1–8, name the single transformation that can replace the
composition of each set of multiple transformations.
1. Translation by 4, 1, followed by 2, 3, followed by
8, 7.
2.
Rotation 60° clockwise, followed by 80° counterclockwise, followed
by 25° counterclockwise all about the same center of rotation
3.
Reflection across vertical line m, followed by reflection across vertical
line n, where n is 8 units to the right of m
4.
Reflection across vertical line p, followed by reflection across horizontal
line q
5.
Reflection across vertical line n, followed by reflection across vertical
line m, where n is 8 units to the right of m
6.
Reflection across horizontal line q, followed by reflection across vertical
line p
7. Translation by 6, 0, followed by reflection across the y-axis
8. Reflection across the y-axis, followed by translation by 6, 0
In Exercises 9–11, copy the figure onto your paper and use your
geometry tools to perform the given transformation. R
9. Locate P, the reflected image across OR, and P, the reflected image P
of Pacross OT. Find mROT and give a single transformation that
O
maps P to P.
T
10.
Locate P, the reflected image across k, and P, the reflected image
of Pacross . Find the distance between and k and give a single
transformation that maps P to P.
11. Draw five glide-reflected images of the triangle.
Pk
Discovering Geometry Practice Your Skills CHAPTER 7 49
©2008 Key Curriculum Press
Lesson 7.4 • Tessellations with Regular Polygons
Name
Period Date
1. Find n.
2. Find n.
Regular
10-gon
Equilateral
Regular
n-gon
triangle
Regular
Square
n-gon
Regular
pentagon
3.
What is a regular tessellation? Sketch an example to illustrate
your explanation.
4.
What is a 1-uniform tiling? Sketch an example of a 1-uniform tiling
that is not a regular tessellation.
5. Give the numerical name for the tessellation at right.
6. Use your geometry tools to draw the 4.82 tessellation.
50 CHAPTER 7 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lessons 7.5–7.8 • Tessellations
Name
Period Date
1.
Trace the quadrilateral at right (or draw a similar one). Make the
outline dark. Set another piece of paper on top of the quadrilateral
and, by tracing, create a tessellation. (Hint: Trace vertices and use a
straightedge to connect them.)
2.
On dot paper, draw a small concave quadrilateral (vertices on dots).
Allow no more than three dots inside the figure. Tessellate the entire
paper with your quadrilateral. Color and shade your tessellation.
3.
In non-edge-to-edge tilings, the vertices of the polygons do not have
to coincide, as in these wooden deck patterns. Use graph paper to
create your own non-edge-to-edge tiling.
4.
Use your geometry tools to draw a parallelogram. Draw squares on
each side. Create a tessellation by duplicating your parallelogram
and squares.
Discovering Geometry Practice Your Skills CHAPTER 7 51
©2008 Key Curriculum Press
Lesson 8.1 • Areas of Rectangles and Parallelograms
Name
Period Date
In Exercises 1–4, find the area of the shaded region.
1.
4 cm
8 cm
4 cm
12 cm
2.
1 cm
9 cm
3. 17 cm
4.
1.5 cm
4 cm4 cm
5 cm
2 cm
8 cm
13 cm
2 cm
5. Rectangle ABCD has area 2684 m2 and width 44 m. Find its length.
6. Draw a parallelogram with area 85 cm2 and an angle with measure 40°.
Is your parallelogram unique? If not, draw a different one.
7. Find the area of PQRS.
8. Find the area of ABCDEF.
y
y
S(–4, 5)
(15, 7)E
D(21, 7)
(4, 2)
x
F
R(7, –1)
C(10, 2)
P(–4, –3)
x
B(10, –5)
A(4, –5)
Q(7, –9)
9.
Dana buys a piece of carpet that measures 20 square yards. Will she be
able to completely cover a rectangular floor that measures 12 ft 6 in. by
16 ft 6 in.? Explain why or why not.
52 CHAPTER 8 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 8.2 • Areas of Triangles, Trapezoids, and Kites
Name Period Date
In Exercises 1–4, solve for the unknown measures.
1. Area 64 ft2, h ______. 2. Area ______.
h
8 ft
2 cm 2 cm
8 cm
2 cm
3. Area 126 in2 4. AB 6 cm, AC 8 cm, and BC 10 cm.
b _____. Find AD.
bB
D
9 in.
16 in.
C
A
5. Find the area of the shaded region.
6. TPis tangent to circles M and N. TP 16 cm. The radius of
N MPT
N is 7 cm and the radius of M is 4 cm. Find the area of NMPT.
12 cm
20 cm
4 cm
7. Find the area of TRI. 8. ABCD is a parallelogram, ABDE is a kite,
y AD 18 cm, and BE 10 cm. Find the
I (12, 12) area of ABCDE.
B
C
(0, 6) R
x
A
D
T (6, 0)
E
Discovering Geometry Practice Your Skills CHAPTER 8 53
©2008 Key Curriculum Press
Lesson 8.3 • Area Problems
Name
Period Date
1
18 ft
6 ft
1. A bundle of hardwood flooring contains 14 ft2 and
2
costs $39.90. 3 ft
a.
How many square feet of flooring is needed to cover the
kitchen and family room? Exclude the fireplace, hearth, and
3 ft 4 ft
18 ft
Kitchen
Tile
Family Fireplace
room and hearth
4 ft
14 ft
13 ft
tiled area.
b.
You should buy 5% extra flooring to account for waste.
How many bundles of flooring should you buy? What will
7 ft
be the cost?
2.
Bert’s Bigtime Bakery has baked the world’s largest chocolate cake. It is
a rectangular sheet cake that is 600 cm by 400 cm by 180 cm high. Bert
wants to apply frosting to the four sides and the top. How many liters
of frosting does he need if 1 liter of frosting covers about 1200 cm2?
3.
For a World Peace Day celebration the students at Cabot Junior/Senior
High School are making a 6 m-by-8 m flag. Each of the six grades will
create a motif to honor the people of the six inhabited continents.
Sketch three possible ways to divide the flag: one into six congruent
triangles; one into six triangles with equal area but none congruent;
and one into six congruent trapezoids. Give measurements or markings
on your sketches so each class knows it has equal area.
4.
Kit and Kat are building a kite for the big kite festival. Kit has already
cut his sticks for the diagonals. He wants to position P so that he will
have maximum kite area. He asks Kat for advice. What should Kat
tell him?
P
54 CHAPTER 8 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 8.4 • Areas of Regular Polygons
Name
Period Date
In Exercises 1–3, the polygons are regular.
1.
s 12 cm 2. s 4.2 cm 3. a 6 cm
a 14.5 cm A 197 cm2 A 130.8 cm2
A _____ a _____
p _____
s
s
a
s
12-gon
a
a
4. In a regular n-gon, s 4.8 cm, a 7.4 cm, and A 177.6 cm2. Find n.
5.
Draw a regular pentagon so that it has 6. Use a compass and straightedge to
perimeter 20 cm. Use the Regular Polygon construct a regular octagon and
Area Conjecture and a centimeter ruler to its apothem. Use a centimeter ruler to
find its approximate area. measure its side length and apothem, and
use the Regular Polygon Area Conjecture to
find its approximate area.
7. Find the area of the shaded region between the square and
s
the regular octagon. s 5 cm. r 3 cm.
r r
Discovering Geometry Practice Your Skills CHAPTER 8 55
©2008 Key Curriculum Press
Lesson 8.5 • Areas of Circles
Name Period Date
In Exercises 1–4, write your answers in terms of .
1. If r 9 cm, A _____. 2. If d 6.4 cm, A _____.
3. If A 529 cm2, r _____. 4. If C 36 cm, A _____.
In Exercises 5–8, round your answers to the nearest 0.01 unit.
5. If r 7.8 cm, A _____. 6. If A 136.46, C _____.
7. If d 3.12, A _____. 8. If C 7.85, A _____.
For Exercises 9 and 10, refer to the figure of a circle inscribed
in an equilateral triangle. Round your answers to the nearest
0.1 unit.
9. Find the area of the inscribed circle.
10. Find the area of the shaded region.
In Exercises 11 and 12, find the area of the shaded region. Write your
answers in terms of .
11. ABCD is a square. 12. The three circles are tangent.
14.0 cm
a
a 4.04 cm
A
C (0, 8)
D (8, 0) B
x
y
5 cm
56 CHAPTER 8 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 8.6 • Any Way You Slice It
Name Period Date
In Exercises 1–6, find the area of the shaded region. Write your answers in
terms of and rounded to the nearest 0.01 cm2.
1.
5 cm30
2.
120
4 cm
3.
2 cm
4 cm
4. 135 5.
8 cm
3 cm
6.
12 cm
8 cm
45
7. Shaded area is 40 cm2. 8. Shaded area is 54 cm2. 9. Shaded area is 51 cm2.
Find r. Find x. The diameter of the larger
circle is 20 cm. Find r.
144
12 cm
x
r
r
Discovering Geometry Practice Your Skills CHAPTER 8 57
©2008 Key Curriculum Press
Lesson 8.7 • Surface Area
Name
Period Date
In Exercises 1–8, find the surface area of each solid. All quadrilaterals are
rectangles, and all measurements are in centimeters. Round your answers
to the nearest 0.1 cm2.
1. 2. 3.
8.5
12.4
6
5
12
13
7
6
2
4.
5. Base is a regular hexagon. 6.
s 6, a 5.2, and l 9.
13
a
l
5
13
s
10
4
3
4
3
3
7. Both bases are squares.
8. A square hole in a round peg
6
8
4
13
12
14
9.
Ilsa is building a museum display case. The sides and bottom
will be plywood and the top will be glass. Plywood comes in
1
_
1 ft
4 ft-by-8 ft sheets. How many sheets of plywood will she need
2
to buy? Explain. Sketch a cutting pattern that will leave her with
the largest single piece possible.
_1
2 ft
2
3 ft
3 ft
2 ft
58 CHAPTER 8 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 9.1 • The Theorem of Pythagoras
Name
Period Date
Give all answers rounded to the nearest 0.1 unit.
1. a _____ 2. p _____
3. x _____
75 cm
p
a
14 cm
72 cm
x
8 cm 21 cm
26 ft
24 ft
6 ft
4. Area 39 in2 5. Find the area.
6. Find the coordinates of C
and the radius of circle A.
h _____
y
6 in.
h
B (11, –4)
A
(7, –1)
C
7 ft
6 ft
7. Find the area.
8. RS 3 cm. Find RV.
R
7.7 cm
6.8 cm
V
S
T
U
13.4 cm
9. Base area 16 cm2 and slant height 10. Given PQR, with mP 90°, PQ 20 in.,
3 cm. What’s wrong with this picture?
and PR 15 in., find the area of PQR,
the length of the hypotenuse, and the
altitude to the hypotenuse.
Discovering Geometry Practice Your Skills CHAPTER 9 59
©2008 Key Curriculum Press
Lesson 9.2 • The Converse of the Pythagorean Theorem
Name Period Date
All measurements are in centimeters. Give answers rounded to the nearest
0.01 cm.
In Exercises 1–4, determine whether a triangle with the given side lengths
is a right triangle.
1. 76, 120, 98 2. 221, 204, 85 3. 5.0, 1.4, 4.8 4. 80, 82, 18
5. Find the area of ABC. 6. What’s wrong with this picture?
5
6.5 6
95
C B
7
A
7. Find x. Explain your method. 8. Find the area of ABCD.
D 22 C
C
72545 6 8
A
B
xB A
24 D 32
In Exercises 9–11, determine whether ABCD is a rectangle and justify your
answer. If not enough information is given, write “cannot be determined.”
9. AB 3, BC 4, and AC 6. AD
10. AB 3, BC 4, DA 4, and AC 5. BC
11. AB 3, BC 4, CD 3, DA 4, and AC BD.
60 CHAPTER 9 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 9.3 • Two Special Right Triangles
Name Period Date
Give your answers in exact form unless otherwise indicated.
All measurements are in centimeters.
In Exercises 1–3, find the unknown lengths.
1. a ______ 2. a ______, b ______ 3. a ______, b ______
a
60
a
6
14
12 3 b
30
b
a
4.
Find the area of rectangle 5. Find the perimeter and 6. AC _____, AB _____,
ABCD. area of KLMN. and area ABC _____.
AB
4530
N 7 MC
12
16
60
45 30
60
L
K
A
B
D
C
7.
Find the area of an isosceles trapezoid if the bases have lengths 12 cm
and 18 cm and the base angles have measure 60°.
In Exercises 8 and 9, find the coordinates of C.
8. y
9. y
120
C
x
(0, 12)
(12, 0)
210
C
x
(1, 0)
(0, 1)
10. Sketch and label a figure to demonstrate that 18
is
equivalent to 32.
Discovering Geometry Practice Your Skills CHAPTER 9 61
©2008 Key Curriculum Press
Lesson 9.4 • Story Problems
Name
Period Date
1.
