C H A P T E R
6 Congruence
6.1 Constructing Congruent Triangles or NotConstructing Triangles | p. 311
6.2 Congruence TheoremsSSS, SAS, ASA, and AAS | p. 319
6.3 Right Triangle Congruence TheoremsHL, LL, HA, and LA | p. 329
6.4 CPCTCCorresponding Parts of Congruent
Triangles are Congruent | p. 337
6.5 Isosceles Triangle TheoremsIsosceles Triangle Base Theorem,
Vertex Angle Theorem, Perpendicular
Bisector Theorem, Altitude to
Congruent Sides Theorem, and
Angle Bisector to Congruent Sides
Theorem | p. 343
6.6 Direct Proof vs. Indirect ProofInverse, Contrapositive, Direct Proof,
and Indirect Proof | p. 349
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Victorian houses in San Francisco, known as “Painted Ladies” for their bright exterior
colors, often share basic designs and floor plans. As a result, while the houses in this
image are decorated differently, their third floor dormers form congruent triangles,
triangles that have the same shape and the same size. You will explore the properties
of congruent triangles and use them as the basis for constructing mathematical proofs.
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Introductory Problem for Chapter 6
Is Congruence a Special Case of Similarity or is Similarity a Special Case of Congruence?
Similar triangles are triangles that have the same shape.
A C
B
1. Construct �DEF so that �ABC � �DEF.
2. Describe the steps taken to construct �DEF.
3. Describe the relationship between the corresponding angles of the
similar triangles.
4. Describe the relationship between the corresponding sides of the
similar triangles.
Congruent triangles are triangles that have the same shape and the same size.
A
B
C
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Chapter 6 | Introductory Problem for Chapter 6 309
5. Construct �GHI so that �ABC � �GHI.
6. Describe the steps taken to construct �GHI.
7. Describe the relationship between the corresponding angles of the
congruent triangles.
8. Describe the relationship between the corresponding sides of the
congruent triangles.
9. Cessia says that all similar triangles are congruent. Ricky says that all
congruent triangles are similar. Who is correct? Explain.
Be prepared to share your solutions and methods.
6
Lesson 6.1 | Constructing Congruent Triangles or Not 311
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Two polygons are congruent if they are the same shape and size.
1. Construct �ABC using the two line segments shown. Write the steps.
A
A C
B
PROBLEM 1 Construction
A Triangle Given Two Line Segments
Constructing Congruent Triangles or NotConstructing Triangles
6.1
OBJECTIVEIn this lesson you will:l Construct triangles to determine uniqueness.
6
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2. Classify �ABC based on the angles and the sides.
3. Compare the triangle that you constructed with the triangles that your
classmates constructed. What do you observe? Why?
4. Name the included angle for sides AB and AC.
5. Measure the included angle for sides AB and AC. Compare the measure of
your included angle with the measures of the included angles of your
classmates. What do you observe?
6. How does the measure of the included angle affect the length of side BC
of �ABC?
7. How many different triangles can be determined given the length of two sides
of a triangle?
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PROBLEM 2 Construction
A Triangle Given Three Line Segments 1. Construct �ABC using the three line segments shown. Write the steps.
A
B C
A C
B
2. Classify �ABC based on the angles and the sides.
3. Compare the triangle that you constructed with the triangles that your
classmates constructed. What do you observe? Why?
4. How many different triangles can be determined given the length of three sides
of a triangle?
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6 1. Construct �ABC using the two line segments and included angle shown.
Write the steps.
A
A
A
C
B
2. Classify �ABC based on the angles and the sides.
3. Compare the triangle that you constructed with the triangles that your class-
mates constructed. What do you observe? Why?
4. Could everyone construct an identical triangle if they were given �C or �B, the
angles that are not included? Explain.
PROBLEM 3 Construction
A Triangle Given Two Line Segments and the Included Angle
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5. How many different triangles can be determined given:
a. The length of two sides of a triangle and the included angle?
b. The length of two sides of a triangle and the angle not included?
PROBLEM 4 Construction
A Triangle Given Three Angles 1. Construct �ABC using the three angles shown. Write the steps.
AB
C
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2. Compare the triangle that you constructed with the triangles that your class-
mates constructed. What do you observe? Why?
3. Could everyone construct an identical triangle if they were given only two angles
of a triangle? Explain.
4. How many different triangles can be determined given three interior angles of
a triangle?
PROBLEM 5 Construction
A Triangle Given Two Angles and One Line Segment 1. Construct a triangle using the two angles and the line segment shown.
Write the steps.
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2. Compare the triangle that you constructed with the triangles that your
classmates constructed. What do you observe? Why?
3. How many different triangles can be determined given the measure of two
angles of a triangle and the length of one side? Explain your reasoning.
1. List all combinations of givens that determine a unique triangle.
2. List all combinations of givens that determine multiple triangles.
3. Did you use inductive or deductive reasoning to answer Questions 1 and 2?
Be prepared to share your solutions and methods.
