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4
Congruent Triangles
CHAPTER FOCUS Learn about some of the Common Core State Standards that
you will explore in this chapter. Answer the preview questions. As you complete each
lesson, return to these pages to check your work.
What You Will Learn
Preview Question
Lesson 4.1: Modeling: Two-Dimensional Figures
G.MG.1 Use geometric shapes, their measures,
and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).
SMP 4 The end of a shaft in a machine is a square
with side length 1 inch. The end of the shaft fits into
a hole in a circular disc. The disc has a radius of 3
inches. Draw a model of the disc and then determine
the area of the circular face of the disc.
Lesson 4.2: Proving Theorems About Triangles
G.CO.10 Prove theorems about triangles.
G.MG.1 Use geometric shapes, their measures,
SMP 2 Explain why the acute angles of a right
triangle are supplementary.
and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).
Lesson 4.3: Proving Triangles Congruent – SSS, SAS
G.CO.10 Prove theorems about triangles.
G.CO.8 Explain how the criteria for triangle
114 CHAPTER 4 Congruent Triangles
D
C
Copyright © McGraw-Hill Education
congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
G.SRT.5 Use congruence and similarity
criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.CO.7 Use the definition of congruence in
terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides
and corresponding pairs of angles are congruent.
SMP 7 Write a two-column proof
to prove the following.
B
Given: ABC and CDA
¯
AB ¯
CD
Prove: ABC CDA A
What You Will Learn
Preview Question
Lesson 4.4: Proving Triangles Congruent – ASA, AAS
G.SRT.5 Use congruence and similarity
criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.CO.8 Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
G.CO.10 Prove theorems about triangles.
G.CO.7 Use the definition of congruence in
terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides
and corresponding pairs of angles are congruent.
YZ || ¯
QR . X is the midpoint of ¯
YQ.
SMP 1 ¯
Prove that XYZ XQR.
Y
R
X
Z
Q
Lesson 4.5: Congruence in Right and Isosceles Triangles
G.CO.10 Prove theorems about triangles.
G.SRT.5 Use congruence and similarity
criteria for triangles to solve problems and to prove
relationships in geometric figures.
SMP 3 Jane says you can prove ABC DBC
using SAS. Flynn says you can prove ABC DBC
using LL. Who is correct? Explain.
B
Copyright © McGraw-Hill Education
A
C
D
Lesson 4.6: Triangles and Coordinate Proof
G.GPE.4 Use coordinates to prove simple
geometric theorems algebraically.
G.CO.10 Prove theorems about triangles.
SMP 1 Prove that the triangle with vertices
A(1, 6), B(12, 3), and C(2, 1) is a right triangle.
CHAPTER 4 Chapter Focus 115
4.1 Modeling: Two-Dimensional Figures
STANDARDS
Objectives
Content: G.MG.1
Practices: 1, 2, 4, 6, 8
Use with Lesson 1–6
• Use two-dimensional figures to model real-world objects and
situations on and off the coordinate plane.
• Solve problems involving perimeter and area.
You can use two-dimensional figures and their properties to model real-world objects and
to solve problems.
EXAMPLE 1
Model Area Using Two-Dimensional Figures
G.MG.1
A design for a medium t-shirt can be modeled as six pieces of material.
a. CALCULATE ACCURATELY Find the area of the front and
back of the t-shirt. Round to the nearest square
centimeter. 20 cm
SMP 6
Rectangle area = (
)(
1
semicircle area = ___
π(
2
area of front =
6 cm
)=
)2 =
−
cm2;
π cm2;
π ≈
8 cm
12 cm
46 cm
cm2.
34 cm
b. CALCULATE ACCURATELY Find the area of each of the four sleeve pieces as the sum
of the areas of a triangle and a rectangle. SMP 6
c. CALCULATE ACCURATELY Find the area of material for each t-shirt. SMP 6
e. PLAN A SOLUTION This design of t-shirt comes in small (S), medium (M), large (L),
extra large (XL), and extra-extra large (XXL) sizes. Multiplying or dividing each
dimension by 1.1 creates larger or smaller sizes. Explain how to find the amount of
material needed for S and XXL sizes of t-shirt. SMP 1
116 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
d. COMMUNICATE PRECISELY Recall that the scale factor is the ratio of the lengths of
the corresponding sides of two similar polygons. For a 5 cm square, a scale factor
increases the perimeter to 60 cm. How did the area of this square change? Describe in
terms of the scale factor. SMP 6
EXAMPLE 2
Model with Two-Dimensional Figures on a
G.MG.1
Coordinate Plane
Lake Superior has been superimposed on a coordinate grid. Each grid unit represents
27 miles.
y
5
O
5
10
x
a. USE A MODEL Use points with whole-number coordinates to create a polygon that
approximates the outline of Lake Superior. SMP 4
b. CALCULATE ACCURATELY Divide the polygon into shapes whose area you can
calculate. Find the approximate area of Lake Superior to the nearest thousand. SMP 6
c. CALCULATE ACCURATELY Using the distance formula where necessary, find the
approximate length of the shoreline of Lake Superior. Explain your method. SMP 6
Copyright © McGraw-Hill Education
d. USE A MODEL Atma has a motorboat with a 27 gallon fuel tank. His boat gets
15 miles per gallon. Can he travel the perimeter of Lake Superior on a full tank of gas?
Show your work or justify your answer. SMP 4
e. EVALUATE REASONABLENESS Which approximation do you think is more reliable,
the area or the shoreline length? Explain. SMP 8
4.1 Modeling: Two-Dimensional Figures 117
PRACTICE
1. a. USE A MODEL The model shows the dimensions of a sofa. Draw a diagram to show
how to calculate the total surface area of the sofa that would be covered by a fitted
G.MG.1, SMP 4
cover. Explain your technique. 26 in.
20 in.
32 in.
8 in.
30 in.
66 in.
b. REASON QUANTITATIVELY How much material is needed for a fitted sofa cover? 2. a. REASON QUANTITATIVELY Miguel is planning to renovate
his living room. How much finish will he need for the
hardwood floor? Assume 1 L of finish covers 4.5 m2
and round to the nearest tenth. G.MG.1, SMP 2
G.MG.1, SMP 2
3.5 m
2.8 m
2.0 m
4.2 m
7.5 m
2.2 m
b. CALCULATE ACCURATELY The height of the room is 2.6 m. Approximate how much
paint is needed for the walls. Assume 1 L of paint covers 7.5 m2 and round to the
nearest tenth. G.MG.1, SMP 6
118 CHAPTER 4 Congruent Triangles
G.MG.1, SMP 8
Copyright © McGraw-Hill Education
c. EVALUATE REASONABLENESS Why might Miguel adapt your answers in practice? 3. USE A MODEL A two-lane running track is made by
connecting two parallel straightaways with semicircular
curves on each end. Each lane of the track has width
1.1 meters and the length of each straightaway is
100 meters. G.MG.1, SMP 4
a. If the radius of the semicircle made by the inside of the
100
first lane is ______
π meters, draw a model of the track
labeling the information provided. Starting from the
same spot on the track, find the distance that a runner
must travel to complete a full lap in each of the two
lanes. Measure from the inside of each lane.
b. Are the distances to run a full lap equal for each lane? If
not, how could you make a lap for each runner the same
distance without changing their lanes or the finish line?