A 20 ft ladder reaches a window 18 ft high. How far is the foot of the
ladder from the base of the building? How far must the foot of the
ladder be moved to lower the top of the ladder by 2 ft?
2.
Robin and Dovey have four pet pigeons that they train to race. They
release the birds at Robin’s house and then drive to Dovey’s to collect
them. To drive from Robin’s to Dovey’s, because of one-way streets,
they go 3.1 km north, turn right and go 1.7 km east, turn left and go
2.3 km north, turn right and go 0.9 km east, turn left and go 1.2 km
north, turn left and go 4.1 km west, and finally turn left and go 0.4 km
south. How far do the pigeons have to fly to go directly from Robin’s
house to Dovey’s house?
3.
Hans needs to paint the 18 in.-wide trim around the roof eaves and
gable ends of his house with 2 coats of paint. A quart can of paint
covers 175 ft2 and costs $9.75. A gallon can of paint costs $27.95. How
much paint should Hans buy? Explain.
18 ft
28 ft
42 ft
18 in.
9.5 ft
4. What are the dimensions of the largest 30°-60°-90° triangle that will fit
inside a 45°-45°-90° triangle with leg length 14 in.? Sketch your solution.
62 CHAPTER 9 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 9.5 • Distance in Coordinate Geometry
Name
Period Date
In Exercises 1–3, find the distance between each pair of points.
1. (5, 5), (1, 3)
2. (11, 5), (5, 7) 3. (8, 2), (7, 6)
In Exercises 4 and 5, use the distance formula and the slope of segments to
identify the type of quadrilateral. Explain your reasoning.
4. A(2, 1), B(3, 2), C(8, 1), D(3, 4) 5. T(3, 3), U(4, 4), V(0, 6), W(5, 1)
For Exercises 6 and 7, use ABC with coordinates A(4, 14), B(10, 6), and
C(16, 14).
6.
Determine whether ABC is scalene, isosceles, or equilateral. Find the
perimeter of the triangle.
7. Find the midpoints M and N of ABand AC, respectively. Find the
slopes and lengths of MN. How do the slopes compare? How
and BC
do the lengths compare?
8. Find the equation of the circle with center (1, 5) and radius 2.
9. Find the center and radius of the circle whose equation is x2 (y 2)2 25.
10. P is the center of the circle. What’s wrong with this picture?
P(10, 1)
A(4, 6)
C(16, –3) B(5, –5)
y
x
Discovering Geometry Practice Your Skills CHAPTER 9 63
©2008 Key Curriculum Press
Lesson 9.6 • Circles and the Pythagorean Theorem
Name
Period Date
In Exercises 1 and 2, find the area of the shaded region in each figure. All
measurements are in centimeters. Write your answers in terms of and
rounded to the nearest 0.1 cm2.
1. AO 5. AC 8.
2. Tangent PT,QM 12, mP 30°
C
O
T
B
M
Q
P
A
S
3.
AP 63 cm. Radius of circle O 37 cm. P
How far is A from the circumference of
the circle?
O
A
4.
Two perpendicular chords with lengths 12.2 cm and 8.8 cm have a
common endpoint. What is the area of the circle?
5. ABCD is inscribed in a circle. ACis a diameter. If AB 9.6 cm,
BC 5.7 cm, and CD 3.1 cm, find AD.
31
6. Find ST.
7. The coordinate of point M is , .
22
P
T
S
O
R
63
30°
Find the measure of AOM.
Oy
(0, 1)
M
A
x
(1, 0)
64 CHAPTER 9 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 10.1 • The Geometry of Solids
Name
Period Date
For Exercises 1–14, refer to the figures below.
T
A
C
I
CBJ
H
D
E
G
R
F
B
A
OP
Q
1. The cylinder is (oblique, right).
2. OPis ________________ of the cylinder.
3. TRis ________________ of the cylinder.
4. Circles O and P are ________________ of the cylinder.
5. PQis ________________ of the cylinder.
6. The cone is (oblique, right).
7. Name the base of the cone.
8. Name the vertex of the cone.
9. Name the altitude of the cone.
10. Name a radius of the cone.
11. Name the type of prism.
12. Name the bases of the prism.
13. Name all lateral edges of the prism.
14. Name an altitude of the prism.
In Exercises 15–17, tell whether each statement is true or false. If the
statement is false, give a counterexample or explain why it is false.
15. The axis of a cylinder is perpendicular to the base.
16. A rectangular prism has four faces.
17. The bases of a trapezoidal prism are trapezoids.
For Exercises 18 and 19, draw and label each solid. Use dashed lines to
show the hidden edges.
18.
A right triangular prism with height 19. An oblique trapezoidal pyramid
equal to the hypotenuse
Discovering Geometry Practice Your Skills CHAPTER 10 65
©2008 Key Curriculum Press
Lesson 10.2 • Volume of Prisms and Cylinders
Name
Period Date
In Exercises 1–3, find the volume of each prism or cylinder.
All measurements are in centimeters. Round your answers to the
nearest 0.01.
1. Right triangular prism 2. Right trapezoidal prism
6
6
6
14
10
853
In Exercises 4–6, use algebra to express the volume of each solid.
4.
Right rectangular prism 5. Right cylinder;
base circumference p
4y
h
2x 3
7.
You need to build a set of solid cement steps for the entrance
to your new house. How many cubic feet of cement do
you need?
3. Regular hexagonal prism
4
10
6. Right rectangular prism
and half of a cylinder
2x
y
3x
6 in.
3 ft
8 in.
66 CHAPTER 10 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 10.3 • Volume of Pyramids and Cones
Name Period Date
In Exercises 1–3, find the volume of each solid. All measurements are in
centimeters. Round your answers to two decimal places.
1. Rectangular pyramid; OP 6 2. Right hexagonal pyramid 3. Half of a right cone
P
8 5
O
9
25
14
6
In Exercises 4–6, use algebra to express the volume of each solid.
4.
25x7x
5.
6. The solid generated by
spinning ABC about
30xb
2a
the axis
A
3x
2yB
C
In Exercises 7–9, find the volume of each figure and tell which volume is larger.
7. A.
4
B.
12
8
6
8. A.
3 5
B.
2
2
5
3
9. A. B.
9
x3
x
Discovering Geometry Practice Your Skills CHAPTER 10 67
©2008 Key Curriculum Press
Lesson 10.4 • Volume Problems
Name
Period
1. A cone has volume 320 cm3 and height 16 cm. Find the radius of the
base. Round your answer to the nearest 0.1 cm.
2.
How many cubic inches are there in one cubic foot? Use your answer
to help you with Exercises 3 and 4.
3.
Jerry is packing cylindrical cans with diameter 6 in. and height 10 in.
tightly into a box that measures 3 ft by 2 ft by 1 ft. All rows must
contain the same number of cans. The cans can touch each other. He
then fills all the empty space in the box with packing foam. How many
cans can Jerry pack in one box? Find the volume of packing foam he
uses. What percentage of the box’s volume is filled by the foam?
4.
A king-size waterbed mattress measures 72 in. by 84 in. by 9 in. Water
weighs 62.4 pounds per cubic foot. An empty mattress weighs
35 pounds. How much does a full mattress weigh?
5.
Square pyramid ABCDE, shown at right, is cut out of a cube
with base ABCD and shared edge DE. AB 2 cm. Find the volume
and surface area of the pyramid.
6.
In Dingwall the town engineers have contracted for a new water
storage tank. The tank is cylindrical with a base 25 ft in diameter
and a height of 30 ft. One cubic foot holds about 7.5 gallons of
water. About how many gallons will the new storage tank hold?
7.
The North County Sand and Gravel Company stockpiles sand
to use on the icy roads in the northern rural counties of the
state. Sand is brought in by tandem trailers that carry 12 m3
each. The engineers know that when the pile of sand, which is in
the shape of a cone, is 17 m across and 9 m high they will have
enough for a normal winter. How many truckloads are needed to
build the pile?
Date
E
A
D
B
68 CHAPTER 10 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 10.5 • Displacement and Density
Name
Period Date
1.
A stone is placed in a 5 cm-diameter graduated cylinder, causing the
water level in the cylinder to rise 2.7 cm. What is the volume of
the stone?
2.
A 141 g steel marble is submerged in a rectangular prism with base
5 cm by 6 cm. The water rises 0.6 cm. What is the density of the steel?
3.
A solid wood toy boat with a mass of 325 g raises the water level of a
50 cm-by-40 cm aquarium 0.3 cm. What is the density of the wood?
4.
For Awards Night at Baddeck High School, the math club is
designing small solid silver pyramids. The base of the pyramids will
be a 2 in.-by-2 in. square. The pyramids should not weigh more than
21 pounds. One cubic foot of silver weighs 655 pounds. What is the
2
maximum height of the pyramids?
5.
While he hikes in the Gold Country of northern California, Sid
dreams about the adventurers that walked the same trails years ago.
He suddenly kicks a small bright yellowish nugget. Could it be gold?
Sid quickly makes a balance scale using his walking stick and finds that
the nugget has the same mass as the uneaten half of his 330 g nutrition
bar. He then drops the stone into his water bottle, which has a 2.5 cm
radius, and notes that the water level goes up 0.9 cm. Has Sid struck
gold? Explain your reasoning. (Refer to the density chart in
Lesson 10.5 in your book.)
Discovering Geometry Practice Your Skills CHAPTER 10 69
©2008 Key Curriculum Press
Lesson 10.6 • Volume of a Sphere
Name
Period Date
In Exercises 1–6, find the volume of each solid. All measurements are in
centimeters. Write your answers in exact form and rounded to the nearest
0.1 cm3.
1.
6
2.
3
3.
6
4.
2
90°
5.
6. Cylinder with hemisphere
6 taken out of the top
6
6
9
4
7. A sphere has volume 221
6
5
cm3. What is its diameter?
8. The area of the base of a hemisphere is 225 in2. What is its volume?
9.
Eight wooden spheres with radii 3 in. are packed snugly into a square
box 12 in. on one side. The remaining space is filled with packing
beads. What is the volume occupied by the packing beads? What
percentage of the volume of the box is filled with beads?
10.
The radius of Earth is about 6378 km, and the radius of Mercury is
about 2440 km. About how many times greater is the volume of Earth
than that of Mercury?
70 CHAPTER 10 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 10.7 • Surface Area of a Sphere
Name
Period Date
In Exercises 1–4, find the volume and total surface area of each solid.
All measurements are in centimeters. Round your answers to the nearest
0.1 cm.
1.
7.2
2.
7
4
3.
5
4.
3
8
3 3
5. If the surface area of a sphere is 48.3 cm2, find its diameter.
6. If the volume of a sphere is 635 cm3, find its surface area.
7.
Lobster fishers in Maine often use spherical buoys to mark their lobster
traps. Every year the buoys must be repainted. An average buoy has a
12 in. diameter, and an average fisher has about 500 buoys. A quart of
marine paint covers 175 ft2. How many quarts of paint does an average
fisher need each year?
Discovering Geometry Practice Your Skills CHAPTER 10 71
©2008 Key Curriculum Press
Lesson 11.1 • Similar Polygons
Name
Period Date
All measurements are in centimeters.
1. HAPIE NWYRS
2. QUAD SIML
120°
6
A
85°
S
AP _____ P
SL _____
H
8
4
5
L
A
EI
I
EI _____
MI _____
W
D
18 24
M
SN _____
mD _____
75°
N
Y
YR _____
mU _____
S 21 R
mA _____
In Exercises 3–6, decide whether or not the figures are similar. Explain why
13
Q 20 U
or why not.
3. ABCD and EFGH
H
G
120°
120°
AB
3 60°
60°
9
D 5 C
120°
60°
E 15 F
5. JKON and JKLM
5
K
J
106 8
O
N
12 14
L
M 20
7. Draw the dilation of ABCD by a scale
1
factor of . What is the ratio of the
2perimeter of the dilated quadrilateral
to the perimeter of the original
quadrilateral?
y
5
5
B(4, 6)
C(3, 3)
A(0, 2)
D(4, 1)
x
4. ABC and ADE
A
42 3
DE
3
2
C
B
8
6. ABCD and AEFG
AE 3 B
4
4
4 F
7
G
3
C
D 7
8.
Draw the dilation of DEF by a scale factor
of 2. What is the ratio of the area of the
dilated triangle to the area of the original
triangle?
y
E (4, 1)
D(1, 2)
F (2, 1)
5
5
x
72 CHAPTER 11 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 11.2 • Similar Triangles
Name Period Date
All measurements are in centimeters.
1. TAR MAC
MC _____
R
2. XYZ QRS 3. ABC EDC 4. TRS TQP
Q _____ A _____ TS _____
QR _____ CD _____ QP _____
QS _____ AB _____
17
T
8
SR
YB _1E
20
30
4
6 16
20
12
C
P
Q
XZ _1
9
28 A 22
2
R
8
S
Q
D
For Exercises 5 and 6, refer to the figure at right. A
5. Explain why CAT and DAG are similar.
3
2
7 T
A
M C
48
12
16
T
GD
6. CA _____ C
In Exercises 7–9, identify similar triangles and explain why they are similar.