PROBLEM 6 Summary
Lesson 6.2 | Congruence Theorems 319
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6OBJECTIVESIn this lesson you will:l Identify congruent parts of triangles by SSS,
SAS, ASA, and AAS.l List the corresponding congruent parts of two
triangles from a congruence statement.l Distinguish between the Similarity Postulates
and the Congruence Theorems.
KEY TERMSl Side-Side-Side (SSS)
Congruence Theoreml Side-Angle-Side (SAS)
Congruence Theoreml Angle-Side-Angle (ASA)
Congruence Theoreml Angle-Angle-Side (AAS)
Congruence Theorem
6.2
PROBLEM 1 Making ConjecturesYou have identified various cases when a unique triangle can be constructed using
given sides or angles. This demonstrates that there is a congruent relationship between
two constructed triangles.
1. Write a conjecture for each congruent triangle relationship.
a. Given three sides.
b. Given two sides and the included angle.
c. Given two angles and one specified side.
These conjectures provide the basis for four theorems.
Congruence TheoremsSSS, SAS, ASA, and AAS
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The Side-Side-Side (SSS) Congruence Theorem states: “If three sides of one
triangle are congruent to the corresponding sides of another triangle, then the
triangles are congruent.”
1. Complete the two-column proof of the SSS Congruence Theorem.
B
AC
E
DF
Given: ___
AC � ___
DF , ___
AB � ___
DE , ___
BC � ___
EF
Prove: �ABC � �DEF
Statements Reasons
1. ___
AC � ___
DF , ____
AB � ____
DE , BC � ___
EF 1.
2. AC � , � DE, BC � 2. Definition of Congruence
3. AC ____ DF
� AB ____ DE
� BC ____ EF
� 3. Division Property of Equality
4. ____ DF
� AB ____ � BC ____ 4. Transitive Property of Equality
5. �ABC � �DEF 5.
6. �A � , ��E, �C � 6. Definition of similar triangles
7. �ABC � �DEF 7. Definition of congruent triangles
2. What is the difference between the SSS Similarity Postulate and the SSS
Congruence Theorem?
PROBLEM 2 Congruence Theorems
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The Side-Angle-Side (SAS) Congruence Theorem states: “If two sides and the
included angle of one triangle are congruent to the corresponding two sides and the
included angle of a second triangle, then the two triangles are congruent.”
3. Create a two-column proof of the SAS Congruence Theorem.
A C
B E
D F
Given: ___
AB � ___
DE , ___
AC � ___
DF , and �A � �D
Prove: �ABC � �DEF
Statements Reasons
4. What is the difference between the SAS Similarity Postulate and the SAS
Congruence Theorem?
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The Angle-Side-Angle (ASA) Congruence Theorem states: “If two angles and the
included side of one triangle are congruent to the corresponding two angles and the
included side of another triangle, the triangles are congruent.”
You will complete the proof of the ASA Congruence Theorem in the assignments for
this lesson.
5. Mark the appropriate sides and/or angles to prove �RWX � �CMT by ASA.
X T
R C
W M
6. Draw �ABC � �ABD by ASA and include appropriate markers.
The Angle-Angle-Side (AAS) Congruence Theorem states: “If two angles and the
non-included side of one triangle are congruent to the corresponding two angles and
the non-included side of another triangle, the triangles are congruent.”
You will complete the proof of the AAS Congruence Theorem in the assignments for
this lesson.
7. Mark the appropriate sides and/or angles to prove �RWX � �CMT by AAS.
X T
R C
W M
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8. Draw �ABC � �ABD by AAS and include appropriate markers.
9. Ricardo said the AAS method for proving two triangles congruent is really the
ASA method in disguise. Is Ricardo correct? Explain.
10. Is SAA a method for proving two triangles congruent? Explain.
11. What is the ratio of corresponding sides for two congruent triangles? Explain.
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1. Draw �JRB � �EMS. List the six pairs of congruent corresponding parts.
2. List the six pair of congruent corresponding parts if �GNP � �WCA.
3. Based on each hypothesis, is there enough information to conclude the
triangles shown are congruent? Explain your reasoning. Name the Congruence
Theorem, if applicable.
a. If ___
AB � ___
CD and ___
AE � ___
CE , is there enough information to determine whether
�ABE � �CDE?
A
E D
C
B
PROBLEM 3 Applying the Congruence Theorems
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b. If ___
RF � ___
BP and ___
BF � ___
RP , is there enough information determine whether
�RFP � �BPF?
R B
T
F P
c. If ____
WN � ___
HK , is there enough information to determine whether
�WNZ � �HKZ?
W
N
Z
K
H
d. If ___
JA � ____
MY and ____
YM bisects �JYA, is there enough information to determine
whether �JYM � �AYM?
J M
Y
A
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e. If ___
ST � ___
AR and ___
RS � ___
TA , is there enough information to determine whether
�STR � �ART?
S T
R A
f. If ____
GU bisects �BGD and ___
GB � ____
GD , is there enough information to determine
whether �GUD � �GUB?
G
D
B
U
g. If �CKM � �EKV, ___
CK � ___
EK , and ___
KV � ____
KM , is there enough information to
determine whether �KCV � �KEM?
M
V
E
C
K
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h. If �X � �Y and ___
BX � ___
BY , is there enough information to determine whether
�BXE � �BYT?