4. Main Street and 1st Street are perpendicular and intersect at a traffic light. The library
is on 1st street and the community center is on Main Street. The distance from the
traffic light to the library is 9 miles. The length of a direct path between the library and
the community center is 17 miles. A city planner wants to put a bike path alongside the
streets from the library to the traffic light to the community center and back to the
G.MG.1
library. a. USE A MODEL What shape best represents the bike
path? Draw a model of the bike path. SMP 4
Copyright © McGraw-Hill Education
b. CALCULATE ACCURATELY The bike path is 8 feet
wide along the entire route. Using the perimeter of
your model, find the number of square feet, to the
nearest square foot, of blacktop that the city planner
will have to pour to cover the entire bike path. Explain
your reasoning. SMP 6
c. CALCULATE ACCURATELY Based on the perimeter of your model, find the area of
the city, to the nearest tenth of a mile, that will be enclosed by the bike path.
Describe your solution process. SMP 6
4.1 Modeling: Two-Dimensional Figures 119
4.2 Proving Theorems About Triangles
STANDARDS
Objectives
Content: G.CO.10, G.MG.1
Practices: 1, 2, 3, 4, 5, 6, 8
Use with Lesson 4–2
• Prove and apply theorems about the angles of triangles.
• Use theorems about the angles of triangles to model real-world
situations.
B
One important characteristic of all triangles is presented in the following
theorem.
Triangle Angle-Sum Theorem The sum of the measures of the
three interior angles of a triangle is 180. In the figure,
m∠A + m∠B + m∠C = 180.
EXAMPLE 1
A
Prove the Triangle Angle-Sum Theorem
C
G.CO.10
a. USE TOOLS Use tracing paper to verify the
Triangle Angle-Sum Theorem. Describe
your method and include a sketch in the
SMP 5
space provided. b. CONSTRUCT ARGUMENTS Complete the paragraph proof. SMP 3
Given: ABC.
Prove: m∠1 + m∠2 + m∠3 = 180
⟷
BC using the
Postulate. ∠4 and ∠BAD form a linear pair. By the
Draw AD || ¯
B
by the definition of supplementary angles. m∠BAD =
by the Angle Addition Postulate, so by the Substitution Property of Equality
∠4 ≈ ∠1 and ∠5 ≈ ∠3, so
,
and
by Definition of
Congruent Angles. Therefore, m∠1 + m∠2 + m∠3 = 180 by the Substitution
Property of Equality.
120 CHAPTER 4 Congruent Triangles
D
2 5
1
3
C
, so m∠4 + m∠BAD =
m∠4 + m∠2 + m∠5 = 180. By
A
Copyright © McGraw-Hill Education
Supplement Theorem, ∠4 and ∠BAD are
4
EXAMPLE 2
B
(2x + 1)°
Apply the Triangle Angle-Sum Theorem
a. REASON QUANTITATIVELY Use the Vertical Angles Theorem to write an algebraic
expression for m∠AEB. Explain. SMP 2
59°
A
E
(3x)°
67° C
D
b. REASON ABSTRACTLY Use the Triangle Angle-Sum Theorem to write and solve an
equation to find the value of x. Justify each step of your solution. G.CO.10, SMP 2
c. CALCULATE ACCURATELY Use your answers to parts a and b to find m∠AEB and
m∠CDE. Use properties or theorems to support each step of your solution. SMP 6
A
Exterior Angle Theorem The measure of an exterior angle of a triangle equals
the sum of the measures of the two remote interior angles. In the figure, m∠A
+ m∠B = m∠1.
Copyright © McGraw-Hill Education
EXAMPLE 3
Use the figure above to prove the Exterior Angle Theorem. Statements
1. m∠A + m∠B + m∠ACB = 180
2. 1
C
Prove the Exterior Angle Theorem
form a linear pair.
B
G.CO.10, SMP 3
Reasons
Triangle Angle-Sum Thm.
Def. of a linear pair
3. m∠1 + m∠ACB = 180
4. Substitution
5. m∠1 = m∠A + m∠B
4.2 Proving Theorems About Triangles 121
EXAMPLE 4
Apply Theorems about Triangles
G.MG.1
The Flatiron Building in New York City is one of America’s oldest skyscrapers, completed
in 1902. Its floorplan is approximately a right triangle. In the figure below, 5th Avenue is
perpendicular to East 22nd Street, and m∠B is 10 less than 3 times m∠C.
a. REASON QUANTITATIVELY Find the angle measures in the floorplan.
Justify your reasoning. SMP 2
A
Ea
Broad
wa
5t h
Flatiron
Building
y
Av
en
ue
D
C
st 2
2n B
dS
tre
e
N
W
E
S
t
b. Find m∠BCD in two ways. Explain each method.
A
Corollary 1 to Triangle Angle-Sum Theorem The acute angles of a right
triangle are complementary. In the figure, ∠C is a right angle, so ∠A and ∠B are
complementary. By the definition of complementary angles, m∠A + m∠B = 90.
C
Corollary 2 to Triangle Angle-Sum Theorem There can be at most one right
or obtuse angle in a triangle. In the figure, if ∠D is a right or obtuse angle, then
∠E and ∠F are acute angles. That is, if m∠D ≥ 90, then m∠E < 90 and m∠F < 90.
D
F
EXAMPLE 5
Prove Corollary 2 to the Triangle Angle-Sum Theorem
B
E
G.CO.10, SMP 8
Fill in each missing statement or reason in the flow proof below.
Given: In ABC, ∠A is a right or obtuse angle.
m∠A ≥ 90
Prove: ∠B is acute.
Given
m∠A + m∠B + m∠C ≥ 90 + m∠B + m∠C
a.
Triangle Angle-Sum Thm.
m∠C > 0
Property of Angles
180 ≥ 90 + m∠B + m∠C
Substitution
m∠B + m∠C > m∠B
c.
Addition
Subtraction
90 > m∠B
Substitution
122 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
b.
The Triangle Angle-Sum Theorem leads to a useful theorem relating the angles of two
triangles.
Third Angles Theorem If two angles of one triangle are congruent to two angles of a
second triangle, then the third angles of the triangles are congruent.
EXAMPLE 6
Prove the Third Angles Theorem
G.CO.12, SMP 3
Fill in the reason for each statement in the following
2-column proof.
P
Given: ∠P ∠X and ∠Q ∠Y
Prove: ∠R ∠Z
Proof:
Z
Y
X
Q
Statements
R
Reasons
1. ∠P ∠X, ∠Q ∠Y
1.