7. B 8.
R
Q
P
9. M
N
K
TL
O
C
E
A
S
D
Discovering Geometry Practice Your Skills CHAPTER 11 73
©2008 Key Curriculum Press
Lesson 11.3 • Indirect Measurement with Similar Triangles
Name
Period Date
1.
At a certain time of day, a 6 ft man casts a 4 ft shadow. At the same
time of day, how tall is a tree that casts an 18 ft shadow?
2.
Driving through the mountains, Dale has to go up and over a high
mountain pass. The road has a constant incline for 73 miles to the
4
top of the pass. Dale notices from a road sign that in the first mile
he climbs 840 feet. How many feet does he climb in all?
3.
Sunrise Road is 42 miles long between the edge of Moon Lake
and Lake Road and 15 miles long between Lake Road and
Sunset Road. Lake Road is 29 miles long. Find the length of
Moon Lake.
4.
Marta is standing 4 ft behind a fence 6 ft 6 in. tall. When
she looks over the fence, she can just see the top edge of a
building. She knows that the building is 32 ft 6 in. behind
the fence. Her eyes are 5 ft from the ground. How tall is
the building? Give your answer to the nearest half foot.
4 ft
5.
You need to add 5 supports under the ramp, in addition to
the 3.6 m one, so that they are all equally spaced. How long
should each support be? (One is drawn in for you.)
32 ft 6 in.
6 ft 6 in.
5 ft
Fence
Building
Moon
Lake
Sunset Road
Sunrise Road
Lake
Road
3.6 m
9.0 m
Support
Ramp
74 CHAPTER 11 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 11.4 • Corresponding Parts of Similar Triangles
Name Period Date
All measurements are in centimeters.
1. ABC PRQ. M and N 2. The triangles are similar.
are midpoints. Find h and j. Find the length of each side of the
R smaller triangle to the nearest 0.01.
1.5
1.2
h
Q
P
N
B
3.2
j2.4
C
A
M
10
12
8
3
3. ABC WXY 4. Find x and y.
WX _____ AD _____
10
DB _____ YZ _____
XZ _____
5
C
x
y
8
28
14
B
D
16
12
W
Y
Z
A
X
24
5. Find a, b, and c. 6. Find CB, CD, and AD.
c
b
a
4
3
6
A
25
7
B
CD
Discovering Geometry Practice Your Skills CHAPTER 11 75
©2008 Key Curriculum Press
Lesson 11.5 • Proportions with Area
Name
Period Date
All measurements are in centimeters unless otherwise indicated.
Area of circle O4
1. ABC DEF. Area of ABC 15 cm2. 2. .
Area of circle P 9Area of DEF _____ a _____
AD
3 cm
5 cm
F
E
B
6 cm
P
O
a
Area of square SQUA
Area of circle P
3.
_____ 4. _____ 5. RECT ANGL
Area of square LRGE
Area of circle O
Area of RECT
_____
E
3
UQ
S AG
PO
2
12.
Area of ANGL
TA
L G
N4
C
10
R
E
L
R
5
6.
The ratio of the corresponding midsegments of two similar trapezoids
is 4:5. What is the ratio of their areas?
7.
The ratio of the areas of two similar pentagons is 4:9. What is the ratio
of their corresponding sides?
8. If ABCDE FGHIJ, AC 6 cm, FH 10 cm, and area of
ABCDE 320 cm2, then area of FGHIJ _____.
9.
Stefan is helping his mother retile the kitchen floor. The tiles are
4-by-4-inch squares. The kitchen is square, and the area of the floor is
144 square feet. Assuming the tiles fit snugly (don’t worry about
grout), how many tiles will be needed to cover the floor?
76 CHAPTER 11 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 11.6 • Proportions with Volume
Name
Period Date
All measurements are in centimeters unless otherwise indicated.
In Exercises 1 and 2, decide whether or not the two solids are similar.
1.
2.
4
20
3
16
8
5
32
12
2
1.56
8
3. The triangular prisms are similar and the ratio of a to b is 5.
2Volume of large prism 250 cm3
Volume of smaller prism _____
a
b
4.
The right cylinders are similar and r 10 cm.
Volume of large cylinder 64 cm
Volume of small cylinder 8 cm
R _______
R
r
5.
The corresponding heights of two similar cylinders is 2:5. What is the
ratio of their volumes?
6.
A rectangular prism aquarium holds 64 gallons of water. A similarly
shaped aquarium holds 8 gallons of water. If a 1.5 ft2 cover fits on the
smaller tank, what is the area of a cover that will fit on the larger tank?
Discovering Geometry Practice Your Skills CHAPTER 11 77
©2008 Key Curriculum Press
Lesson 11.7 • Proportional Segments Between Parallel Lines
Name
Period Date
All measurements are in centimeters.
?
?
1. x _____
2. Is XY BC3. Is XY MK
AA
M
8 23
40
6
XY
X
30
4.5
3
x
9
K
O
60 Y 48
C
B
P
T
4.
NE _____ 5. PR _____ 6. a _____
N
12.5
PQ _____ b _____
n
RI _____
M
4.5
T 15
5
M 3
P
Q
b
5
10
T
3
E
P 8 A 12
a
I
A
R
7.
RS _____ 8. x _____ 9. p _____
EB _____ y _____ q _____
R
2 q15 10
E
B
3
4
5
S
30 T
9
12
y1512 x
4 p
78 CHAPTER 11 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 12.1 • Trigonometric Ratios
Name
Period Date
In Exercises 1–4, give each answer as a fraction in terms of p, q, and r.
1.
sin P _____ 2. cos P _____ P
q
3. tan P _____ 4. sin Q _____
Rp
Q
In Exercises 5–8, give each answer as a decimal accurate to the nearest 0.001.
5. sin T _____ 6. cos T _____
7. tan T _____ 8. sin R _____
T
For Exercises 9–11, solve for x. Express each answer accurate to the
nearest 0.01.
x 12.1
x
9. cos 64° .
10. sin 24° 11. tan 51°
28
x 14.8
For Exercises 12–14, find the measure of each angle to the nearest degree.
4
12. sin A 0.9455 13. tan B .
14. cos C 0.8660
3
For Exercises 15–17, write a trigonometric equation you can use to solve
for the unknown value. Then find the value to the nearest 0.1.
15. w _____ 16. x _____
17. y _____
r
86
R
w
40°
y
x
28 cm
17°
73 cm
28°
14 cm
For Exercises 18–20, find the value of each unknown to the nearest degree.
18. a _____ 19. t _____
20. z _____
14 cm
26 cm
a
11 in.
15 in.
t
25 cm
z
12 cm
Discovering Geometry Practice Your Skills CHAPTER 12 79
©2008 Key Curriculum Press
Lesson 12.2 • Problem Solving with Right Triangles
Name
Period Date
For Exercises 1–3, find the area of each figure to the nearest square unit.
1. Area _____ 2. Area _____
3. Area _____
50°
28°
28 ft
13 in.
140°
2.0 cm
For Exercises 4–9, find each unknown to the nearest tenth of a unit.
4. Area 88 cm2 5. y _____
6. a _____
x _____
y28 ft
14 in.
72°
x
8 in.
a
8 in.
17 ft
14 in.
16 cm
7. PS and PT are tangents. 8. Right cone 9. Right rectangular prism
Diameter _____ _____ mABC _____
S
10 in.
65°
22 cm
T
P
B
A
5 ft
24 in.
13 ft
C
14 in.
In Exercises 10–12, give each answer to the nearest tenth of a unit.
10.
A ladder 7 m long stands on level ground and makes a 73° angle with
the ground as it rests against a wall. How far from the wall is the base
of the ladder?
11.
To see the top of a building 1000 feet away, you look up 24° from the
horizontal. What is the height of the building?
12. A guy wire is anchored 12 feet from the base of a pole. The wire makes
a 58° angle with the ground. How long is the wire?
80 CHAPTER 12 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 12.3 • The Law of Sines
Name
Period Date
In Exercises 1–3, find the area of each figure to the nearest square unit.
1. Area _____ 2. Area _____
3. Area _____
15 cm
70°
8 ft
17°
5 ft
21 cm
14 ft
55°
8.7 m
In Exercises 4–6, find each length to the nearest centimeter. All lengths
are in centimeters.
4. m _____ 5. p _____
6. q _____
m
32°
51°
12
21
22°
17
p
40°
67°
76°
q
In Exercises 7–9, find the measure of each angle to the nearest degree.
7. mB _____ 8. mP _____ 9. mK _____
mC _____ mQ _____ mM _____
B
RM
48°
16
78°
105
26
81°
29
106
A
K
C
P
Q
32
L
10.
A large helium balloon is tethered to the ground by two taut lines. One
line is 100 feet long and makes an 80° angle with the ground. The
second line makes a 40° angle with the ground. How long is the second
line, to the nearest foot? How far apart are the tethers?
Discovering Geometry Practice Your Skills CHAPTER 12 81
©2008 Key Curriculum Press
Lesson 12.4 • The Law of Cosines
Name
Period Date
In Exercises 1–3, find each length to the nearest centimeter. All lengths are
in centimeters.
1. t _____
2. b _____ 3. w _____
U
OMw
O
118
b
16
t
87
25°
114
39°
B
Y
152
12°
P
T
21
W
In Exercises 4–6, find each angle measure to the nearest degree.
4. mA _____ 5. mA _____ 6. mS _____
mB _____ mP _____ mU _____
mC _____ mS _____ mV _____
C
PV
48
7030
51
192
265
S
A
45
B
A
62
S
U
201
7.
A circle with radius 12 in. has radii drawn to the endpoints of a
5 in. chord. What is the measure of the central angle?
8.
A parallelogram has side lengths 22.5 cm and 47.8 cm. One angle
measures 116°. What is the length of the shorter diagonal?
9.
The diagonals of a parallelogram are 60 in. and 70 in. and intersect
at an angle measuring 64°. Find the length of the shorter side of
the parallelogram.
82 CHAPTER 12 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 12.5 • Problem Solving with Trigonometry
Name
Period Date
1.
While floating down a river with a 2.75 mi/h current, Alicia decides
to swim directly toward the river bank. She can swim 0.75 mi/h in still
water. What is the actual speed at which she moves toward the bank?
At what angle will she approach the bank, measured with respect to
the bank?
2. Find the measure of each angle to the nearest hundredth of a degree.
xyB(4, 7)
abcA(15, 9)
C(10, 0)
3.
Two fire watchtowers 8.4 km apart spot a fire at the same time. Tower 1
reports the fire at a 36° angle measure from its line of sight to Tower 2.
Tower 2 reports a 68° angle measure between the fire and Tower 1.
How far is the fire from each tower?
4.
Two airplanes leave O’Hare Airport in Chicago at the same time. One
plane flies 280 mi/h at bearing 55°. The other plane flies 350 mi/h at
bearing 128°. How far apart are the two planes after 2 hours 15 minutes?
5.
Carla needs to fence her triangular plot of land. The angle between
the two shorter sides measures 83°. The shortest side is 122 ft and
the longest is 215 ft. How much fencing does Carla need? What
is the area of her plot of land?
Discovering Geometry Practice Your Skills CHAPTER 12 83
©2008 Key Curriculum Press
Lesson 13.1 • The Premises of Geometry
Name
Period Date
1.
Provide the missing property of equality or arithmetic as a reason for
each step to solve the equation.
Solve for x:5(x 4) 2x 17
Solution: 5(x 4) 2x 17 a. ________________
5x 20 2x 17 b. ________________
3x 20 17 c. ________________
3x 37 d. ________________
37
x e. ________________
3
In Exercises 2–4, identify each statement as true or false. If the statement is
true, tell which definition, property, or postulate supports your answer. If
the statement is false, give a counterexample.
2. If AM BM, then M is the midpoint of AB.
3. If P is on AB
and D is not, then mAPD mBPD 180°.
4. If PQ STand PQ KL, then ST KL.
5.
Complete the flowchart proof.
Given: AB CD,PB QD
,AP CQ.
Show: AB CD
Flowchart Proof
P
B
A
Q
C
D
____________ ____________
ABP CDQGiven
AB CDAB CD____________
AP CQ
____________ CA Postulate ____________
84 CHAPTER 13 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 13.2 • Planning a Geometry Proof
Name Period Date
For these exercises, you may use theorems added to your theorem list
through the end of Lesson 13.2.
In Exercises 1–3, write a paragraph proof or a flowchart proof for
each situation.