X
D
E YB
T
4. Draw two congruent triangles that share a common side and write the
congruence statement.
5. Draw two congruent triangles that share a common angle and write the
congruence statement.
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6. Six teams of students have the task of locating a specific point on an outdoor
basketball court. Determine whether each team can locate the point and
explain your reasoning.
a. The first team was given the distance between the posts for the basketball
hoops and the distance from each post to the point.
b. The second team was given the distance between the posts for the
basketball hoops and the measure of the angles from each post to the point.
c. The third team was given the distance between the posts for the basketball
hoops, the distance from one post to the point, and the measure of the
angle from the other post to the point.
d. The fourth team was given the distance between the posts for the basketball
hoops, the distance from one post to the point, and the measure of the
angle from that post to the point.
e. The fifth team was given the distance between the posts for the basketball
hoops, the measure of the angle from one post to the point, and the
measure of the angle from that point to the posts.
f. The sixth team was given the measures of all the angles.
Be prepared to share your solutions and methods.
Lesson 6.3 | Right Triangle Congruence Theorems 329
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1. List all of the Triangle Congruence Theorems.
2. How many pairs of corresponding parts are used in each congruence theorem?
The Congruence Theorems apply to all triangles. There are also theorems that only
apply to right triangles. Methods for proving two right triangles congruent are
somewhat shorter. You can prove two right triangles congruent using only two pairs
of corresponding parts.
3. Explain why only two pairs of corresponding parts are necessary to prove
two right triangles are congruent. What is special about right triangles that will
shorten the steps to prove two are congruent?
Let’s explore these methods.
OBJECTIVESIn this lesson you will:l Use given information to show two
right triangles are congruent.
l Prove the HL Congruence Theorem.
l Prove the LL Congruence Theorem.
l Prove the HA Congruence Theorem.
l Prove the LA Congruence Theorem.
KEY TERMSl Hypotenuse-Leg (HL) Congruence
Theoreml Leg-Leg (LL) Congruence Theoreml Hypotenuse-Angle (HA) Congruence
Theoreml Leg-Angle (LA) Congruence Theorem
6.3
PROBLEM 1 Right Triangle Congruence Theorems
Right Triangle Congruence TheoremsHL, LL, HA, and LA
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The Hypotenuse-Leg (HL) Congruence Theorem states: “If the hypotenuse and
leg of one right triangle are congruent to the hypotenuse and leg of another right
triangle, then the triangles are congruent.”
4. Create a two-column proof of the HL Congruence Theorem
A F E
C B D
Given: �C and �F are right angles
___
AC � ___
DF
___
AB � ___
DE
Prove: �ABC � �DEF
Statements Reasons
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The Leg-Leg (LL) Congruence Theorem states: “If two legs of one right triangle are
congruent to two legs of another right triangle, then the triangles are congruent.”
5. Create a two-column proof of the LL Congruence Theorem.
A F E
C B D
Given: �C and �F are right angles
___
AC � ___
DF
___
CB � ___
FE
Prove: �ABC � �DEF
Statements Reasons
The Hypotenuse-Angle (HA) Congruence Theorem states: “If the hypotenuse and
an acute angle of one right triangle are congruent to the hypotenuse and acute
angle of another right triangle, then the triangles are congruent.”
6. Create a two-column proof of the HA Congruence Theorem.
A F E
C B D
Given: �C and �F are right angles
___
AB � ___
DE
�A � �D
Prove: �ABC � �DEF
Statements Reasons
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The Leg-Angle (LA) Congruence Theorem states: “If a leg and an acute angle of
one right triangle are congruent to a leg and an acute angle of another right triangle,
then the triangles are congruent.”
7. Create a two-column proof of the LA Congruence Theorem.
A F E
C B D
Given: �C and �F are right angles
___
AC � ___
DF
�A � �D
Prove: �ABC � �DEF
Statements Reasons
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Determine if there is enough information to prove the two triangles are congruent.
If so, name the Congruence Theorem used.
1. If ___
CS � ___
SD , ____
WD � ___
SD , and P is the midpoint of ____
CW , is �CSP � �WDP?
C
S
P D
W
2. If ___
RF � ___
FP , ___
BP � ___
FP , and ___
RP and ___
FB bisect each other, is �RFP � �BPF?
F P
BR
T
PROBLEM 2 Applying Right Triangle Congruence Theorems
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3. Pat always trips on the third step and she thinks that step may be a different
size. The contractor told her that all the treads and risers are perpendicular
to each other. Is that enough information to state that the steps are the same
size? In other words, if ____
WN � ___
NZ and ___
ZH � ___
HK , is �WNZ � �ZHK?
4. If ___
JA � ____
MY and ___
JY � ___
AY , is �JYM � �AYM?
J M
Y
A
5. If ___
ST � ___
SR , ___
AT � ___
AR , and �STR � �ATR, is �STR � �ATR?
S
R
A
T
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It is necessary to make a statement about the presence of right triangles when you
use the Right Triangle Congruence Theorems. If you have previously identified them,
the reason is the definition of right angles.