2. m∠P = m∠X, m∠Q = m∠Y
2.
3. m∠P + m∠Q + m∠R = 180
m∠X + m∠Y + m∠Z = 180
3.
4. m∠P + m∠Q + m∠R = m∠X + m∠Y + m∠Z
4.
5. m∠X + m∠Y + m∠R = m∠X + m∠Y + m∠Z
5.
6. m∠R = m∠Z
6.
7. ∠R ∠Z
7.
PRACTICE
J
1. REASON QUANTITATIVELY Find the angle measures in KLM. Justify
G.CO.10, SMP 2
your calculations. K
81°
a. Find m∠KML.
N
67°
M
L
Copyright © McGraw-Hill Education
b. Find m∠L. P
2. REASON QUANTITATIVELY Find the angle measures in PQR .
Justify your calculations. G.CO.10, SMP 2
77°
22°
S
a. Find m∠PRQ.
37°
R
Q
b. Find m∠QPR.
4.2 Proving Theorems About Triangles 123
3. PLAN A SOLUTION Find the measure of the indicated angle.
Show your work, and state any theorems you use. G.CO.10, SMP 1
A
a. Find m∠C.
C
B
(5x)°
(2x - 10)°
(x + 15)°
27°
E
D
b. Find m∠A.
4. CONSTRUCT ARGUMENTS Prove Corollary 1 to the Triangle Angle-Sum Theorem.
Given: ABC with ∠C a right angle. Prove: m∠A + m∠B = 90 G.CO.10, SMP 3
5. USE A MODEL Cassie, a real estate agent, is assessing a
triangular plot of land next to a ravine. She has determined that
m∠1 = 64 and m∠4 = 154. Find m∠2. Which theorem did you
G.MG.1, SMP 4
use? 1
Ravine
36°
3
4
A
a. Find m∠D and m∠ACB. Justify each step. D
124 CHAPTER 4 Congruent Triangles
C
B
Copyright © McGraw-Hill Education
6. CONSTRUCT ARGUMENTS In ACD, m∠DAC = 36 and ∠D ∠ACD.
G.CO.10, SMP 3
In ABC, m∠B > 18. 2
b. Give a reason why ∠B is acute. Then write and solve an inequality describing the
measure of ∠B
7. REASON QUANTITATIVELY To navigate around a peninsula a
ship sails from Port A at a bearing 58° west of due north and
when it reaches point B, it changes course to the bearing 20°
west of due south to reach Port C. The ship’s route is shown in the
figure. G.MG.1, SMP 2
B
20°
58°
E
C
D
A
a. Find m∠DAB, ∠DBA, and m∠BCD. Explain your reasoning.
b. Find m∠BCE. State any theorems you use to determine your answer.
8. REASON QUANTITATIVELY A wall panel needs to be
cut to fill a space in transition to a staircase as shown
in the figure. If m∠DFE = 60, find m∠AFD and m∠FDC,
the angles at which the panel should be cut. A
B
F
G.MG.1, SMP 2
staircase
Copyright © McGraw-Hill Education
E
D
C
9. REASON QUANTITATIVELY In the two triangles ABC and DEF, ∠A ∠D and
∠C ∠F. Find the value of x if m∠B = 3x - 5 and m∠E = x + 27. Justify each step
of your solution. G.CO.10, SMP 2
4.2 Proving Theorems About Triangles 125
4.3 Proving Triangles Congruent—SSS, SAS
STANDARDS
Objectives
Content: G.CO.7, G.CO.8,
G.CO.10, G.SRT.5
Practices: 1, 2, 3, 4, 6, 7
Use with Lesson 4-4
• Show that the SSS and SAS criteria for triangle congruence follow
from the definition of congruence in terms of rigid motions.
• Use congruence criteria for triangles to prove relationships in figures.
Recall that two figures are congruent if each pair of corresponding parts is congruent.
The converse is also true. If corresponding parts of two different figures are congruent,
then the figures are congruent. In this lesson you will explore triangle congruence in
terms of rigid motion using two postulates: SSS (side-side-side) Congruence and SAS
(side-angle-side) Congruence. To show that SSS is sufficient to show triangle congruence,
the Perpendicular Bisector Theorem and its converse will be used. These theorems will be
explored further in a later chapter.
KEY CONCEPT
Perpendicular Bisector Theorem
Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of the segment.
Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints
of a segment, then it is on the perpendicular bisector of the segment.
EXAMPLE 1
Explore SSS Congruence
EXPLORE In ABC and DEF, ¯
AB ¯
DE, ¯
BC ¯
EF,
¯
¯
and CA FD. Use rigid motion transformations to
show that ABC DEF.
G.CO.8
A
E
B
a. PLAN A SOLUTION Describe a sequence of right
motion transformations that would map ¯
CA to ¯
FD as shown in the figure
at the right. Label the figure with all known information. SMP 1
C
F
D
E
F
C
D
A
c. REASON ABSTRACTLY Which rigid motion transformation maps E to B? 126 CHAPTER 4 Congruent Triangles
B
SMP 2
Copyright © McGraw-Hill Education
b. REASON ABSTRACTLY Given that ¯
EF ¯
BC, what can you conclude
¯
EB? Explain. about the relationship between FD and ¯
SMP 2
d. CONSTRUCT ARGUMENTS Explain how your observations complete the argument
that ABC DEF. SMP 3
EXAMPLE 2
Use SSS to Determine Triangle Congruence
G.CO.7
The triangles ABC, DEF, and GHI are placed on the
coordinate grid shown to the right.
a. CALCULATE ACCURATELY Use the distance formula to
find the lengths of the sides of ABC and DEF. Show
your work. SMP 6
(-2, 5) C 6
(-6, 4) A
F (3, 4)
D (1, 1)
(-4, 1) B
-6
y
E (5, 0)
O
6x
H (2, -2)
(-3, -2) G
I (0, -5)
-6
b. REASON QUANTITATIVELY Determine which sides of the two triangles are congruent.
Explain your reasoning. SMP 2
c. CONSTRUCT ARGUMENTS Use the results from part b to conclude that
ABC DEF. SMP 3
Copyright © McGraw-Hill Education
d. USE STRUCTURE Use the distance formula to find the lengths of the sides of GHI. Is
ABC GHI? Explain your reasoning. SMP 7
In Example 1, rigid motion transformations are used to show that SSS is sufficient to
prove triangle congruence. A similar argument can be used to show that SAS is a valid
congruence criterion. This argument uses the Angle Bisector Theorem and its converse.
These theorems will be explored further in a later chapter.
KEY CONCEPT
Angle Bisector Theorem
Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from
the sides of the angle.
Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant
from the sides of the angle, then it is on the bisector of the angle.