1. Given: AB,AP
CD CQ
Show: PAB QCD
BPACQD
2. Given: PQ ST, QPR STU Q
Show: PR UT
RPUST
3. Given: Noncongruent, nonparallel
segments AB, BC, and AC
Show: x y z 180°
y
xAabczB
C
Discovering Geometry Practice Your Skills CHAPTER 13 85
©2008 Key Curriculum Press
Lesson 13.3 • Triangle Proofs
Name Period Date
Write a proof for each situation. You may use theorems added to your
theorem list through the end of Lesson 13.3.
MW
AC AB BD
1. Given: XY ZY,XZ WY2. Given: CD,BD,CD
Show: WXY WZY
Show: ABD ACD
C
D
X
Z
Y
A
B
QM QM
3. Given: MN,NO,
4. Given: AB BC, ACB ECD,
P is the midpoint of MO
R
AB BD
Show: QMN RON
Show: BD CE B
O
E
A
N
P
D
C
M
Q
86 CHAPTER 13 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 13.4 • Quadrilateral Proofs
Name
Period Date
In Exercises 1–6, write a proof of each conjecture on a separate piece of
paper. You may use theorems added to your theorem list through the end
of Lesson 13.4.
1.
The diagonals of a parallelogram bisect each other. (Parallelogram
Diagonals Theorem)
2.
If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram. (Converse of the Parallelogram
Diagonals Theorem)
3.
The diagonals of a rhombus bisect each other and are perpendicular.
(Rhombus Diagonals Theorem)
4.
If the diagonals of a quadrilateral bisect each other and are
perpendicular, then the quadrilateral is a rhombus. (Converse of the
Rhombus Diagonals Theorem)
5.
If the base angles on one base of a trapezoid are congruent, then the
trapezoid is isosceles. (Converse of the Isosceles Trapezoid Theorem)
6.
If the diagonals of a trapezoid are congruent, then the trapezoid is
isosceles. (Converse of the Isosceles Trapezoid Diagonals Theorem)
In Exercises 7–9, decide if the statement is true or false. If it is true, prove
it. If it is false, give a counterexample.
7.
A quadrilateral with one pair of parallel sides and one pair of
congruent angles is a parallelogram.
8.
A quadrilateral with one pair of congruent opposite sides and one pair
of parallel sides is a parallelogram.
9.
A quadrilateral with one pair of parallel sides and one pair of
congruent opposite angles is a parallelogram.
Discovering Geometry Practice Your Skills CHAPTER 13 87
©2008 Key Curriculum Press
Lesson 13.5 • Indirect Proof
Name
Period Date
1.
Complete the indirect proof of the conjecture: In a triangle the side
opposite the larger of two angles has a greater measure.
Given: ABC with mA mB
Show: BC AC
Proof: Assume ________________
A
Case 1: BC AC
If BC AC, then ABC is ________________ by _______________.
By _______________, A B, which contradicts ______________.
So, BC AC.
Case 2: BC AC
If BC AC, then it is possible to construct point D on CAsuch that
CD, by the Segment Duplication Postulate. Construct DB, by
CB
the Line Postulate. DBC is ________________. Complete the proof.
C
BC
D4
3
21
BA
In Exercises 2–5, write an indirect proof of each conjecture.
AB BC
Show: DAC BAC
2. Given: AD,DC.
A
D
B
C
3.
If two sides of a triangle are not congruent, then the angles
opposite them are not congruent.
4.
If two lines are parallel and a third line in the same plane intersects
one of them, then it also intersects the other.
88 CHAPTER 13 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 13.6 • Circle Proofs
Name
Period Date
Write a proof for each conjecture or situation. You may use theorems
added to your theorem list through the end of Lesson 13.6.
1. If two chords in a circle are congruent, then their arcs are congruent.
2. Given:
Regular pentagon ABCDE inscribed in circle O, with diagonals
ACand AD
Show: AC trisect BAE
and AD
A
3. Given: Two circles externally tangent at R, common external tangent
SRT
segment TS.
Show: TRS is a right angle
EDCO
B
4. Given:
Two circles internally tangent at T with chords TD
and TBof the larger circle intersecting the smaller
circle at C and A
Show: AC BD
ACDBT
Discovering Geometry Practice Your Skills CHAPTER 13 89
©2008 Key Curriculum Press
Lesson 13.7 • Similarity Proofs
Name
Period Date
Write a proof for each situation. You may use theorems added to your
theorem list through the end of Lesson 13.7.
1. Given: ABC with A BCD
Show: BC2 AB BD
A
2.
The diagonals of a trapezoid divide each other into segments with
lengths in the same ratio as the lengths of the bases.
3.
In a right triangle the product of the lengths of the two legs equals the
product of the lengths of the hypotenuse and the altitude to the
hypotenuse.
4.
If a quadrilateral has one pair of opposite right angles and one pair of
opposite congruent sides, then the quadrilateral is a rectangle.
CBD
90 CHAPTER 13 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
ANSWERS ANSWERS
LESSON 1.1 • Building Blocks of Geometry
1. S 2. 9 cm 3. SN 4. endpoint
5. NS 6. PQ 7. SP
LM LO
8. KN KL,NM,NO
9. E(14, 15)
10.
C
A 3.2 cm B
M
2.0 cm
D
11.
1.5 cm
1.5 cm
3 cm3 cmA
B
C
ED
12. AB,AC,AD,AF,BDB
,AE,BC, C
BE,BF,CD,CF,DF
,CE,DE,
EF
(15 lines)
A
D
FE
13. Possible coplanar set: {C, D, H, G}; 12 different sets
LESSON 1.2 • Poolroom Math
1. vertex 2. bisector 3. side
4. 126° 5. DAE 6. 133°
7. 47° 8. 63° 9. 70°
M
10. N
15°
O
12. 160°
11. R
z
90°
G
13. 90° 14. 120° 15. 75°
16. B
A
C
D
17. R
T
Q
Z
P
LESSON 1.3 • What’s a Widget?
1. d 2. c 3. e 4. i
5. f 6. b 7. h 8. a
9. g
10. They have the same measure, 13°. Because
mQ 77°, its complement has measure 13°.
So mR 13°, which is the same as mP.
11.
1
12. R
P
2
Q
3
13.
C
B
X
A
Y
LESSON 1.4 • Polygons
Polygon
name
Number of
sides
Number of
diagonals
1. Triangle 3 0
2. Quadrilateral 4 2
3. Pentagon 5 5
4. Hexagon 6 9
5. Heptagon 7 14
6. Octagon 8 20
7. Decagon 10 35
8. Dodecagon 12 54
9. E 10. Q
N
D
U
P
A
T
A
Discovering Geometry Practice Your Skills ANSWERS 91
©2008 Key Curriculum Press
11. AC,AD,AE For Exercises 6–10, 12, and 13, answers may vary.
Possible answers are shown.
12. Possible answer: ABand BC
6.
13. Possible answer: A and B
14. Possible answer: ACand FD
b
15. 82° 16. 7.2 17. 61° 18. 16.1
G
19. 6.2 cm
a b
7. ACFD 8. EFHG 9. BFJD
10. BFHD
LESSON 1.5 • Triangles
11. D(0, 3) 12. E(0, 5) 13. G(16, 3)
For Exercises 1–7, answers will vary. Possible answers
are shown.
LESSON 1.7 • Circles
1. AB.
BI
GH2. EF
1. 48° 2. 132° 3. 228° 4. 312°
3. CG FH 4. DEG and GEF
5.
6.
5. DEG and GEF
6.
T 7. C
M
L
I
R
A
S
N
7. (8, 2); (3, 7); (3, 3)
8. A
9. R
40°
8.
A
OCD
B
H
G
C
D
For Exercises 10–12, answers may vary. Possible answers
are shown.
R A
N
a b
b
10. F(8, 2)
9. The chord goes through the center, P. (It is
11. D(4, 3)
a diameter.)
12. G(10, 2)
LESSON 1.6 • Special Quadrilaterals
Q
1. PG 2. S
P
10.
180°
AR
50°
E
R
3. R
4. TR
290°
M
H
O
P
A
E
5. E
T K
I
92 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
11. Kite 4. Possible answers:
P
B
A
Q
a. b. c.
LESSON 1.8 • Space Geometry
1.
2.
3. Rectangular prism 4. Pentagonal prism
5.
6.
7. 18 cubes 8. x 2, y 1
LESSON 1.9 • A Picture Is Worth a Thousand Words
1. Possible locations
Gas
12 m
10 m
5
5
A
Power
2. Dora, Ellen, Charles, Anica, Fred, Bruce
D
1
1
43
3
ECAF B
3.
Triangles
Acute triangles
Isosceles triangles Scalene triangles
LESSON 2.1 • Inductive Reasoning
1. 20, 24
4. 1, 1
7. 91, 140
8.
2. 121
2,61
4
5. 72, 60
3. 5
4,2
6. 243, 729
9.
10. yy
(3, 1)
x
x
(1, –3)
11
11. False; 2
24
12. False; 11 10 110, 11 12 132
13. True
LESSON 2.2 • Finding the nth Term
1. Linear 2. Linear 3. Not linear
4. Linear
5.
6.
7. f(n) 4n 5; f(50) 205
8. f(n) 5n 11; f(50) 239
9. f(n) 1
2n 6; f(50) 31
n 1 2 3 4 5
g (n) 10 18 26 34 42
n 1 2 3 4 5
f (n) 5 2 9 16 23
Discovering Geometry Practice Your Skills ANSWERS 93
©2008 Key Curriculum Press
10.
n 123 4 5 . . . n ... 50
Number
of triangles
159 13 17 ... 4n 3 ... 197
11.
n 1 2 3 4 5 ... n ... 50
Area
of figure 1416 64 256 ... 4n1 ... 449
LESSON 2.3 • Mathematical Modeling
1.
a. 240
b. 1350
c. f(n) 2n(n 2), or f(n) 2n2 4n
2.
H
HHH
H
T
HHT
H
H
HTH
T
T
HTT
H
THH
H
T
THT
T
H
TTH
T
T
TTT
a. 8 sequences
b.
3 sequences have 1 tail.
3
c.
8
3. 66 different pairs. Use a dodecagon showing sides
and diagonals.
4. Answers will vary. Possible answers:
b
12
2c
34516
10 8
97b
a
3c
7
e
4
d5
d
a
6
ef
5 teams, 10 games 6 teams, 7 games
a
b
1
f
6
4 3e
5
2
c
d
6 teams, 6 games
LESSON 2.4 • Deductive Reasoning
1. No. Explanations will vary. Sample explanation:
Because ABC is equilateral, AB BC. Because C
lies between B and D, BD BC, so BD is not equal
to AB. Thus ABD is not equilateral, by deductive
reasoning.
2. Answers will vary. mE mD (mE
mD 90°); deductive
3. a, e, f; inductive
4. Deductive
a. 4x 3(2 x) 8 2x The original equation.
4x 6 3x 8 2x Distributive property.
x 6 8 2x Combining like terms.
3x 6 8 Addition property of
equality.
3x 2 Subtraction property of
equality.
x 2
3 Division property of
equality.
94 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
19 2(3x 1)
b. x 2 The original equation.
5
19 2(3x 1) 5(x 2) Multiplication property
of equality.
19 6x 2 5x 10 Distributive property.
21 6x 5x 10 Combining like terms.
21 11x 10 Addition property of
equality.
11 11x Subtraction property of
equality.
1 x Division property of
equality.
5. a. 16, 21; inductive
b. f(n) 5n 9; 241; deductive
LESSON 2.5 • Angle Relationships
1. a 68°, b 112°, c 68°
2. a 127°
3. a 35°, b 40°, c 35°, d 70°
4. a 90°, b 90°, c 42°, d 48°, e 132°
5. a 20°, b 70°, c 20°, d 70°, e 110°
6. a 70°, b 55°, c 25°
7. Sometimes 8. Always 9. Never
10. Sometimes 11. acute 12. 158°
13. 90° 14. obtuse 15. converse
LESSON 2.6 • Special Angles on Parallel Lines
1. a 54°, b 54°, c 54°
2. a 115°, b 65°, c 115°, d 65°
3. a 72°, b 126° 4. 1 2
5. 1 .
2 6. cannot be determined
7. a 102°, b 78°, c 58°, d 122°, e 26°,
f 58°
8. x 80°
9. x 20°, y 25°
LESSON 3.1 • Duplicating Segments and Angles
1.
P
Q
R S
A
B
2. XY 3PQ 2RS
X
Y
3. Possible answer: 4. B.
128° 35° 93°
B
DC
CD
5.
6.
C
D
D
LESSON 3.2 • Constructing Perpendicular Bisectors
1.
2. Square
Discovering Geometry Practice Your Skills ANSWERS 95
©2008 Key Curriculum Press
5
3. XY AB
4
M
B
A
A
M
B
B
X
WY
4. ABC is not unique.
A
C
5. ABC is not unique.
C
A
QC
P
B
A.
B
6. BD AD CD
C
D
A
B
7. a. A and B
b. A, B, and C
c.