6. Create a two-column proof of the following.
Given: ____
GU � ___
DB
D
UG
B
___
GB � ____
GD
Prove: �GUD � �GUB
Statements Reasons
7. Create a two-column proof of the following.
Given: ____
GU is the � bisector of ___
DB
D
UG
B
Prove: �GUD � �GUB
Statements Reasons
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8. A friend wants to place a post in a lake 20 feet straight out from the dock.
What is the minimum information you need to make sure the angle formed
by the edge of the dock and the post is a right angle?
Be prepared to share your solutions and methods.
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6OBJECTIVESIn this lesson you will:l Identify corresponding parts of
congruent triangles.l Use corresponding parts of congruent triangles
are congruent to prove angles and segments are congruent.
l Use corresponding parts of congruent triangles are congruent to prove the Isosceles Triangle Base Angle Theorem.
l Use corresponding parts of congruent triangles are congruent to prove the Isosceles Triangle Base Angle Converse Theorem.
KEY TERMSl corresponding parts of congruent
triangles are congruent (CPCTC)l Isosceles Triangle Base
Angle Theoreml Isosceles Triangle Base
Angle Converse Theorem
6.4 CPCTCCorresponding Parts of Congruent Triangles are Congruent
PROBLEM 1 CPCTCIf two triangles are congruent, then each part of one triangle is congruent to the
corresponding part of the other triangle. “Corresponding parts of congruent triangles are congruent,” abbreviated as CPCTC, is often used for a reason in
proof problems. CPCTC states that corresponding angles or sides in two congruent
triangles are congruent. This reason can only be used after you have proven that the
triangles are congruent.
To use CPCTC in a proof, follow these steps:
Step 1: Identify two triangles in which segments or angles are
corresponding parts.
Step 2: Prove the triangles congruent.
Step 3: State the two parts are congruent using CPCTC as the reason.
6
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1. Create a two-column proof.
Given: ____
CW and ___
SD bisect each other
Prove: ___
CS � ____
WD
Statements Reasons
2. Create a two-column proof.
Given: ___
SU � ___
SK , ___
SR � ___
SH
Prove: �U � �K
Statements Reasons
C
S
P D
W
S H K
R
U
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CPCTC makes it possible to prove other theorems. For example:
The Isosceles Triangle Base Angle Theorem states: “If two sides of a triangle are
congruent, then the angles opposite these sides are congruent.”
To prove the Isosceles Triangle Base Angle Theorem, you need to add an auxiliary
line
to an isosceles triangle that bisects the vertex angle as shown.
1. Create a two-column proof.
Given: ___
GB � ____
GD
Prove: �B � �D
D
UG
B
Statements Reasons
PROBLEM 2 Isosceles Triangle Base Angle Theorem and Its Converse
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The Isosceles Triangle Base Angle Converse Theorem states: “If two angles of a
triangle are congruent, then the sides opposite these angles are congruent.”
To prove the Isosceles Triangle Base Angle Converse Theorem, you need to add an
auxiliary line to an isosceles triangle that bisects the vertex angle as shown.
2. Create a two-column proof. D
UG
B
Given: �B � �D
Prove: ___
GB � ____
GD
Statements Reasons
1. How wide is the horse’s pasture?
PROBLEM 3 Applications
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2. Calculate AP if the perimeter of �AYP is 43 cm.
P
Y
A 13 cm
70°
70°
3. Lighting booms on a Ferris wheel consist of four steel beams that have cabling
with light bulbs attached. These beams along with 3 shorter beams form the
edges of three congruent isosceles triangles as shown. Maintenance crews
are installing new lighting along the four beams. Calculate the total length of
lighting needed.
4. Calculate m�T.
W M
117°
T
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5. What is the width of the river?
6. Given: ___
ST � ___
SR , ___
TA � ___
RA
Explain why �T � �R.
S
R
A
T
Be prepared to share your solutions and methods.
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OBJECTIVESIn this lesson you will:l Prove the Isosceles Triangle
Base Theorem.l Prove the Isosceles Triangle Vertex
Angle Theorem.l Prove the Isosceles Triangle
Perpendicular Bisector Theorem.l Prove the Isosceles Triangle Altitude to
Congruent Sides Theorem.l Prove the Isosceles Triangle Angle
Bisector to Congruent Sides Theorem.
KEY TERMSl vertex anglel Isosceles Triangle Base Theoreml Isosceles Triangle Vertex Angle Theoreml Isosceles Triangle Perpendicular
Bisector Theoreml Isosceles Triangle Altitude to Congruent
Sides Theoreml Isosceles Triangle Angle Bisector to
Congruent Sides Theorem
6.5 Isosceles Triangle TheoremsIsosceles Triangle Base Theorem, Vertex Angle Theorem, Perpendicular Bisector Theorem, Altitude to Congruent Sides Theorem, and Angle Bisector to Congruent Sides Theorem
PROBLEM 1 Isosceles Triangle TheoremsYou will prove theorems related to isosceles triangles. These proofs involve altitudes,
perpendicular bisectors, angle bisectors, and vertex angles. The vertex angle is the
angle formed by the two congruent legs in an isosceles triangle.