4.3 Proving Triangles Congruent—SSS, SAS 127
EXAMPLE 3
Explore SAS Congruence
In ABC and DEF ¯
AC ¯
DF, ¯
BC ¯
EF, and
∠BCA ∠EFD. Use rigid motion transformations
to show that ABC DEF.
G.CO.8
A
a. PLAN A SOLUTION Describe a sequence of rigid
motion transformations that would map ¯
CA to ¯
FD
as shown in the figure at the right. Label the figure
with all known information.
SMP 1
D
E
B
F
C
D
A
E
b. REASON ABSTRACTLY Given that ¯
EF ¯
BC, what can you conclude about
¯
¯
the relationship between FD and EB ? Explain. SMP 2
B
F
C
c. REASON ABSTRACTLY Which rigid motion transformation maps E to B
and ¯
DE to ¯
AB? SMP 2
d. REASON ABSTRACTLY Since ∠BCA ∠EFD, what can be concluded about ¯
FD? SMP 2
e. REASON ABSTRACTLY What can you conclude about ∠EDF and ∠BAC? Explain. SMP 2
f. CONSTRUCT ARGUMENTS Explain how your observations complete the argument
that ABC DEF. SMP 3
Determine Triangle Congruence
G.CO.10, G.SRT.5
In the figure, ¯
AC ¯
AD.
a. INTERPRET PROBLEMS Suppose you know ∠C ∠D. Can you prove
SMP 1
that ABC ABD? Why or why not? C
A
B
D
128 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
EXAMPLE 4
⟶
b. CONSTRUCT ARGUMENTS Suppose you are given that AB bisects ∠CAD. Write a
paragraph proof to show that ABC ABD. SMP 3
KEY CONCEPT
SSS and SAS Congruence Postulates
Complete each congruence postulate. Then mark each figure to show an example of given information that
would allow you to use the postulate to prove the triangles are congruent.
Postulate
Example
D
A
Side-Side-Side (SSS) Congruence
If
F
C
,
E
B
then the triangles are congruent.
Side-Angle-Side (SAS) Congruence
D
A
If
F
C
,
E
B
then the triangles are congruent.
PRACTICE
J
1. CONSTRUCT ARGUMENTS In the figure, point P is the
JL and the midpoint of ¯
KM. midpoint of ¯
M
P
G.CO.10, G.SRT.5, SMP 3
K
L
Copyright © McGraw-Hill Education
a. Write a paragraph proof to show that ∠K ∠M.
b. What can you conclude about JPK and LPM regarding rigid motions?
4.3 Proving Triangles Congruent—SSS, SAS 129
2. CONSTRUCT ARGUMENTS Write a two-column proof for the
following. G.CO.10, G.SRT.5, SMP 3
C
B
Given: ¯
AB is the perpendicular bisector of ¯
CD.
D
A
Prove: ∠C ∠D
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
INTERPRET PROBLEMS In Exercises 3– 6, explain whether there is enough information
given in the figure to prove that the triangles are congruent using SAS or SSS. G.CO.10, G.SRT.5, SMP 1
3.
4.
H
Q
R
G
L
J
T
S
K
5.
6.
A
D
C
130 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
B
7. USE A MODEL An engineer is designing a new cell phone tower. Part of the
tower is shown in the figure. The engineer makes sure that line m is parallel to
AB ¯
CD. Can she prove that ABC DCB? Explain why or
line n and that ¯
why not. G.CO.10, G.SRT.5, SMP 4
m
n
A
C
B
D
8. REASON ABSTRACTLY For each pair of triangles, describe a sequence of rigid
motions attempting to map ABC to DEF that could be used to determine whether
or not the triangles are congruent. G.CO.7, SMP 2
a.
D
b.
A
D
A
B
C
E
B
E
F
F
9. CALCULATE ACCURATELY Triangles ABC and DEF are placed
on the coordinate grid with vertices A(-5, 2), B(-1, -1), C(1, 3),
D(-3, -2), E(1, 2), and F(3, -3). G.CO.7, SMP 6
y
(1, 3) C
(-5, 2) A
a. Use the distance formula to find the lengths of the sides of ABC
and DEF.
Copyright © McGraw-Hill Education
C
E (1, 1)
O
(-3, -2) D
x
B (-1, -1)
(3, -3) F
b. Which sides of the two triangles are congruent?
c. Are the two triangles congruent? Explain your reasoning.
d. What can you conclude about ABC and DEF regarding rigid motions?
4.3 Proving Triangles Congruent—SSS, SAS 131
4.4 Proving Triangles Congruent—ASA, AAS
STANDARDS
Objectives
Content: G.CO.7, G.CO.8,
G.CO.10, G.SRT.5
Practices: 1, 2, 3, 4, 5, 7
Use with Lesson 4-5
• Show that the ASA criterion for triangle congruence follows from
the definition of congruence in terms of rigid motions.
• Use congruence criteria for triangles to prove relationships
in figures.
An included side is the side located between two consecutive angles of a
PQ is the included side between
polygon. In the triangle shown at the right, ¯
∠P and ∠Q. If two angles and the included side of one triangle are congruent
to two angles and the included side of a second triangle, then the triangles
are congruent by the Angle-Side-Angle (ASA) Congruence Postulate.
EXAMPLE 1
Explore ASA Congruence Postulate
included side between
∠P and ∠Q
P
Q
R
G.CO.8
Follow these steps to show how the ASA Congruence Postulate follows from the
rigid-motion definition of congruence.
a. PLAN A SOLUTION In the figure at right, ∠A ∠D, ∠B ∠E,
and ¯
AB ¯
DE. Mark this information on the figure, then describe a
AB
sequence of rigid-motion transformations that would map ¯
¯
to DE as shown in the figure below. SMP 1
C
A
B
F
E
b. CONSTRUCT ARGUMENTS Julia states that if¯
AB ¯
DE there will
AB
always exist a sequence of rigid-motion transformations that map ¯
DE even if the transformations are not easy to determine. Do you agree?
to ¯
Explain your reasoning. SMP 3
D
F
E
B
d. CONSTRUCT ARGUMENTS What rigid transformation maps C to F?
Justify your answer. SMP 3
132 CHAPTER 4 Congruent Triangles
C
Copyright © McGraw-Hill Education
c. REASON ABSTRACTLY In the figure at right, what is the relationship
between ¯
AB and ∠FEC? ¯
AB and ∠FDC? SMP 2
D
A
⟶
f. USE STRUCTURE What can you say about the image of BC under this reflection? Why? SMP 7
g. USE REASONING Explain how your answer to parts e and f complete the argument
that ABC DEF. SMP 3
KEY CONCEPT
ASA Congruence Postulate
Complete the congruence postulate. Then mark the figure to show an example of given information
that would allow you to use the postulate to prove the triangles are congruent.
Postulate
Example
Angle-Side-Angle (ASA) Congruence
D
A
If
F
C
,
EXAMPLE 2
E
B
then the triangles are congruent.