A and B and from C and D (but not from
B and C)
d. A and B and from D and E
LESSON 3.3 • Constructing Perpendiculars to a Line
1. False. The altitude from A coincides with the side so
it is not shorter.
A
C
B
2. False. In an isosceles triangle, an altitude and
median coincide so they are of equal length.
3. True
4. False. In an acute triangle, all altitudes are inside. In
a right triangle, one altitude is inside and two are
sides. In an obtuse triangle, one altitude is inside
and two are outside. There is no other possibility so
exactly one altitude is never outside.
5. False. In an obtuse triangle, the intersection of the
perpendicular bisectors is outside the triangle.
6.
7.
S R
P Q
8.
9. WX YZ
P
Q
W
Y
X
Z
96 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
LESSON 3.4 • Constructing Angle Bisectors 5. 5.
O
1. a. 1 and 2 M
b. 1, 2, and 3
c. 2, 3, and 4
R
H
d. 1 and 2 and from 3 and 4
6. Possible answer:
2. AP is the bisector of CAB
3. x 20°, mABE 50°
O
B
I
4.
Z
AD
LESSON 3.6 • Construction Problems
5. They are concurrent.
1. Possible answer:
E
T
6.
I
K
Perimeter
7. RN GN and RO HO
H
O M
N
R
G
A
U
LESSON 3.5 • Constructing Parallel Lines
S
Q
1.
P
2.
3. 4.
P
2. Possible answer:
P
S R
Q
A R
P T
m
Discovering Geometry Practice Your Skills ANSWERS 97
©2008 Key Curriculum Press
3. Possible answers:
LESSON 3.7 • Constructing Points of Concurrency
R E
CT
R
E
C
1. Circumcenter
2. Locate the power-generation plant at the incenter.
Locate each transformer at the foot of the perpendi-
T
cular from the incenter to each side.
3.
4. Possible answer:
T
E
O B
S
U
5. Possible answer:
B
4. Possible answer: In the equilateral triangle, the
centers of the inscribed and circumscribed circles
A
C.
are the same. In the obtuse triangle, one center
is outside the triangle.
6. Possible answer:
S1
S1
S2
A
A
S2
7. Possible answer:
S1
B
B
AS1
A
5. Possible answer: In an acute triangle, the circumcenter
is inside the triangle. In a right triangle, it
is on the hypotenuse. In an obtuse triangle, the
circumcenter is outside the triangle. (Constructions
not shown.)
90° B
B
8.
S2
S3
S1
98 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
LESSON 3.8 • The Centroid
1.
2.
G
C
3. CP 3.3 cm, CQ 5.7 cm, CR 4.8 cm
R
C
10 cm
6 cm
Q
P
8 cm
4. (3, 4)
5. PC 16, CL 8, QM 15, CR 14
6. a. Incenter b. Centroid
c. Circumcenter d. Circumcenter
e. Orthocenter f. Incenter
g. Centroid
LESSON 4.1 • Triangle Sum Conjecture
1. p 67°, q 15° 2. x 82°, y 81°
3. a 78°, b 29°
4. r 40°, s 40°, t 100°
5. x 31°, y 64° 6. y 145°
7. s 28° 8. m 721
2°
9. mP a 10. mQPT 135°
11. 720°
12. The sum of the measures of A and B is 90°
because mC is 90° and all three angles must be
180°. So, A and B are complementary.
13. mBEA mCED because they are vertical
angles. Because the measures of all three angles in
each triangle add to 180°, if equal measures are
subtracted from each, what remains will be equal.
LESSON 4.2 • Properties of Isosceles Triangles
1. mT 64° 2. mG 45°
3. x 125°
4. mA 39°, perimeter of ABC 46 cm
5. LM 163 m, mM 50°
6. mQ 44°, QR 125
7. a. DAB ABD BDC BCD
b. ADB CBD
BC
c. ADby the Converse of the AIA Conjecture.
8. x 21°, y 16° 9. mQPR 15°
10. mPRQ 55° by VA, which makes mP 55° by
the Triangle Sum Conjecture. So, PQR is isosceles
by the Converse of the Isosceles Triangle Conjecture.
LESSON 4.3 • Triangle Inequalities
1. Yes
2. No
17 km 9 km
28 km
3. 19 x 53 4. b a c
5. b c a 6. a c d b
7. x 76° 8. x 79°
9. The interior angle at A is 60°. The interior angle at
B is 20°. But now the sum of the measures of the
triangle is not 180°.
10. By the Exterior Angles Conjecture,
2x x mPQS.So, mPQS x. So, by the
Converse of the Isosceles Triangle Conjecture,
PQS is isosceles.
11. Not possible. AB BC AC
12. P
Q
R
LESSON 4.4 • Are There Congruence Shortcuts?
1. SAA or ASA 2. SSS 3. SSS
4. BQM (SAS) 5. TIE (SSS)
Discovering Geometry Practice Your Skills ANSWERS 99
©2008 Key Curriculum Press
6. Cannot be determined, as shown by the figure.
Y
AC
Z
B
X
7. TNO (SAS)
8. Cannot be determined, as shown by the figure.
Q
U
R
T
S
9. DOG (SAS)
10. Only one triangle because of SSS.
T
U
S
11. Two possible triangles.
A
A
B
B
CC
12. Only one triangle because of SAS.
Z
XY
LESSON 4.5 • Are There Other Congruence Shortcuts?
1. Cannot be determined
2. XZY (SAA) 3. ACB (ASA or SAA)
4. PRS (ASA) 5. NRA (SAA)
6. GQK (ASA or SAA)
7. Yes, QRS MOL by SSS.
8. No, corresponding sides TV are not
and WV
congruent.
9. All triangles will be congruent by ASA. Possible
triangle:
R
PQ
10. All triangles will be congruent by SAA. Possible
procedure: Use A and C to construct B and
then copy A and B at the ends of AB.
C
A
B
LESSON 4.6 • Corresponding Parts of Congruent
Triangles
1. SSS, SAS, ASA, SAA
2. YZ WX, AIA Conjecture
XY
3. WZ, AIA Conjecture
4. ASA 5. CPCTC
ZX
6. YWM ZXM by SAS. YWby CPCTC.
BD
7. ACD BCD by SAS. ADby CPCTC.
8. Possible answer: DE and CF are both the distance
between DC. Because the lines are parallel,
and AB
the distances are equal. So, DE CF.
9. Possible answer: DE CF(see Exercise 8).
DEF CFE because both are right angles,
EF FEbecause they are the same segment. So,
DEF CFE by SAS. EC FDby CPCTC.
10. Possible answer: It is given that TP RA and
PTR ART, and TR RTbecause they
are the same segment. So PTR ART
by SAS and TA RPby CPCTC.
LESSON 4.7 • Flowchart Thinking
1. (See flowchart proof at bottom of page 101.)
2. (See flowchart proof at bottom of page 101.)
3. (See flowchart proof at bottom of page 101.)
LESSON 4.8 • Proving Special Triangle Conjectures
1. AD 8 2. mACD 36°
3. mB 52°, CB 13 4. mE 60°
5. AN 17 6. Perimeter ABCD 104
100 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
7. (See flowchart proof at bottom of page 102.) 3. 170°; 36 sides 4. 15 sides
8. Flowchart Proof 5. x 105° 6. x 18°
CD is a median 7. mE 150°
Given C
AC BC AD BD CD CD
150°85°
125°
110°
70°
D
B
Given Definition of Same segment
median
A
E
ADC BDC
SSS Conjecture
ACD BCD
CPCTC
CD bisects
ACB
Definition of
bisect
LESSON 5.1 • Polygon Sum Conjecture
1. a 103°, b 103°, c 97°, d 83°, e 154°
2. a 92°, b 44°, c 51°, d 85°, e 44°, f 136°
Lesson 4.7, Exercises 1, 2, 3
1.
PQ SR
Given
LESSON 5.2 • Exterior Angles of a Polygon
1. 12 sides 2. 24 sides 3. 4 sides 4. 6 sides
2
5. a 64°, b 1383° 6. a 102°, b 9°
7. a 156°, b 132°, c 108°
8. a 135°, b 40°, c 105°, d 135°
9.
PQS RSQ PQS RSQPQ SR
SP QR
Given AIA Conjecture
SAS Conjecture CPCTC
QS QS
Same segment
2.
KE KI
Given
TE TIKITE is a kite
ETK ITK
CPCTC
KT bisects EKI
KET KIT
and ETI
Given Definition of SSS Conjecture
EKT IKT
Definition of
kite
bisect
CPCTC
KT KT
Same segment
3.
ABD CDB
AB CD AIA Conjecture
Definition of
parallelogram
ABCD is a parallelogram
BDA DBCBD DB
A C
Given
Same segment
ASA Conjecture CPCTC
AD CB
Definition of
ADB CBD
parallelogram
AIA Conjecture
Discovering Geometry Practice Your Skills ANSWERS 101
©2008 Key Curriculum Press
LESSON 5.3 • Kite and Trapezoid Properties
1. x 30 2. x 124°, y 56°
3. x 64°, y 43° 4. x 12°, y 49°
5. PS 33 6. a 11
7.
8.
D
B
D C
BA
AC
9. Possible answer:
Paragraph proof: Draw AE PTwith E on TR.
TEAP is a parallelogram. T AER because
they are corresponding angles of parallel lines.
T R because it is given, so AER R,
because both are congruent to T. Therefore,
AER is isosceles by the Converse of the Isosceles
Triangle Conjecture. TP EAbecause they are
opposite sides of a parallelogram and AR EA
because AER is isosceles. Therefore, TP RA
because both are congruent to EA.
LESSON 5.4 • Properties of Midsegments
1. a 89°, b 54°, c 91°
2. x 21, y 7, z 32
3. x 17, y 11, z 6.5
4. Perimeter XYZ 66, PQ 37, ZX 27.5
5. M(12, 6), N(14.5, 2); slope AB1.6,
slope MN
1.6
6. Pick a point P from which A and B can be viewed
over land. Measure AP and BP and find the
midpoints M and N. AB 2MN.
Lesson 4.8, Exercise 7
CD CD
Same segment
CD is an altitude
ADC and BDC
are right angles
A
M N
B
P
7. AMNO is a parallelogram. By the Triangle Mid-
segment Conjecture, ON and MN
AM AO.
Flowchart Proof
OC 1AC_
2
MN 1AC_
2
Definition of Midsegment
midpoint Conjecture
SAS Conjecture
ONC MBN
CA Conjecture
NMB A
CA Conjecture
CON A
Both congruent to A
CON NMB
Definition of
midpoint
MB AB1_
2
Midsegment
Conjecture
ON AB1_
2
Both congruent to AC
OC MN
1_
2
Both congruent to AB1_
2
ON MB
8. Paragraph proof: Looking at FGR, HI FGby
the Triangle Midsegment Conjecture. Looking at
PQR, FG PQfor the same reason. Because
FG PQ, quadrilateral FGQP is a trapezoid and
DEis the midsegment, so it is parallel to FG
and PQ. Therefore, HI FG DE PQ.
CD is a median
AD BDADC BDC ADC BDC
Given
Both are right angles SAA Conjecture CPCTC Definition of
Definition of altitude
median
AC BC
A B
Given Converse of IT
102 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
LESSON 5.5 • Properties of Parallelograms
1. Perimeter ABCD 82 cm
2. AC 22, BD 14
3. AB 16, BC 7
4. a 51°, b 48°, c 70°
5. AB 35.5
6. a 41°, b 86°, c 53°
7. AD 75
8.
Resultant
F2
vector
F1 F2
F1
9. a 38°, b 142°, c 142°, d 38°, e 142°,
f 38°, g 52°, h 12°, i 61°, j 81°, k 61°
10. No
A
B
D
C
D
A
C
B
4. c, d, f, g 5. d, e, g
6. f, g 7. h
8. c, d, f, g 9. c, d, f, g
10. d, g 11. d, g
12. None 13. Parallelogram
14.
D
A
B
C
LESSON 5.7 • Proving Quadrilateral Properties
1. (See flowchart proof at bottom of page.)
2. Flowchart Proof
AB CB
Given
AD CD A CABD CBD
Given
SSS Conjecture CPCTC
DB DB
Same segment
3. Flowchart Proof
LESSON 5.6 • Properties of Special Parallelograms
1. OQ 16, mQRS 90°, PR 32
2. mOKL 45°, mMOL 90°,
perimeter KLMN 32
3. OB 6, BC 11, mAOD 90°
Lesson 5.7, Exercise 1
APR CQR
AO CO
Diagonals of parallelogram
bisect each other
AB CB CD AD BO DO
Definition of rhombus Diagonals of a parallelogram
bisect each other
ABO CBO CDO ADO
SSS Conjecture
AIA Conjecture
AR CR
CPCTC
AC and QP
DC AB
APR CQR
bisect each other
Given
AP CQ
Given ASA Conjecture
Definition of bisect
PR QR
CPCTC
PAR QCR
AIA Conjecture
Discovering Geometry Practice Your Skills ANSWERS 103
©2008 Key Curriculum Press
4. Flowchart Proof
AXD CYB
SAA Conjecture
DX BY
CPCTC
DAX BCY
AIA Conjecture
AD CB
Opposite sides of
parallelogram
AXD CYB
Both are 90°
AD BC
Definition of parallelogram
LESSON 6.1 • Tangent Properties
1. w 126°
2. mBQX 65°
3. a. mNQP 90°, mMPQ 90°
b. Trapezoid. Possible explanation: MP
and NQare both perpendicular to PQ, so they are
parallel to each other. The distance from M to
PQis MP, and the distance from N to PQis
NQ. But the two circles are not congruent, so
MP NQ. Therefore, MN
is not a constant
distance from PQand they are not parallel.