1. Given: Isosceles �ABC with ___
CA � ___
CB .
Construct altitude ___
CD from the vertex angle to the base.
A B
C
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a. Create a flow chart proof to prove ___
AD � ___
BD .
b. Create a two-column proof.
Given: Isosceles �ABC with ___
CA � ___
CB
Prove: ___
AD � ___
BD
Statements Reasons
Congratulations! You have just proven a theorem!
The Isosceles Triangle Base Theorem states: “The altitude to the base of an
isosceles triangle bisects the base.” You can now use this theorem as a valid reason
in proofs.
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2. In isosceles �ABC, with altitude ___
CD , explain how to prove �ACD � �BCD.
Congratulations! You have just explained how to prove another theorem!
The Isosceles Triangle Vertex Angle Theorem states: “The altitude to the base of
an isosceles triangle bisects the vertex angle.” You can now use this theorem as a
valid reason in proofs.
3. In isosceles �ABC, explain how to prove altitude ___
CD is the � bisector of ___
AB .
Congratulations! You have just explained how to prove another theorem!
The Isosceles Triangle Perpendicular Bisector Theorem states: “The altitude from
the vertex angle of an isosceles triangle is the perpendicular bisector of the base.”
You can now use this theorem as a valid reason in proofs.
The Isosceles Triangle Altitude to Congruent Sides Theorem states: “In an
isosceles triangle, the altitudes to the congruent sides are congruent.”
1. Draw and label a diagram for proving this theorem.
2. State the “Given” and “Prove” statements.
Given:
Prove:
PROBLEM 2 More Isosceles Triangle Theorems
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3. Write a paragraph proof of the Isosceles Triangle Altitude to Congruent
Sides Theorem.
4. Create a two-column proof of the Isosceles Triangle Altitude to Congruent
Sides Theorem.
Statements Reasons
The Isosceles Triangle Angle Bisector to Congruent Sides Theorem states:
“In an isosceles triangle, the angle bisectors to the congruent sides are congruent.”
5. Draw and label a diagram to prove this theorem.
6. State the “Given” and “Prove” statements.
Given:
Prove:
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7. Write a paragraph proof of the Isosceles Triangle Angle Bisector to Congruent
Sides Theorem.
8. Create a two-column proof of the Isosceles Triangle Angle Bisector to
Congruent Sides Theorem.
Statements Reasons
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1. Solve for the width of the dog house.
___
CD � ___
AB
___
AC � ___
BC
CD � 12�
AC � 20�
PROBLEM 4 Summary
PROBLEM 3 Dog House
Use the theorems you have just proven to answer each question about
isosceles triangles.
1. What can you conclude about an altitude drawn from the vertex angle to
the base?
2. What can you conclude about the altitudes to the congruent sides?
3. What can you conclude about the angle bisectors to the congruent sides?
Be prepared to share your solutions and methods.
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6OBJECTIVESIn this lesson you will:l Write the inverse of a conditional
statement.l Differentiate between direct and
indirect proof.l Use indirect proof.
KEY TERMSl inversel contrapositivel direct proofl indirect proof or proof by contradictionl Hinge Theoreml Hinge Converse Theorem
6.6 Direct Proof vs. Indirect ProofInverse, Contrapositive, Direct Proof, and Indirect Proof
PROBLEM 1 The Inverse and ContrapositiveEvery conditional statement written in the form “If p, then q” has three additional
conditional statements associated with it: converse, contrapositive, and inverse.
Recall from previous lessons, to state the converse, reverse the hypothesis, p,
and the conclusion, q. To state the inverse, negate both parts. To state the
contrapositive, negate each part and reverse them.
Conditional Statement If p, then q.
Converse If q, then p.
Inverse If not p, then not q.
Contrapositive If not q, then not p.
For each conditional statement written in propositional form, identify the hypothesis p
and the conclusion q. Identify the negation of the hypothesis and conclusion, and
then write the inverse and contrapositive of the conditional statement.
1. If a quadrilateral is a square, then the quadrilateral is a rectangle.
a. Hypothesis p:
b. Conclusion q:
c. Is the conditional statement true? Explain.
d. Not p:
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e. Not q:
f. Inverse:
g. Is the inverse true? Explain.
h. Contrapositive:
i. Is the contrapositive true? Explain.
2. If an integer is even, then the integer is divisible by two.
a. Hypothesis p:
b. Conclusion q:
c. Is the conditional statement true? Explain.
d. Not p:
e. Not q:
f. Inverse:
g. Is the inverse true? Explain.
h. Contrapositive:
i. Is the contrapositive true? Explain.
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3. If a polygon has six sides, then the polygon is a pentagon.
a. Hypothesis p:
b. Conclusion q:
c. Is the conditional statement true? Explain.
d. Not p:
e. Not q:
f. Inverse:
g. Is the inverse true? Explain.
h. Contrapositive:
i. Is the contrapositive true? Explain.
4. If two lines intersect, then the lines are perpendicular.
a. Hypothesis p:
b. Conclusion q:
c. Is the conditional statement true? Explain.
d. Not p:
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e. Not q:
f. Inverse:
g. Is the inverse true? Explain.
h. Contrapositive:
i. Is the contrapositive true? Explain.