Use ASA to Determine Triangle Congruence
G.CO.10, G.SRT.5
Pamela is studying Native American arrowheads. She has drawn a diagram to
model aspects of the shape of a certain arrowhead and wants to know if AB AD.
She drew a dashed line AC through the diagram and found that AC bisects ∠BAD
and ∠BCD.
‾
‾
a. PLAN A SOLUTION Mark the congruent angles on the figure. SMP 1
b. CONSTRUCT ARGUMENTS Write a two-column proof using the ASA
Congruence Postulate. SMP 3
Copyright © McGraw-Hill Education
A
‾ ‾
B
C
D
Given: ¯
AC bisects ∠BAD and ∠BCD.
Prove: ¯
AB ¯
AD
Statements
Reasons
1.
2.
3.
4.
5.
4.4 Proving Triangles Congruent—ASA, AAS 133
c. REASON ABSTRACTLY Describe a rigid motion that would map ABC to ADC.
SMP 2
EXAMPLE 3
Prove the Angle-Angle-Side (AAS)
Congruence Theorem
G.CO.10, G.SRT.5
M
Follow these steps to prove the AAS Congruence Theorem.
J
Given: ∠J ∠M, ∠K ∠N, and ¯
JL ¯
MP
Prove: JKL MNP
P
L
a. PLAN A SOLUTION Mark the given information on the figure.
Then explain how you can prove the AAS Congruence Theorem
by using one of the three congruence postulates you have
SMP 1
already established. N
K
b. CONSTRUCT ARGUMENTS Write a two-column proof of the AAS Congruence
SMP 3
Theorem. Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
KEY CONCEPT
AAS Congruence Theorem
Theorem
Copyright © McGraw-Hill Education
Complete the congruence theorem. Then mark the figure to show an example of given information
that would allow you to use the theorem to prove the triangles are congruent.
Example
Angle-Angle-Side (AAS) Congruence
A
D
If
C
,
then the triangles are congruent.
134 CHAPTER 4 Congruent Triangles
B
F
E
EXAMPLE 4
Use the AAS Congruence Theorem
G.CO.10, G.SRT.5, SMP 3
Althea used a kit to build a picnic table for her yard. The side
view is shown in the figure. Althea made sure the tabletop is
parallel to the ground and she checked that BC DC. She wants
to know if she can conclude that AC EC.
‾ ‾
A
B
‾ ‾
C
a. CONSTRUCT ARGUMENTS Write a two-column proof using
the AAS Congruence Theorem.
D
Given: ¯
AB || ¯
DE, ¯
BC ¯
DC
E
Prove: ¯
AC ¯
EC
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
b. CRITIQUE REASONING Althea said she proved that ¯
AC ¯
EC using a different congruence
criterion. Do you think this is possible? If so, explain how. If not, explain why not.
PRACTICE
1. In the figure, ¯
JK is perpendicular to ¯
LM, and ¯
JK bisects ∠LKM. L
Copyright © McGraw-Hill Education
G.CO.10, G.SRT.5
a. CONSTRUCT ARGUMENTS Write a paragraph proof to show that
LKJ MKJ. SMP 3
J
K
M
b. REASON ABSTRACTLY What rigid-motion transformation maps LKJ to MKJ? SMP 2
4.4 Proving Triangles Congruent—ASA, AAS 135
2. CONSTRUCT ARGUMENTS Write a two-column proof for the following. Given: ¯
AC is parallel to ¯
BD. Point D is the midpoint
CE. ∠CAD ∠DBE
of ¯
G.CO.10, G.SRT.5, SMP 3
A
AD ¯
BE
Prove: ¯
C
B
D
Statements
E
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
3. USE TOOLS Use a compass and straightedge and the ASA
Congruence Postulate to construct a triangle congruent to PQR.
Show your work in the space at the right. G.SRT.5, SMP 5
P
Q
R
INTERPRET PROBLEMS In Exercises 4 and 5, explain whether there is enough
information given in the figure to prove that the triangles are congruent. If so, describe
a sequence of rigid motions mapping one triangle to the other. G.CO.10, G.SRT.5, SMP 1
4.
5.
F
P
D
G
E
136 CHAPTER 4 Congruent Triangles
M
N
Q
Copyright © McGraw-Hill Education
H
6. USE A MODEL Dylan came to a river during a hike and he wanted
AB ,
to estimate the distance across it. He held his walking stick, ¯
vertically on the ground at the edge of the river and sighted along
the top of the stick across the river to the base of a tree, T. Then
he turned, without changing the angle of his head, and sighted
along the top of the stick to a rock, R, located on his side of the
river. G.CO.10, G.SRT.5, SMP 4
A
B
T
R
a. Explain why ABT ABR.
b. Dylan finds that it takes 27 paces for him to walk from his current location to the
rock. He also knows that each of his paces is 14 inches long. Explain how he can use
this information to estimate the distance across the river.
7. CRITIQUE REASONING Raj says that he can draw two triangles that have two sides
and a nonincluded angle congruent and that the two triangles are congruent. G.CO.10, G.SRT.5, SMP 3
a. Raj says that there must be a SSA Congruence Theorem to justify the triangle he
constructed. Do you agree? Explain.
Copyright © McGraw-Hill Education
b. Provide a counterexample to disprove Raj’s conjecture.
c. Two triangles have two sides and a nonincluded angle congruent. Prove that if any
other pair of angles of the two triangles is congruent, then the two triangles are
congruent.
4.4 Proving Triangles Congruent—ASA, AAS 137
4.5 Congruence in Right and Isosceles Triangles
STANDARDS
Objectives
Content: G.CO.10, G.SRT.5
Practices: 3, 4, 5, 6, 7, 8
Use with Extend 4–5, Lesson 4–6
• Use congruence criteria for right triangles.
• Prove that base angles of an isosceles triangle are congruent.
• Apply the Isosceles Triangle Theorem and its converse.
In Lessons 4.3 and 4.4, you developed and applied the SSS, SAS, ASA, and AAS
Congruence Criteria. Now you will investigate SSA congruence.
EXAMPLE 1
Investigate SSA Congruence
G.SRT.5
EXPLORE Follow these steps to use a ruler, protractor, and compass to create triangles.
Compare the triangles you draw with those of other students.
. Then use the protractor
a. USE TOOLS In the space at the left below, draw a ray, AX
, so that m∠A = 30°. Mark point B on AY
so that AB = 6 cm. Finally,
to draw a ray, AY
−−
draw BC so that BC = 4 cm. To do this, open your compass to 4 cm, place the point on B,
. Label the triangle. Is there more than one way to
and draw an arc that intersects AX
draw ABC? If so, draw it a second way in the space at the right below. SMP 5
b. MAKE A CONJECTURE Is there an SSA Congruence Criterion? Explain. 138 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
c. USE TOOLS Repeat part a in the space
at the right. Use the same dimensions,
but this time draw the triangle so that
BC = 3 cm. SMP 5
SMP 3
d. COMMUNICATE PRECISELY Describe how the situation in part c is different from the
SMP 6
situation in part a. e. MAKE A CONJECTURE Is there ever a time when the SSA Congruence Criterion
works? If so, when? SMP 3
The congruence criteria that you worked with in Lessons 4.3 and 4.4 (SSS, SAS, ASA, AAS)
all hold for right triangles, and can be given special names using the parts of a right
triangle. In addition, there is an SSA Congruence Theorem for right triangles.