Exactly one pair of sides is parallel, so MNQP
is a trapezoid.
1
4. y x 10
3
5. Possible answer: Tangent segments from a point to a
circle are congruent. So, PA PB,PB PC, and
PC PD. Therefore, PA PD.
6. a. 4.85 cm
b. 11.55 cm
7.
A
T
P
LESSON 6.2 • Chord Properties
1. a 95°, b 85°, c 47.5°
2. v cannot be determined, w 90°
104 ANSWERS
3. z 45°
4. w 100°, x 50°, y 110°
5. w 49°, x 122.5°, y 65.5°
6. x 16 cm, y cannot be determined
ONcongruent chords ABand ACare the same distance
from the center. AM because they are halves
7. Kite. Possible explanation: OM because
ANof congruent chords. So, AMON has two pairs of
adjacent congruent sides and is a kite.
8. The perpendicular segment from the center of the
circle bisects the chord, so the chord has length
12 units. But the diameter of the circle is 12 units,
and the chord cannot be as long as the diameter
because it doesn’t pass through the center of the
circle.
9. P(0,1), M(4, 2)
10. mAB49°, mABC253°, mBAC156°,
mACB311°
11. Possible answer: Fold and crease to match the
endpoints of the arc. The crease is the perpendicular
bisector of the chord connecting the endpoints. Fold
and crease so that one endpoint falls on any other
point on the arc. The crease is the perpendicular
bisector of the chord between the two matching
points. The center is the intersection of the two
creases.
Center
LESSON 6.3 • Arcs and Angles
1. mXNM 40°, mXN180°, mMN100°
2. x 120°, y 60°, z 120°
3. a 90°, b 55°, c 35°
4. a 50°, b 60°, c 70°
5. x 140°
6. mA 90°, mAB72°, mC 36°, mCB108°
7. mAD140°, mD 30°, mAB60°,
mDAB200°
8. p 128°, q 87°, r 58°, s 87°
9. a 50°, b 50°, c 80°, d 50°, e 130°,
f 90°, g 50°, h 50°, j 90°, k 40°,
m 80°, n 50°
Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
LESSON 6.4 • Proving Circle Conjectures
4. Flowchart Proof
Construct radii AO,OB, and OD
,OC.
1. Flowchart Proof
AB CD
Given
XC XD
XA XD
Tangent Segments
Conjecture
XC XA
Tangent Segments
Conjecture
AOB COD
Definition of arc
measure
Transitivity
AO CO
BO DO
AB bisects CD at X
Radii of same circle
Radii of same circle
Definition of segment
bisector
AOB COD
2. Angles are numbered for
reference.
SAS Conjecture
A
O
1
234
B
E
D
Paragraph Proof
It is given that OE AD
, so 2 1 by the CA
Conjecture. Because OAand OD
are radii, they are
congruent, so AOD is isosceles. Therefore
4 1 by the IT Conjecture. Both 2 and
4 are congruent to 1, so 2 4. By the AIA
Conjecture, 4 3, so 2 3. The measure
of an arc equals the measure of its central angle, so
because their central angles are congruent,
DE BE .
3. Flowchart Proof
PX RX
Tangent Segments
Conjecture
XQ XS
Tangent Segments
Conjecture
PX XQ RX XS
AB CD
CPCTC
LESSON 6.5 • The Circumference/Diameter Ratio
1. C 21 cm 2. r 12.5 cm
3. C 60 cm 4. d 24 cm
5. C 30.2 cm
6. d 42.0 cm, r 21.0 cm
7. C 37.7 in.
8. Yes; about 2.0 in.
5 in.
2 in.
Addition Property of
9. C 75.4 cm
Equality
10. Press the square against the tree as shown. Measure
PX XQ PQ RX XS RS
the tangent segment on the square. The tangent
Segment addition
Segment addition
segment is the same length as the radius. Use
C 2r to find the circumference.
PQ RS
PQ RS
Definition of
congruent segments
Tree
11. 4 cm
Discovering Geometry Practice Your Skills ANSWERS 105
©2008 Key Curriculum Press
Transitivity
LESSON 6.6 • Around the World
1. At least 7 olive pieces
2. About 2.5 rotations
(2 4.23 107)
3. 3085 m/s (about 3 km/s or just
(60 60 23.93)
under 2 mi/s)
4. 6.05 cm or 9.23 cm
(2 1.4957 1011103)
5. Sitting speed
(364.25 24)
107,500 km/h
LESSON 6.7 • Arc Length
1. 4 2. 4
3. 30 4. 35
9
5. 80
9
6. 6.25 or 25
4
7. 10
9
0
8. 31.5
9. 22 10. 396
EXPLORATION • Intersection Secants, Tangents, and
Chords
1. x 21°
2. mDC 70°, mED 150°
3. mDC 114°, mDEC 66°
4. mBCE 75°, mBAC 210°
5. x 80°, y 110°, z 141°
6. x 34°, y 150°, z 122°
7. x 112°, y 68°, z 53°
8. x 28°, y 34.5°
LESSON 7.1 • Transformations and Symmetry
1.
I
I
T
T
RR
2.
L
R
A
PR
P
QA
3.
E
T.
A
N
TP
AN
PE
4. Possible answers: The two points where the figure
and the image intersect determine . Or connect
any two corresponding points and construct the
perpendicular bisector, which is .
B
BDD
E
C
C
E
A
5. 3-fold rotational symmetry, 3 lines of reflection
6. 2-fold rotational symmetry
7. 1 line of reflection
8. 1 line of reflection
9. 2-fold rotational symmetry, 2 lines of reflection
C
10. 2-fold rotational symmetry
C
11. 1 line of reflection
12. 4-fold rotational symmetry, 4 lines of reflection
C
106 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
LESSON 7.2 • Properties of Isometries
1. Rotation
y
(2, 3)
(5, 1)
(2, 1)
x
(–2, –1)
(–5, –1)
(–2, –3)
2. Translation
y
(–2, 6)
(–5, 4) (–2, 4)
(2, 0)
x
(–1, –2)
(2, –2)
3. Reflection
y
(4, 5)
(0, 5)
x
(–2, –2)
(6, –2)
(0, –2) (4, –2)
4.
S
Fence
S
SS
F
5.
SS
S
F
Fence 1
Fence 2
6. (x, y) . (x 13, y 6); translation; B(8, 8),
C(8, 4)
7. (x, y) . (x, y); reflection across the y-axis;
P(7, 3), R(4, 5)
8. (x, y) . (y, x); reflection across the line y x;
T (7, 0), R(0, 3)
LESSON 7.3 • Compositions of Transformations
1. Translation by 2, 5
2. Rotation 45° counterclockwise
3. Translation by 16, 0
4. Rotation 180° about the intersection of the two lines
5. Translation by 16, 0
6. Rotation 180° about the intersection of the two lines
7. Reflection across the line x 3
8. Reflection across the line x 3
9. mROT 50°; rotation 100° clockwise about O
P.
R
P
O
T
P
10. Translation 2 cm along the line perpendicular to
k and in the direction from k to .
P
1 cm
P
P
k
11.
Discovering Geometry Practice Your Skills ANSWERS 107
©2008 Key Curriculum Press
LESSON 7.4 • Tessellations with Regular Polygons 4. Sample answer: 4. Sample answer:
1. n 15 2. n 20
3. Possible answer: A regular tessellation is a
tessellation in which the tiles are congruent
regular polygons whose edges exactly match.
4. Possible answer: A 1-uniform tiling is a tessellation
in which all vertices are identical.
5. 3.42.63.6.3.6
6.
LESSONS 7.5–7.8 • Tessellations
1.
2. Sample answer:
3. Sample answer:
LESSON 8.1 • Areas of Rectangles and Parallelograms
1. 112 cm2 2. 7.5 cm2 3. 110 cm2 4. 81 cm2
5. 61 m
6. No. Possible answer:
2.5 cm40
34 cm
40
17 cm
5 cm
7. 88 units2 8. 72 units2
9. No. Carpet area is 20 yd2 180 ft2. Room area
is (21.5 ft)(16.5 ft) 206.25 ft2. Dana will be
1
26 ft2 short.
4
LESSON 8.2 • Areas of Triangles, Trapezoids, and Kites
1. 16 ft 2. 20 cm2
3. b 12 in. 4. AD 4.8 cm
5. 40 cm2 6. 88 cm2 7. 54 units2 8. 135 cm2
LESSON 8.3 • Area Problems
1. a. 549.5 ft2 b. 40 bundles; $1596.00
2. 500 L
3. Possible answer:
8
2
2
2
35
3 2
3 2 3
2
2
2
53
108 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
4. It is too late to change the area. The length of the 6. 10 cm2 31.42 cm2
diagonals determines the area.
7. r 10 cm 8. x 135° 9. r 7 cm
LESSON 8.4 • Areas of Regular Polygons
LESSON 8.7 • Surface Area
1. A 696 cm2 2. a 7.8 cm
3. p 43.6 cm 4. n 10
5. s 4 cm, a 2.8 cm, A 28 cm2
1. 136 cm2
4. 796.4 cm2
7. 468 cm2
2. 240 cm2
5. 255.6 cm2
8. 1055.6 cm2
3. 558.1 cm2
6. 356 cm2
11
22
; bottom rectangle: 3 2 6;
9. 1 sheet: front rectangle: 3 1
4
; back rectangle:
11
22
side trapezoids:22
3 2
7
221
121
8; total 26 ft2
2
Area of 1 sheet 4 8 32 ft2. Possible pattern:
.
6. Possible answer (s will vary): s
3.1 cm, a
3.7 cm, 1
_
1ft
A
45.9 cm2 2
2 ft
3 ft
1ft1_
2
2ft1_
2
1ft1_
2
2ft1_
2Back
Bottom
Left over
Front Side
Side
2 ft
3 ft
3 ft
LESSON 9.1 • The Theorem of Pythagoras
1. a 21 cm 2. p
23.9 cm
7. Approximately 31.5 cm2: area of square 36; area 3. x 8 ft 4. h
14.3 in.
2 ft
1
_
2ft
2
of square within angle
8
3
36 13.5; area of
5. Area
19.0 ft2 6. C(11, 1); r 5
octagon 120; area of octagon within angle
3
8
120 45; shaded area 45 13.5 31.5 cm2 7. Area
49.7 cm2 8. RV
15.4 cm
LESSON 8.5 • Areas of Circles
1. 81 cm2 2. 10.24 cm2
4. 324 cm2 5. 191.13 cm2
7. 7.65 cm2 8. 4.90 cm2
25
10. 33.5 or 33.6 cm2
11. (64 128) square units
12. 25 cm2
LESSON 8.6 • Any Way You Slice It
12
9. If the base area is 16 cm2, then the radius is 4 cm.
The radius is a leg of the right triangle; the slant
height is the hypotenuse. The leg cannot be longer
than the hypotenuse.
10. Area 150 in2; hypotenuse QR 25 in.; altitude to
the hypotenuse 12 in.
LESSON 9.2 • The Converse of the Pythagorean Theorem
1. No 2. Yes 3. Yes 4. Yes
5. Area
21.22 cm2
6. The top triangle is equilateral, so half its side length
is 2.5. A triangle with sides 2.5, 6, and 6.5 is a right
cm2 6.54 cm2
triangle because 2.52 62 6.52. So, the angle
3. 23 cm
6. 41.41 cm
9. 51.3 cm2
1.
7. x
44.45 cm. By the Converse of the Pythagorean
3. 12 cm2 37.70 cm2
Theorem, ADC is a right triangle, and ADC is a
4. (16 32) cm2 18.27 cm2 right angle. ADC and BDC are supplementary,
so BDC is also a right triangle. Use the
5. 13.5 cm2 42.41 cm2
Pythagorean Theorem to find x.
marked 95° should be 90°.
cm2 33.51 cm2
32
2.
3
Discovering Geometry Practice Your Skills ANSWERS 109
©2008 Key Curriculum Press
8. 129.6 cm2
22
6
4.8
8
6
32 10
9. No. Because AB 2 BC2 AC 2, B of ABC is
not a right angle.