5. What do you notice about the truth value of a conditional statement and the
truth value of its inverse?
6. What do you notice about the truth value of a conditional statement and the
truth value of its contrapositive?
Lesson 6.6 | Direct Proof vs. Indirect Proof 353
6
All of the proofs up to this point were direct proofs. A direct proof begins with the
given information and works to the desired conclusion directly through the use of
givens, definitions, properties, postulates, and theorems.
An indirect proof is different and may be shorter than a direct proof. An indirect proof, or proof by contradiction, uses the contrapositive. If you prove the contrapositive
true, then the statement is true. Begin by assuming the conclusion is false and use
this assumption to show one of the given statements is false, thereby creating a
contradiction.
In an indirect proof:
• State the assumption; use the negation of the conclusion or prove statement.
• Write the givens.
• Write the negation of the conclusion.
• Use the assumption, in conjunction with definitions, properties, postulates, and
theorems, to prove a given statement is false, thus creating a contradiction.
Hence, your assumption leads to a contradiction; therefore, the assumption must be
false. This proves the contrapositive.
Let’s look at an example of an indirect proof.
Given: In �CHT, ___
CH � ___
CT
___
CA does not bisect ___
HT
Prove: �CHA � �CTA
H A
C
T
Statements Reasons
1. �CHA � �CTA 1. Assumption
2. ___
CA does not bisect ___
HT 2. Given
3. ___
HA � ___
TA 3. CPCTC
4. ___
CA bisects ___
HT 4. Defi nition of bisect
5. �CHA � �CTA is false 5. This is a contradiction.
Step 4 contradicts step 2;
the assumption is false
6. �CHA � �CTA is true 6. Proof by contradiction
In step 5, the “assumption” is stated as “false.” The reason for making this
statement is “contradiction.”
PROBLEM 2 Proof by Contradiction©
201
0 C
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Lear
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, Inc
.
Take NoteNotice, you are
trying to prove
�CHA � �CTA.
You assume the
negation of this
statement,
�CHA � �CTA.
This becomes the
first statement in
your proof and the
reason for making
this statement is
“assumption.”
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Now try one yourself!
1. Given: ___
BR bisects �ABN A
B
R
N
�BRA � �BRN
Prove: ___
AB � ___
NB
Statements Reasons
2. When writing an indirect proof, it is often easier to write it as a paragraph proof.
Write the proof in Question 1 as a paragraph proof.
3. Use a paragraph proof model to write an indirect proof proving a triangle
cannot have more than one right angle.
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The Hinge Theorem states: “If two sides of one triangle are congruent to two sides
of another triangle and the included angle of the first pair is larger than the included
angle of the second pair, then the third side of the first triangle is longer than the
third side of the second triangle.”
In the two triangles shown, notice that RS � DE, ST � EF, and �S � �E. The Hinge
Theorem guarantees that RT � DE.
R
S T100°
D
E F80°
1. Use an indirect proof to prove the Hinge Theorem.
Begin by restating the Hinge Theorem using �ABC and �DEF.
If sides AB � DE and AC � DF, and the included angle at A is larger than the
included angle at D, then BC � EF.
A
B
C
D
EF
Given: AB � DE
AC � DF
m�A � m�D
Prove: BC � EF
PROBLEM 3 Hinge Theorem and Its Converse
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This theorem must be proven for two cases:
Case 1: BC � EF
Case 2: BC � EF
a. Write the indirect proof for Case 1.
Statements Reasons
b. Write the indirect proof for Case 2.
Statements Reasons
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The Hinge Converse Theorem states: “If two sides of one triangle are congruent to
two sides of another triangle and the third side of the first triangle is longer than the
third side of the second triangle, then the included angle of the first pair of sides is
larger than the included angle of the second pair of sides.”
In the two triangles shown, notice that RT � DF, RS � DE, and ST � EF. The Hinge
Converse Theorem guarantees that m�R � m�D.
S
R
T
10
E
D
8
F
2. Create an indirect proof to prove the Hinge Converse Theorem.
A
C
B
D
E F
Given: AB � DE
AC � DF
BC � EF
Prove: m�A � m�D
This theorem must be proven for two cases:
Case 1: m�A � m�D
Case 2: m�A � m�D
a. Create an indirect proof for Case 1.
Statements Reasons
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b. Create an indirect proof for Case 2.
Statements Reasons
3. Use the Hinge Theorem and its converse to answer each.
a. Matthew and Jeremy’s families are going
camping for the weekend. Before heading out
of town, they decide to meet at Al’s Diner for
breakfast. During breakfast, the boys try to
decide which family will be further away from
the diner “as the crow flies.” “As the crow flies”
is an expression based on the fact that crows,
generally fly straight to the nearest food supply.
Matthew’s family is driving 35 miles due north
and taking an exit to travel an additional
15 miles northeast. Jeremy’s family is driving
35 miles due south and taking an exit to travel
an additional 15 miles southwest. Use the
diagram shown to determine which family is
further from the diner. Explain your reasoning.