KEY CONCEPT
Right Triangle Congruence
Complete each congruence theorem. Then mark the figure to show an example of given information that
would allow you to use the theorem to prove the right triangles are congruent.
Theorem
Example
A
Leg-Leg (LL) Congruence
If
,
C
B
D
then the triangles are congruent.
F
E
A
Hypotenuse-Angle (HA) Congruence
If
C
B
D
,
then the triangles are congruent.
F
E
A
Leg-Angle (LA) Congruence
If
C
B
Copyright © McGraw-Hill Education
D
,
then the triangles are congruent.
F
E
A
Hypotenuse-Leg (HL) Congruence
If
C
B
D
,
then the triangles are congruent.
E
F
4.5 Congruence in Right and Isosceles Triangles 139
EXAMPLE 2
Use Right Triangle Congruence
G.CO.10, G.SRT.5
−− −−
−−
−−
In the figure, AC AD and AB is perpendicular to CD .
A
a. CONSTRUCT ARGUMENTS Mark the given information
on the figure. Then write a paragraph proof that
SMP 3
ABC ABD. C
D
B
b. COMMUNICATE PRECISELY What type of triangle is CAD? What must be true about
∠C and ∠D? Why? SMP 6
vertex angle
In an isosceles triangle, the congruent sides are called the legs of the triangle. The
angle whose sides are the legs of the triangle is the vertex angle. The side opposite
the vertex angle is the base of the triangle. The two angles formed by the base and
the congruent sides are the base angles.
leg
leg
base
base angles
The Key Concept box summarizes a relationship that you may have discovered in
Example 2.
KEY CONCEPT
Isosceles Triangles
Complete each example.
Theorem
Example
Isosceles Triangle Theorem
P
R
−− −−
Example: If PQ PR , then
Converse of the Isosceles Triangle Theorem
If two angles of a triangles are congruent, then the sides opposite
those angles are congruent.
K
L
J
Example: If ∠K ∠L, then
140 CHAPTER 4 Congruent Triangles
.
.
Copyright © McGraw-Hill Education
If two sides of a triangle are congruent, then the angles opposite
those sides are congruent.
Q
Prove the Isosceles Triangle Theorem
EXAMPLE 3
G.CO.10, G.SRT.5
A
Follow these steps to prove the Isosceles Triangle Theorem.
−− −−
Given: AB AC
B
Prove: ∠B ∠C
a. USE STRUCTURE The first steps of the proof are to let P be the midpoint of
−−
−−
BC and to draw the auxiliary line segment AP . Why are these steps justified? b. CONSTRUCT ARGUMENTS Write a two-column proof for the theorem. Statements
SMP 7
SMP 3
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
EXAMPLE 4
C
Use the Isosceles Triangle Theorem
G.CO.10
Julia works for a company that makes lounge chairs. As shown in the figure, the back of
each chair is an isosceles triangle that can be adjusted so the person sitting on the chair
can recline.
Q
Copyright © McGraw-Hill Education
P
Q
R
S
P
R
S
a. CONSTRUCT ARGUMENTS Suppose the chair is adjusted so that m∠Q = 50. What is
m∠QRS? Write a paragraph proof to justify your answer. SMP 3
4.5 Congruence in Right and Isosceles Triangles 141
b. DESCRIBE A METHOD Julia would like a general method that she can use to find
m∠QRS if she knows m∠Q. Write an expression for m∠QRS when m∠Q = x. Explain. SMP 8
c. CRITIQUE REASONING Manuel says that he can use the Exterior Angle Theorem to
get the result shown in part b. Is he correct? Explain. SMP 3
PRACTICE
−−
1. CONSTRUCT ARGUMENTS In the figure, BD is the perpendicular
−−
−− −−
bisector of AC , and AB CD . Write a paragraph proof to show that
AEB CED. G.CO.10, G.SRT.5, SMP 3
B
A
C
E
D
M
2. CONSTRUCT ARGUMENTS Write a two-column proof of the converse
G.CO.10, G.SRT.5, SMP 3
of the Isosceles Triangle Theorem. Given: ∠N ∠P
−−− −−
Prove: MN MP
N
Statements
Reasons
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
Copyright © McGraw-Hill Education
1.
142 CHAPTER 4 Congruent Triangles
P
3. Each of the triangles shown below is isosceles. G.CO.10, G.SRT.5
a. USE TOOLS Use a ruler to find the midpoint of each side of each triangle. Then
draw the triangle formed by connecting the midpoints of each side. SMP 5
b. MAKE A CONJECTURE Look for patterns in your drawings. Make a conjecture
about what you notice. SMP 3
A
c. CONSTRUCT ARGUMENTS In the isosceles triangle at the
−− −−
right, AB AC . Use the figure to help you explain why the
SMP 3
conjecture you made in part b is true. F
E
B
4. A boat is traveling at 25 mi/h parallel to a straight section
−−
of the shoreline, XY, as shown. An observer in a lighthouse
L spots the boat when the angle formed by the boat, the
lighthouse, and the shoreline is 35° and again when this
angle is 70°. G.CO.10
Copyright © McGraw-Hill Education
a. USE STRUCTURE Explain how you can prove that
BCL is isosceles. SMP 7
B
D
C
C
70°
35°
X
L
Y
Shoreline
b. USE A MODEL It takes the boat 15 minutes to travel from point B to point C. When
the boat is at point C, what is its distance to the lighthouse? SMP 4
5. CRITIQUE REASONING Anisa says that if two exterior angles of a triangle are
G.CO.10, SMP 3
congruent, then the triangle is isosceles. Do you agree? Explain. 4.5 Congruence in Right and Isosceles Triangles 143
4.6 Triangles and Coordinate Proof
STANDARDS
Objectives
• Use coordinates to prove simple geometric theorems algebraically.
Content: G.CO.10, G.GPE.4
Practices: 1, 2, 3, 4, 6, 7, 8
Use with Lesson 4-8
• Prove that the segment joining the midpoints of two sides of
a triangle is parallel to the third side and half the length of the
third side.