10. Cannot be determined. The length of CD is
unknown. One possible quadrilateral is shown.
A
D
B
C
11. Yes. Using SSS, ABC BAD CDA
DCB. That means that the four angles of the
quadrilateral are all congruent by CPCTC. Because
the four angles must sum to 360° and they are all
congruent, they must be right angles. So, ABCD is
a rectangle.
LESSON 9.3 • Two Special Right Triangles
1. a 142 cm
2. a 12 cm, b 24 cm
3. a 12 cm, b 63
cm
4. 643 cm2
5. Perimeter 32 62 63
cm;
area 60 183
cm2
6. AC 302 cm;
cm; AB 30 303area 450 4503
cm2
7. 453 cm2
1 3
8. C,
22
9. C(63, 6)
10. Possible answer:
2
18 2
3
3
LESSON 9.4 • Story Problems
1. The foot is about 8.7 ft away from the base of the
building. To lower it by 2 ft, move the foot an
additional 3.3 ft away from the base of the building.
2. About 6.4 km
4.1 km
0.4 km
1.2 km
0.9 km
2.3 km
1.7 km
3.1 km
3. 149.5 linear feet of trim must be painted, or
224.3 feet2. Two coats means 448.6 ft2 of coverage.
Just over 21 quarts of paint is needed. If Hans buys
2
1
3 quarts, he would have almost 2 quart left. It is
slightly cheaper to buy 1 gallon and have about
1
12 quarts left. The choice is one of money versus
conserving. Students may notice that the eaves
extend beyond the exterior walls of the house and
adjust their answer accordingly.
14 28
4. 14 in., in.
8.08 in., in.
16.17 in.
3
3
14
30
LESSON 9.5 • Distance in Coordinate Geometry
1. 10 units 2. 20 units 3. 17 units
4. ABCD is a rhombus: All sides 34
,
33
slope AB, slope BC, so B is not
55
a right angle, and ABCD is not a square.
5. TUVW is an isosceles trapezoid: TU
and VWhave slope 1, so they are parallel. UV
and TWhave length 20
and are not parallel
1
(slope UV2, slope TW
2).
6. Isosceles; perimeter 32 units
4 4
MN 5; BC 10; the slopes are equal;
1
7. M(7, 10); N(10, 14); slope MN3; slope BC3;
MN BC.
2
8. (x 1)2 (y 5)2 4 9. Center (0, 2), r 5
10. The distances from the center to the three points
on the circle are not all the same: AP 61
,
BP 61
, CP 52
110 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
LESSON 9.6 • Circles and the Pythagorean Theorem
1. (25 24) cm2, or about 54.5 cm2
2. (723 24) cm2, or about 49.3 cm2
3. (5338
37) cm 36.1 cm
4. Area 56.57 cm 177.7 cm2
5. AD 115.04cm 10.7 cm
6. ST 937. 150°
15.6
LESSON 10.1 • The Geometry of Solids
1. oblique 2. the axis 3. the altitude
4. bases 5. a radius 6. right
7. Circle C 8. A 9. ACor AC
10. BCor BC 11. Right pentagonal prism
12. ABCDE and FGHIJ
13. AF,BG,DI
,CH,EJ
14. Any of AF,BG,DI or their lengths
,CH,EJ
15. False. The axis is not perpendicular to the base in
an oblique cylinder.
16. False. A rectangular prism has six faces. Four are
called lateral faces and two are called bases.
17. True
18.
19.
LESSON 10.2 • Volume of Prisms and Cylinders
1. 232.16 cm3 2. 144 cm3
3. 415.69 cm3
4. V 4xy(2x 3), or 8x2y 12xy
11
5. V 4p2h 6. V 6 2x2y
7. 6 ft3
LESSON 10.3 • Volume of Pyramids and Cones
1. 80 cm3 2. 209.14 cm3 3. 615.75 cm3
8
4. V 840x3 5. V 3a2b 6. V 4xy2
7. A: 128 cubic units, B: 144 cubic units.
B is larger.
8. A: 5 cubic units, B: 5 cubic units.
They have equal volumes.
9. A: 9x cubic units, B: 27x cubic units. B is larger.
LESSON 10.4 • Volume Problems
1. 4.4 cm 2. 1728 in3
3. 24 cans; 3582 in3 2.07 ft3; 34.6%
4. 2000.6 lb (about 1 ton)
8
5. Note that AE ABand EC BC. V 3 cm3;
SA (8 42
) cm2 13.7 cm2
6. About 110,447 gallons
7. 57 truckloads
LESSON 10.5 • Displacement and Density
All answers are approximate.
1. 53.0 cm3 2. 7.83 g/cm3
3. 0.54 g/cm3 4. 4.94 in.
5. No, it’s not gold (or at least not pure gold). The
mass of the nugget is 165 g, and the volume is
17.67 cm3, so the density is 9.34 g/cm3. Pure gold
has density 19.3 g/cm3.
LESSON 10.6 • Volume of a Sphere
1. 288 cm3, or about 904.8 cm3
2. 18 cm3, or about 56.5 cm3
3. 72 cm3, or about 226.2 cm3
28
4. cm3, or about 29.3 cm3
3
5. 432 cm3, or about 1357.2 cm3
304
6. cm3, or about 318.3 cm3
3
7. 11 cm
8. 2250 in3 7068.6 in3
9. 823.2 in3; 47.6% 10. 17.86
LESSON 10.7 • Surface Area of a Sphere
1. V 1563.5 cm3; S 651.4 cm2
2. V 184.3 cm3; S 163.4 cm2
Discovering Geometry Practice Your Skills ANSWERS 111
©2008 Key Curriculum Press
3. V 890.1 cm3; S 486.9 cm2
4. V 34.1 cm3; S 61.1 cm2
5. About 3.9 cm
6. About 357.3 cm2
7. 9 quarts
LESSON 11.1 • Similar Polygons
1. AP 8 cm; EI 7 cm; SN 15 cm; YR 12 cm
2. SL 5.2 cm; MI 10 cm; mD 120°;
mU 85°; mA 80°
3. Yes. All corresponding angles are congruent. Both
figures are parallelograms, so opposite sides within
each parallelogram are equal. The corresponding
6. CA 64 cm
7. ABC
EDC. Possible explanation: A E
and B D by AIA, so by the AA Similarity
Conjecture, the triangles are similar.
8. PQR
STR. Possible explanation: P S
and Q T because each pair is inscribed in the
same arc, so by the AA Similarity Conjecture, the
triangles are similar.
9. MLK
NOK. Possible explanation:
MLK NOK by CA and K K because
they are the same angle, so by the AA Similarity
Conjecture, the two triangles are similar.
LESSON 11.3 • Indirect Measurement with
Similar Triangles
159
53
4. Yes. Corresponding angles are congruent by the CA 4. About 18.5 ft
Conjecture. Corresponding sides are proportional
5. 0.6 m, 1.2 m, 1.8 m, 2.4 m, and 3.0 m
234
68
sides are proportional
.
1. 27 ft 2. 6510 ft 3. 110.2 mi
.
5. No.
4
=
=
68
1822
6. Yes. All angles are right angles, so corresponding 1. h 0.9 cm; j 4.0 cm
angles are congruent. The corresponding side
4 2. 3.75 cm, 4.50 cm, 5.60 cm
LESSON 11.4 • Corresponding Parts of Similar Triangles
.
lengths have the ratio
, so corresponding side
lengths are proportional.
7
5
3. WX 13
13.7 cm; AD 21 cm; DB 12 cm;
7
6
YZ 8 cm; XZ 6
6.9 cm
7
1
2
x
y
4
4
C (1.5, 1.5)
D(2, 0.5)A(0, 1)
B (2, 3)
7.
50 80
4. x 3.85 cm; y 6.15 cm
1313
5. a 8 cm; b 3.2 cm; c 2.8 cm
6. CB 24 cm; CD 5.25 cm; AD 8.75 cm
LESSON 11.5 • Proportions with Area
1. 5.4 cm2 2. 4 cm 3. 2
9
5 4. 3
1
6
5. 2
4
5 6. 16:25 7. 2:3 8. 8888
9 cm2
9. 1296 tiles
8. 4 to 1
LESSON 11.6 • Proportions with Volume
1. Yes 2. No 3. 16 cm3 4. 20 cm
5. 8:125 6. 6 ft2
LESSON 11.7 • Proportional Segments Between
x
y
5
5 D (2, 4)
E (8, 2)
F (4, 2)
LESSON 11.2 • Similar Triangles
1. MC 10.5 cm
2. Q X; QR 4.8 cm; QS 11.2 cm
3. A E; CD 13.5 cm; AB 10 cm
4. TS 15 cm; QP 51 cm
5. AA Similarity Conjecture
112 ANSWERS
Parallel Lines
1. x 12 cm 2. Yes
3. No 4. NE 31.25 cm
5. PR 6 cm; PQ 4 cm; RI 12 cm
6. a 9 cm; b 18 cm
Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
7. RS 22.5 cm, EB 20 cm
8. x 20 cm; y 7.2 cm
9. p 16
5.3 cm; q 8 2.6 cm
33
LESSON 12.1 • Trigonometric Ratios
pq
1. sin P 2. cos P
rr
pq
3. tan P 4. sin Q
qr
5. sin T 0.800 6. cos T 0.600
7. tan T
1.333 8. sin R 0.600
9. x
12.27 10. x
29.75
11. x
18.28 12. mA
71°
13. mB
53° 14. mC
30°
2
w
8; w
18.0 cm
x
14
15. sin 40°
; x
7.4 cm
16. sin 28°
7. About 24° 8. About 43.0 cm
9. About 34.7 in.
LESSON 12.5 • Problem Solving with Trigonometry
1. About 2.85 mi/h; about 15°
2. mA 50.64°, mB 59.70°, mC 69.66°
3. About 8.0 km from Tower 1, 5.1 km from Tower 2
4. About 853 miles
5. About 530 ft of fencing; about 11,656 ft2
LESSON 13.1 • The Premises of Geometry
1. a. Given
b. Distributive property
c. Subtraction property
d. Addition property
e. Division property
73
y
18. a
28° 19. t 47° 20. z 76° M
LESSON 12.2 • Problem Solving with Right Triangles
A
3. False
4. True; transitive property of congruence and
9. 28.3° 10. About 2.0 m
221.2.Area 2 cmArea 325 ft
23.4.Area 109 in54.0°x
5.6.31.3°7.6 in. Dya
ABP7.8.Diameter 20.5 cm45.2°
definition ofcongruence
11. About 445.2 ft 12. About 22.6 ft
5.
17. cos 17°
; y
76.3 cm
2. False
B
ABP CDQ
CA Postulate
APB CQD
CA Postulate
ABP CDQ
ASA Postulate
CPCTC
AB CD
AB CD
Given
AP CQ
Given
PB QD
Given
LESSON 12.3 • The Law of Sines
1. Area 46 cm2 2. Area 24 m2 3. Area 45 ft2
4. m 14 cm 5. p 17 cm 6. q 13 cm
7. mB 66°, mC 33°
8. mP 37°, mQ 95°
9. mK 81°, mM 21°
10. Second line: about 153 ft, between tethers: about 135 ft
LESSON 13.2 • Planning a Geometry Proof
LESSON 12.4 • The Law of Cosines
Proofs may vary.
1. t 13 cm 2. b 67 cm 1. Flowchart Proof
3. w 34 cm
AB CD
ABP CDQ
PAB QCD
Third Angle
Theorem
4. mA 76°, mB 45°, mC 59° Given CA Postulate
5. mA 77°, mP 66°, mS 37°
6. mS 46°, mU 85°, mV 49°
AP CQ
Given CA Postulate
APQ CQD
Discovering Geometry Practice Your Skills ANSWERS 113
©2008 Key Curriculum Press
2. Flowchart Proof
PQR TSU QPR STU
Given AIA Theorem Given
PQ ST
QRP TUS
Third Angle
Theorem
PR UT
Converse of
AEA Theorem
3. Flowchart Proof
ABC a x
Given VA Theorem
a b c 180°
x b c 180° b y
Triangle Sum Substitution VA Theorem
Theorem
x y c 180°
Substitution
x y z 180°
c z
Substitution VA Theorem
LESSON 13.3 • Triangle Proofs
Proofs may vary.
1. Flowchart Proof
XZ WY
Given
XY ZY
XZY is isosceles
Definition of
isosceles triangle
YM is the altitude
from vertex Y
Given
Definition of altitude
and vertex angle
WY WY
YM is angle
bisector of XYZ
Reflexive
property Isosceles Triangle
Vertex Angle Theorem
WXY WZY
XYM ZYM
SAS Theorem Definition of
angle bisector
2. Proof:
Statement
BD
1. CD
2. BD AB
AC
3. CD
4. AD
is bisector
of CAB
5. CAD BAD
6. ACD is a right
angle
7. ABD is a right
angle
8. ACD ABD
9. ABD ACD
3. Flowchart Proof
MN QM
Given
Reason
1. Given
2. Given
3. Given
4. Converse of Angle
Bisector Theorem
5. Definition of angle
bisector
6. Definition of
perpendicular
7. Definition of
perpendicular
8. Right Angles Are
Congruent Theorem
9. SAA Theorem
NO QM
Given
MN NO
Transitivity
MNO is isosceles
Definition of
isosceles triangle
NMO NOP
IT Theorem
Linear Pair Postulate
Linear Pair Postulate
QMN RON
Supplements of
Congruent Angles
Theorem
4. Proof:
Statement Reason
1. AB BC 1. Given
2.