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b. Which of the following is a possible length for AH: 20 cm, 21 cm, or 24 cm?
Explain your choice.
55°
21 cm
W
E P
61°
A
HR
c. Which of the following is a possible angle measure for �ARH: 54º, 55º or
56º? Explain your choice.
55°
34 mm
W
PE
A
HR
36 mm
Be prepared to share your solutions and methods.
360 Chapter 6 | Congruence
6KEY TERMSl corresponding parts of
congruent triangles are congruent (CPCTC) (6.4)
l angle bisector (6.5)
l perpendicular bisector (6.5)l vertex angle (6.5)l inverse (6.6)l contrapositive (6.6)
l direct proof (6.6)l indirect proof or proof by
contradiction (6.6)
THEOREMSl Side-Side-Side (SSS)
Congruence Theorem (6.2)l Side-Angle-Side (SAS)
Congruence Theorem (6.2)l Angle-Side-Angle (ASA)
Congruence Theorem (6.2)l Angle-Angle-Side (AAS)
Congruence Theorem (6.2)l Hypotenuse-Leg (HL)
Congruence Theorem (6.3)l Leg-Leg (LL) Congruence
Theorem (6.3)
l Hypotenuse-Angle (HA) Congruence Theorem (6.3)
l Leg-Angle (LA) Congruence Theorem (6.3)
l Isosceles Triangle Base Angle Theorem (6.4)
l Isosceles Triangle Base Angle Converse Theorem (6.4)
l Isosceles Triangle Base Theorem (6.5)
l Isosceles Triangle Vertex Angle Theorem (6.5)
l Isosceles Triangle Perpendicular Bisector Theorem (6.5)
l Isosceles Triangle Altitude to Congruent Sides Theorem (6.5)
l Isosceles Triangle Angle Bisector to Congruent Sides Theorem (6.5)
l Hinge Theorem (6.6)l Hinge Converse
Theorem (6.6)
CONSTRUCTIONSl triangle given two line
segments (6.1)l triangle given three line
segments (6.1)
l triangle given two line segments and the included angle (6.1)
l triangle given three angles (6.1)
l triangle given two angles and one line segment (6.1)
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Constructing Triangles
By constructing triangles, you discovered that all combinations of givens that
determine a unique triangle are:
1. Given three sides
2. Given two sides and the included angle
3. Given two angles and a specific side
You discovered that all combinations of givens that determine multiple triangles are:
1. Given two sides
2. Given two or three angles
3. Given two sides and an angle not included
4. Given two angles and a side not specified
Using the Side-Side-Side (SSS) Congruence Theorem
The Side-Side-Side (SSS) Congruence Theorem states: “If three sides of one triangle
are congruent to the corresponding sides of another triangle, then the triangles are
congruent.”
Example:
R
S
T
5 m
6 m
7 m
F
G
H5 m
6 m
7 m
___
FG � ___
RS , ___
GH � ___
ST , and ___
FG � ___
RT , so �FGH � �RST.
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Using the Side-Angle-Side (SAS) Congruence Theorem
The Side-Angle-Side (SAS) Congruence Theorem states: “If two sides and the
included angle of one triangle are congruent to the corresponding two sides and the
included angle of another triangle, then the triangles are congruent.”
Example:
X
10 in.
Y
Z
17 in.44°
L
M
N
10 in.
17 in.
44°
___
XY � ___
LM , �Y � �M, and ___
YZ � ____
MN , so �XYZ � �LMN.
Using the Angle-Side-Angle (ASA) Congruence Theorem
The Angle-Side-Angle (ASA) Congruence Theorem states: “If two angles and the
included side of one triangle are congruent to the corresponding two angles and the
included side of another triangle, the triangles are congruent.”
Example:
J K
L
15 cm
87° 62°
D
E F62°
87°15 cm
�D � �J, ___
DE � ___
JK , and �E � �K, so �DEF � �JKL.
Using the Angle-Angle-Side (AAS) Congruence Theorem
The Angle-Angle-Side (AAS) Congruence Theorem states: “If two angles and the
non-included side of one triangle are congruent to the corresponding two angles and
the non-included side of another triangle, the triangles are congruent.”
Example:
P
Q
R
2 ft
35°
115°
V
W
X
115° 35°2 ft
�P � �V, �Q � �W, and ___
QR � ____
WK , so �PQR � �VWX.
6.2
6.2
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Using the Hypotenuse-Leg (HL) Congruence Theorem
The Hypotenuse-Leg (HL) Congruence Theorem states: “If the hypotenuse and leg of
one right triangle are congruent to the hypotenuse and leg of another right triangle,
then the triangles are congruent.”
Example:
6 in.
A
B
C3 in.
F
D
E6 in.
3 in.
___
BC � ___
EF , ___
AC � ___
DF , and angles A and D are right angles, so �ABC � �DEF.
Using the Leg-Leg (LL) Congruence Theorem
The Leg-Leg (LL) Congruence Theorem states: “If two legs of one right triangle are
congruent to two legs of another right triangle, then the triangles are congruent.”