EXAMPLE 1
Investigate a Triangle Property
G.CO.10
EXPLORE A town is preparing for a 5K run. The race will start at city
hall, C. The course will take runners along straight streets to the
library, L, to the science museum, S, and back to city hall for the
finish. A city employee has been asked to develop a different route
for runners who want a shorter race.
a. CALCULATE ACCURATELY The employee decides to create a shorter
−−
−−
route by locating the midpoint X of CL and the midpoint Y of CS . The
runners will go from C to X to Y and back to C. Use the Midpoint Formula
−−
−−
−−
SMP 6
to locate the midpoints of CL and CS . Draw XY on the figure. y
C
8
6
4
S
2
L
O
2
4
6
8
10
x
2
4
6
x
b. CALCULATE ACCURATELY Use the Slope Formula and Distance Formula to
−− −−
compare XY to LS . What do you notice? SMP 6
c. MAKE A CONJECTURE Draw and label your own triangle, PQR, in the space
at the right. Then find the midpoints, M and N, of two sides and draw the segment
joining these midpoints. Use the Midpoint Formula, Slope Formula, and Distance
Formula to check if the relationship you noticed in part b holds for this figure.
State your observations as a conjecture. SMP 3
y
6
4
2
d. USE A MODEL What is the length of the race course from C to X to Y and back to C?
Explain how you know. SMP 4
144 CHAPTER 4 Congruent Triangles
O
Copyright © McGraw-Hill Education
−2
e. DESCRIBE A METHOD Describe how the employee could find another route that is
half as long as the original route. Explain your reasoning. SMP 8
A coordinate proof uses figures in the coordinate plane and algebra to prove geometric
relationships. Use can use a coordinate proof to prove the conjecture you made in the
previous exploration. The Key Concept box provides suggestions for placing triangles
on the coordinate plane when writing a coordinate proof.
KEY CONCEPT
Placing Triangles on the Coordinate Plane
Step 1 Use the origin as a vertex or center of the triangle.
Step 2 Place at least one side of the triangle on an axis.
Step 3 Keep the triangle within the first quadrant if possible.
Step 4 Use coordinates that make computations as simple as possible.
EXAMPLE 2
Write a Coordinate Proof
G.CO.10, G.GPE.4
P
Follow these steps to write a coordinate proof for the following.
−−
Given: PQR, where M is the midpoint of PQ and N is the
−−
midpoint of PR
‾
‾
‾
N
1
Prove: MN is parallel to QR, and MN = ___
QR.
2
a. REASON ABSTRACTLY Place PQR on the coordinate plane
and assign coordinates to the vertices of the triangle. For
−−
convenience, place vertex Q at the origin and place QR along
the positive x-axis. Since the proof will involve midpoints, it
makes sense to assign coordinates that are multiples of two.
What are appropriate coordinates for R in terms of a? What
are appropriate coordinates for P in terms of b and c? Label
the coordinates of the vertices P, Q, and R in the figure. Copyright © McGraw-Hill Education
M
Q
R
P
M
N
Q
R
SMP 2
b. CALCULATE ACCURATELY Show how to find the coordinates of M and N. SMP 6
−−−
c. PLAN A SOLUTION What theorem or postulate can you use to prove that MN is
−−
SMP 1
parallel to QR ? 4.6 Triangles and Coordinate Proof 145
−−
−−−
d. CONSTRUCT ARGUMENTS Prove that MN is parallel to QR . 1
e. CONSTRUCT ARGUMENTS Prove that MN = __2 QR. EXAMPLE 3
Write a Coordinate Proof
SMP 3
SMP 3
G.CO.10, G.GPE.4
Follow these steps to prove that the midpoint of the hypotenuse of a right
triangle is equidistant from the vertices of the triangle.
Given: JKL is a right triangle with a right angle at ∠K, and M is the
midpoint of JL.
‾
J
M
K
L
Prove: JM = KM = LM
a. REASON ABSTRACTLY Place the triangle above on the coordinate
plane. Label the coordinates of the vertices J, K, and L. SMP 2
b. CALCULATE ACCURATELY Show how to find the coordinates
SMP 6
of M. c. PLAN A SOLUTION What property, theorem, or formula can you use to complete the proof?
SMP 1
Explain. d. CALCULATE ACCURATELY Show how to find each of the following distances. SMP 6
JM =
LM =
e. USE STRUCTURE The segment that joins K and M divides JKL into two smaller
triangles. What types of triangles are these? Explain how you know. SMP 7
146 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
KM =
Some coordinate proofs are based on specific values for the coordinates of the figures.
As in the more general proofs, you can use the Distance Formula, Midpoint Formula, Slope
Formula, and/or congruence criteria to write the proof.
EXAMPLE 4
Write a Coordinate Proof
G.CO.10, G.GPE.4
Andrew is using a coordinate plane to design a quilt. Two of the triangular
patches for the quilt are shown in the figure. Andrew wants to be sure that
∠A and ∠D have the same measure.
5
A
B
Follow these steps to prove that ∠A ∠D.
C
O
-5
a. PLAN A SOLUTION Describe the main steps you can use to prove that
∠A ∠D? SMP 1
5x
D
E
-5
b. CONSTRUCT ARGUMENTS Write a paragraph proof that ∠A ∠D. y
F
SMP 3
PRACTICE
1. CONSTRUCT ARGUMENTS PQR is a right triangle
with a right angle at ∠Q, and M is the midpoint of ¯
PR.
Draw a figure and assign coordinates to prove that the
area of QMR is one-half the area of PQR. Copyright © McGraw-Hill Education
G.CO.10, G.GPE.4, SMP 3
4.6 Triangles and Coordinate Proof 147
2. Complete the following proof. A
G.CO.10, G.GPE.4
Given: ABC is isosceles with ¯
AB ¯
AC. D is the midpoint of ¯
AB, E is
¯
AC.
the midpoint of BC , and F is the midpoint of ¯
D
B
Prove: DEF is isosceles.
F
E
C
a. USE STRUCTURE In the space at the right, show how to
place ABC on a coordinate plane. Show how to assign
coordinates to the vertices of ABC. SMP 7
b. REASON ABSTRACTLY What are the coordinates of
SMP 2
D, E, and F? c. CONSTRUCT ARGUMENTS Explain how to complete the proof. SMP 3
3. CRITIQUE REASONING A student was asked to prove that the segment
joining the midpoints of two sides of a triangle is parallel to the third side.
He set up the figure and coordinates shown at right. Then he gave the
argument shown below the figure. G.CO.10, G.GPE.4, SMP 3
a. Is the student’s proof correct? If not, explain why not and explain
what the student would need to do differently to write a correct
proof. R (0, 2b)
M
S (0, 0)
N
T(2a, 0)
−−
Let M be the midpoint of RS . Then
the coordinates of M are M(0, b).
−−
Let N be the midpoint of ST. Then
the coordinates of N are N(a, 0).
−−− 0 - b
b
__
The slope of MN is ________
a - 0 = -a.
−− 0 - 2b
b
__
The slope of RT is __________
2a - 0 = - a .
148 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
Since the slopes are equal,
b. Sharon argues that the proof in Example 2 makes an assumption
−−− −−
MN || RT.
about the triangle and is therefore invalid. She claims that we
assumed that the triangle lies on the x-axis with one of its vertices
at the origin. In order to prove this theorem in general, we would have
to assume that the triangle has three vertices at (a, b), (c, d), and (e, f).