ABC is isosceles 2. Definition of isosceles
triangle
3. A ACB 3. IT Theorem
4. ACB DCE 4. Given
QMN and NMO
are supplementary
RON and NOP
are supplementary
114 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
5. A DCE
6. AB CE
7. ABD CED
8. AB BD
9.
ABD is a right
angle
10.
CED is a right
angle
11. BD CE
5. Transitivity
6.
Converse of CA
Postulate
7. CA Postulate
8. Given
9.
Definition of
perpendicular
10.
Definition of right
angle, transitivity
11.
Definition of
perpendicular
LESSON 13.4 • Quadrilateral Proofs
Proofs may vary.
1. Given: ABCD is a D
C
parallelogram
Show: ACand BDbisect
A
each other at M
Flowchart Proof
3. AM CM 3. Given
4. AM CM 4. Definition of
congruence
5. DMA BMC 5. VA Theorem
6. AMD CMB 6. SAS Postulate
7. DAC BCA 7. CPCTC
8. AD BC 8. Converse of AIA
Theorem
9. DMC BMA 9. VA Theorem
10. DMC BMA 10. SAS Postulate
11. CDB ABD 11. CPCTC
12. DC AB 12. Converse of AIA
Theorem
13. ABCD is a 13. Definition of
parallelogram parallelogram
M
B
3. Given: ABCD is a rhombus D
Show: ACand BDbisect
each other at M and
AC BD.
A
M
B
Flowchart Proof
ABCD is a
rhombus
BDC DBA
AIA Theorem
CAB ACD
AIA Theorem
ABM CDM
ASA Postulate
Opposite Sides
Theorem
CD AB
Definition of
parallelogram
AB CD
ABCD is a
parallelogram
Given
Given
ABCD is a DAM BAM AD AB
parallelogram
Rhombus Angles Definition of
Definition of Theorem rhombus
rhombus
AC and BD bisect ADM ABM
each other
SAS Postulate
Parallelogram
Diagonals Theorem
AM AM
Reflexive property
AMD and AMB AMD AMB
DM BM AM CM are supplementary
CPCTC
CPCTC CPCTC Linear Pair Postulate
AC and BD bisect
each other at M
AMB is a right
angle
Definition of bisect,
definition of congruence
Congruent and
Supplementary
2. Given: DM BM,
MD C Theorem
AM CM
AC BD
Show: ABCD is a
A
B
parallelogram
Definition of
perpendicular
Proof:
Statement Reason
1. DM BM 1. Given
BM
2. DM 2. Definition of
congruence
Discovering Geometry Practice Your Skills ANSWERS 115
©2008 Key Curriculum Press
4. Given: ACand BDbisect each DC
other at M and AC BD
Show: ABCD is a rhombus
Flowchart Proof A
MB
(See flowchart at bottom of page.)
5. Given: ABCD is a DC
trapezoid with AB CD
and A B
Show: ABCD is isosceles
B
A
E
Proof:
Statement Reason
1.
ABCD is a trapezoid 1. Given
with AB.
CD
2. Construct CE
AD2. Parallel Postulate
3.
AECD is a 3. Definition of
parallelogram parallelogram
4. AD CE 4. Opposite Sides
Congruent Theorem
5. A BEC 5. CA Postulate
6. A B 6. Given
7. BEC B 7. Transitivity
8.
ECB is isosceles 8. Converse of IT
Theorem
9. EC CB 9. Definition of isosceles
triangle
CB
10. AD 10. Transitivity
11.
ABCD is isosceles 11. Definition of isosceles
trapezoid
Lesson 13.4, Exercise 4
6. Given: ABCD is a
trapezoid with AB CD
and AC BD
Show: ABCD is isosceles
Proof:
Statement
1.
ABCD is a trapezoid
with AB
CD
2. Construct BE
AC
3. DC intersect
and BEat F
4.
ABFC is a
parallelogram
5. AC BF
6. AC BD
7. BF BD
8. DFB is isosceles
9. DFB FDB
10. CAB DFB
11. FDB DBA
12. CAB DBA
13. AB AB
14. ACB BDA
BC
15. AD
16. ABCD is isosceles
E
DC F
AB
Reason
1. Given
2. Parallel Postulate
3.
Line Intersection
Postulate
4.
Definition of
parallelogram
5.
Opposite Sides
Congruent Theorem
6. Given
7. Transitivity
8.
Definition of isosceles
triangle
9. IT Theorem
10.
Opposite Angles
Theorem
11. AIA Theorem
12. Transitivity
13. Reflexive property
14. SAS Postulate
15. CPCTC
16.
Definition of isosceles
trapezoid
ADM ABM
SAS Postulate
Ref lexive property
AM AM
CPCTC
AD AB
DMA and BMA
are right angles
Definition of
perpendicular
Given
AC BD
Converse of the
Parallelogram
Diagonals Theorem
ABCD is a
parallelogram
Definition of
rhombus
ABCD is a
rhombus
AC and BD bisect
each other at M
Given
All 4 sides are
congruent
Transitivity
Definition of bisect,
definition of
congruence
DM BM
Opposite Sides
Theorem
AB DC
DMA BMA
Right Angles
Opposite Sides
Theorem
AD BC
Congruent Theorem
116 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
7. False 8. False
9. True
Given: ABCD with D
C
AB CD
and A C
Show: ABCD is a
A
B
parallelogram
Flowchart Proof
A C
Given
D B
Supplements of
Congruent Angles
Theorem
ABCD is a
parallelogram
AB CD
Given
A and D
are supplementary
Interior Supplements
Theorem
C and B
are supplementary
Interior Supplements
Theorem
Converse of Opposite
Angles Theorem
LESSON 13.5 • Indirect Proof
Proofs may vary.
1. Assume BC AC
Case 1: If BC AC, then ABC is isosceles, by the
definition of isosceles. By the IT Theorem,
A B, which contradicts the given that
mA mB.So, BC AC.
Case 2: DBC is isosceles.
C
4
321D
A
B
By the Exterior Angle Theorem, m1 m2
m4, so m1 m4.
By the Angle Sum Postulate, m2 m3
mABC, so m3 mABC.But DBC is
isosceles, so m4 m3 by the IT Theorem.
So, by transitivity, m1 m4 m3 mABC,
or m1 mABC, which contradicts the given
that mA mB.So, BC
AC.
Therefore the assumption, BC AC, is false, so
BC AC.
2. Paragraph Proof: Assume DAC BAC
It is given that AD. By the reflexive property
ABAC AC. So by SAS, ADC ABC. Then DC
BCby CPCTC. But this contradicts the given
that DC.So DAC BAC.
BC
C
3. Given: ABC with AB BC
Show: C A
Paragraph Proof: Assume
A
C A B
If C A, then by the Converse of the IT
Theorem, ABC is isosceles and AB BC. But this
contradicts the given that AB BC. Therefore,
C A.
4. Given: Coplanar lines k, ,
and m, k , and m
k
intersecting k
m
Show: m intersects
Paragraph Proof: Assume m does not intersect
If m does not intersect , then by the definition of
parallel, m . But because k , by the Parallel
Transitivity Theorem, k m. This contradicts the
given that m intersects k. Therefore, m intersects .
LESSON 13.6 • Circle Proofs
A
1. Given: Circle O with
AB CD
Show: AB
CD
B
D
Flowchart Proof
C
O
Construct
OA, OB, OC, OD
Line Postulate
OA OD AB CD OB OC
Definition of circle, Given Definition of circle,
definition of radii
definition of radii
OAB ODC
SSS Postulate
AOB DOC
CPCTC
AB CD
Definition of congruence,
definition of arc measure,
transitivity
Discovering Geometry Practice Your Skills ANSWERS 117
©2008 Key Curriculum Press
2. Paragraph Proof: Chords BC,CDare LESSON 13.7 • Similarity Proofs
, and DE
congruent because the pentagon is regular. By the
1. Flowchart Proof
proof in Exercise 1, the arcs BC,CD, and DEare
congruent and therefore have the same measure.
by the Inscribed Angles Inter
A BCD B B
mDC Given Reflexive property
ABC CBD
AA Similarity
Postulate
as U.
AB
BC
BC
BD
Definition of
similar triangle
BC2 AB BD
Multiplication
property
2. Given: Trapezoid ABCD
with AB CD, and AC
D
Aand BDintersecting at E
Show: D
BE
E C
AE
E D
AB
C
C
B
Flowchart Proof
AB CD
Given
BDC DBA DCA CAB
AIA Theorem AIA Theorem
1
cepting Arcs Theorem. Similarly, mDAC
2
mEAD
mDE
the three angles have the same measure. So, by the
definition of trisect, the diagonals trisect BAE.
3. Paragraph Proof: Construct the common internal
tangent RU
(Line Postulate, definition of tangent).
Label the intersection of the tangent and TS
T
1
2
1
2
S
U
R
and mBAC
mBC
. By transitivity and algebra,
TU RU SUby the Tangent Segments
E
Theorem. TUR is isosceles by definition because
TU RU. So, by the IT Theorem, T TRU.
Call this angle measure x. SUR is isosceles because
RU SU, and by the IT Theorem, S URS.
Call this angle measure y. The angle measures of
TRS are then x, y, and (x y). By the Triangle
Sum Theorem, x y (x y) 180°. By algebra
(combining like terms and dividing by 2), x y
90°. But mTRS x y, so by transitivity and
the definition of right angle, TRS is a right angle.
4. Paragraph Proof: Construct tangent TP
(Line
Postulate, definition of tangent). PTD and TAC
both have the same intercepted arc, TC
. Similarly,
PTD and TBD have the same intercepted arc,
TD
. So, by transitivity, the Inscribed Angles Intercepting
Arcs Theorem, and algebra, TAC and
TBD are congruent. Therefore, by the Converse
of the CA Postulate, AC BD.
CDE ABE
AA Similarity
Postulate
DE CE DC
BE AE AB
Definition of
similarity
T
P
C
A
B
D
118 ANSWERS Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
AB
3. Given: ABC with ACB right, CD
Show: AC BC AB CD
A
D
B
Flowchart Proof
CD AB
Given
ADC is right
Definition of
perpendicular
ADC ACB
ACB is right
Right Angles Are Given
Congruent Theorem
ACB ADC
A A
AA Similarity Reflexive property
Postulate
AC AB
CD BC
Definition of
similarity
AC BC AB CD
Multiplication
property
4. Given: ABCD with right angles A and C, AB DC
Show: ABCD is a rectangle
DC
B
A
Proof:
Statement Reason
1. Construct DB 1. Line Postulate
2. A and C are 2. Given
right angles
3. A C 3. Right Angles Are
Congruent Theorem
4. AB DC 4. Given
5. DB DB 5. Reflexive property
6. DBA BDC 6. HL Congruence
Theorem
7. DBA BDC 7. CPCTC
8. mDBA mBDC 8. Definition of
congruence
9. mADB mDBA 9. Triangle Sum
mA 180° Theorem
10. mA 90° 10. Definition of right
angle
11. mADB mDBA 11. Subtraction property
90°
12. mADB mBDC 12. Substitution
90°
13. mADB mBDC 13. Angle Addition
mADC Postulate
14. mADC 90° 14. Transitivity
15. mC 90° 15. Definition of right
angle
16. mA mABC 16. Quadrilateral Sum
mC mADC Theorem
360°
17. mABC 90° 17. Substitution property
and subtraction
property
18. A ABC C 18. Definition of
ADC congruence
19. ABCD is a rectangle 19. Four Congruent
Angles Rectangle
Theorem
Discovering Geometry Practice Your Skills ANSWERS 119
©2008 Key Curriculum Press
Comment Form
Please take a moment to provide us with feedback about this book. We are eager to read any comments or
suggestions you may have. Once you’ve filled out this form, simply fold it along the dotted lines and drop it
in the mail. We’ll pay the postage. Thank you!
Your Name
School
School Address
City/State/Zip
Phone Email
Book Title
Please list any comments you have about this book.
Do you have any suggestions for improving the student or teacher material?
To request a catalog or place an order, call us toll free at 800-995-MATH or send a fax to 800-541-2242.
For more information, visit Key’s website at www.keypress.com.
Fold carefully along this line.
Attn: Editorial Department
1150 65th Street
Emeryville, CA 94608-9740
Fold carefully along this line.