Is Sharon correct?
4. USE A MODEL A landscape architect is using a coordinate plane to design
a triangular community garden. The fence that will surround the garden is
modeled by ABC.
5
y
A
The architect wants to know if the any of the three angles in the fence
will be congruent. Determine the answer for the architect and give a
coordinate proof to justify your response. G.CO.10, G.GPE.4, SMP 4
O
-5
5x
C
B
-5
5. Complete the following proof. G.CO.10, G.GPE.4
JK, Q is the midpoint
Given: JKL where P is the midpoint of ¯
KL, and R is the midpoint of ¯
JL.
of ¯
Prove: The area of JKL is 4 times the area of PQR.
a. USE STRUCTURE In the space at the right, show how to
place JKL on a coordinate plane. Show how to assign
coordinates to the vertices of JKL. SMP 7
b. CONSTRUCT ARGUMENTS Write the proof. SMP 3
Copyright © McGraw-Hill Education
c. CRITIQUE REASONING Jane approaches this proof differently. She argued that
the four triangles KPQ, QRL, PJR, and RQP are all congruent. Since they
are all congruent, each of them is one fourth of JKL. Is Jane correct? Outline how
this proof would work making use of Example 2.
6. CONSTRUCT ARGUMENTS Write the following proof. AB ¯
AC
Given: ¯
AB, Y is the midpoint of ¯
AC.
X is the midpoint of ¯
BY ¯
CX
Prove: ¯
G.CO.10, G.GPE.4, SMP 3
A
X
B
Y
C
4.6 Triangles and Coordinate Proof 149
Performance Task
Designing a Park
Provide a clear solution to the problem. Show all of your work, including relevant
drawings. Justify your answers.
A town is building a skateboarding park within the region bounded by the pentagon
shown on the map. Each unit on the grid represents 10 meters. The town wants the
designer to do the following:
• Create the park in the shape of an equilateral triangle.
• Have one side be parallel to Jones Avenue.
• Make the perimeter of the park 180 meters.
12
10
y
West Street
Jones
Avenue
A
8
Bay
Road
6
4
2
O
2
4
6
8
10
12 x
Part A
You are told Jones Avenue makes a 120º angle with West Street. Use this information to
construct the side of the park parallel to Jones Avenue from point A as accurately as
possible. Explain your reasoning. Label the other endpoint as point B.
Copyright © McGraw-Hill Education
150 CHAPTER 4 Congruent Triangles
Part B
Construct the rest of the park. Explain what you did. Label the final vertex of the park as
point C.
Part C
Copyright © McGraw-Hill Education
The mayor is impressed with your drawing, but would like to know the coordinates of
points B and C. Give approximate values for each point based on your drawing. Then find
the exact coordinate of point B.
CHAPTER 4 Performance Task 151
Performance Task
Kites and Congruence
Provide a clear solution to the problem. Be sure to show all of your work, include
all relevant drawings, and justify your answers.
In a kite design, four triangular pieces of fabric are sewn together to form a
quadrilateral. Two rods are attached to the kite so that one rod bisects the angles,
as shown.
B
E
A
C
D
Part A
The manufacturer wants to create templates for the triangular shapes that need to be
AC bisects ¯
BD where
cut. If m∠DCA = 50, AB = 20 in., EC = 10 in., m∠ABC = 93, and ¯
BD = 24 in., solve for all angles and side lengths for each triangle in the figure. Label the
angle measures in the figure. Name all the congruent triangles shown within the design.
B
A
E
152 CHAPTER 4 Congruent Triangles
Copyright © McGraw-Hill Education
D
C
Part B
One worker notices that the rods intersect at right angles. Write a paragraph proof to
prove that the rods will always intersect at right angles.
Part C
Copyright © McGraw-Hill Education
The manufacturer will use a bolt of fabric measuring 60 inches wide and 130 yards long.
Draw a model and find how many kite patterns can be cut from this bolt of fabric.
CHAPTER 4 Performance Task 153
Standardized Test Practice
1. The coordinates of the vertices of a triangle are
(0, 0), (a, 0), and (b, c). The area of this triangle
square units. is
5. The triangles below can be shown to be congruent
. by
G.CO.8
G.GPE.4
2. Shelley has drawn two triangles, XYZ and
TUV. She knows that ∠X ∠T and ∠Y ∠U.
In order to prove that XYZ TUV using AAS,
she also needs to know that
. or
G.CO.8
3. Archeologists have found two triangular building
foundations. The diagram shows some
measurements from the foundations. Which of
the following additional measurements would be
sufficient to prove that the two triangles are
congruent? G.MG.1
6. Which of the following statements are true for
the following diagram? G.SRT.5
N
M
E
B
36˚
P
67 ft
A
65 ft
41 ft
O
Q
C
D
MQN MQP PQO OQN
MNP OPN MPO MNO
69˚
41 ft
F
ONQ OPQ QPM QPO
m∠E = 36 and DE = 67 ft
7. Consider the following diagram. BC = 65 ft and DE = 67 ft
G.CO.8
L
M
K
m∠E = 36 and m∠A = 69
J
4. William draws triangle PQR. He rotates,
translates, and reflects the triangle to create
a new triangle with vertices S, T, and V. He
finds that ∠P ∠T, ∠Q ∠S, and ∠R ∠V.
From this, William determines that he can
write the congruence statement
. G.CO.7
154 CHAPTER 4 Congruent Triangles
JKN JN means that ¯
N
by
or
. This
.
8. The HL Congruence Theorem states that in two
triangles, if the
are
are
congruent and corresponding
congruent, the triangles are congruent. G.CO.10
Copyright © McGraw-Hill Education
m∠A = 69 and DE = 67 ft
9. Complete the following proof. G.CO.10
AB ¯
BC
Given: ¯
B
Prove: ∠A ∠C
A
C
Statement
Reason
¯
AB ¯
BC
¯
Draw BD bisecting ∠ABC
Definition of angle bisector
¯
BD ¯
BD
SAS
∠A ∠C
10. Consider the following diagram G.SRT.5
B
D
C
A
E
a. What can you conclude about ABE and EDA? Justify your answer. b. What can you conclude about BCA and DCE? Justify your answer. c. If AE = 8 and the perimeter of EDA is 22, what is the perimeter of DCE? Explain how you know. Copyright © McGraw-Hill Education
11. The vertices of ABC are A(-2, -4), B(-1, 1) and C(3, -2). The vertices of DEF are D(-3, 2), E(2, 3) and
F(6, 0). G.GPE.4, G.CO.7
a. Are the triangles congruent? Justify your answer.
b. The vertices of GHI are G(-2, 0), H(1, 4), and I(3, -1). Is GHI congruent to either of ABC or
DEF? Justify your answer.
CHAPTER 4 Standardized Test Practice 155