Example:
12 ft
11 ftX
Y
Z
12 ft
11 ft
RS
T
___
XY � ___
RS , ___
XZ � ___
RT , and angles X and R are right angles, so �XYZ � �RST.
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Using the Hypotenuse-Angle (HA) Congruence Theorem
The Hypotenuse-Angle (HA) Congruence Theorem states: “If the hypotenuse and an
acute angle of one right triangle are congruent to the hypotenuse and an acute angle
of another right triangle, then the triangles are congruent.”
Example:
D
E
FJ
K
L
10 m
32°
32°10 m
___
KL � ___
EF , �L � �F, and angles J and D are right angles, so �JKL � �DEF.
Using the Leg-Angle (LA) Congruence Theorem
The Leg-Angle (LA) Congruence Theorem states: “If a leg and an acute angle of one
right triangle are congruent to the leg and an acute angle of another right triangle,
then the triangles are congruent.”
Example:
L
M
N
9 mm
51°
9 mm
51°
G
JH
____
GN � ___
LN , �H � �M, and angles G and L are right angles,
so �GHJ � �LMN.
6.3
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Using CPCTC to Solve a Problem
If two triangles are congruent, then each part of one triangle is congruent to the
corresponding part of the other triangle. In other words, “corresponding parts of
congruent triangles are congruent,” which is abbreviated CPCTC. To use CPCTC,
first prove that two triangles are congruent.
Example:
You want to determine the distance between two docks along a river. The docks are
represented as points A and B in the diagram below. You place a marker at point X,
because you know that the distance between points X and B is 26 feet. Then you
walk horizontally from point X and place a marker at point Y, which is 26 feet from
point X. You measure the distance between points X and A to be 18 feet, and so you
walk along the river bank 18 feet and place a marker at point Z. Finally, you measure
the distance between Y and Z to be 35 feet.
From the diagram, segments XY and XB are congruent and segments XA and XZ
are congruent. Also, angles YXZ and BXA are congruent by the Vertical Angles
Congruence Theorem. So, by the Side-Angle-Side (SAS) Congruence Postulate,
�YXZ � �BXA. Because corresponding parts of congruent triangles are congruent
(CPCTC), segment YZ must be congruent to segment BA. The length of segment YZ is
35 feet. So, the length of segment BA, or the distance between the docks, is 35 feet.
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Using the Isosceles Triangle Base Angle Theorem
The Isosceles Triangle Base Angle Theorem states: “If two sides of a triangle are
congruent, then the angles opposite these sides are congruent.”
Example:
F40°
15 yd
G
H
15 yd
___
FH � ____
GH , so �F � �G, and the measure of angle G is 40°.
Using the Isosceles Triangle Base Angle Converse Theorem
The Isosceles Triangle Base Angle Converse Theorem states: “If two
angles of a triangle are congruent, then the sides opposite these angles
are congruent.”
Example:
J
75°
21 m
L
K
75°
�J � �K, ___
JL � ___
KL , and the length of side KL is 21 meters.
Using the Isosceles Triangle Base Theorem
The Isosceles Triangle Base Theorem states: “The altitude to the base of an
isosceles triangle bisects the base.”
Example:
A
100 ft
C
B
100 ft
D75 ft x
CD � AD, so x � 75 feet.
6.4
6.4
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Using the Isosceles Triangle Vertex Angle Theorem
The Isosceles Triangle Base Theorem states: “The altitude to the base of an
isosceles triangle bisects the vertex angle.”
Example:
F
5 in.
J
H
5 in.
G48°
x
m�FGJ � m�HGJ, so x � 48°.
Using the Isosceles Triangle Perpendicular Bisector Theorem
The Isosceles Triangle Perpendicular Bisector Theorem states: “The altitude from the
vertex angle of an isosceles triangle is the perpendicular bisector of the base.”
Example:
W Z Y
X
____
WY � ___
XZ and WZ � YZ
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Using the Isosceles Triangle Altitude to Congruent Sides Theorem
The Isosceles Triangle Perpendicular Bisector Theorem states: “In an isosceles
triangle, the altitudes to the congruent sides are congruent.”
Example:
J 11 m
L
K11 m
M
N
___
KN � ___
JM
Using the Isosceles Triangle Bisector to Congruent Sides Theorem
The Isosceles Triangle Perpendicular Bisector Theorem states: “In an isosceles
triangle, the angle bisectors to the congruent sides are congruent.”
Example:
R
12 cm
T
S
12 cm
V W
____
RW � ___
TV
Stating the Inverse and Contrapositive of Conditional Statements
To state the inverse of a conditional statement, negate both the hypothesis and the
conclusion. To state the contrapositive of a conditional statement, negate both the
hypothesis and the conclusion and then reverse them.
Conditional Statement: If p, then q.
Inverse: If not p, then not q.
Contrapositive: If not q, then not p.
Example:
Conditional Statement: If a triangle is equilateral, then it is isosceles.
Inverse: If a triangle is not equilateral, then it is not isosceles.
Contrapositive: If a triangle is not isosceles, then it is not equilateral.
6.5
6.5
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Writing an Indirect Proof
In an indirect proof, or proof by contradiction, first write the givens. Then, write the
negation of the conclusion. Then, use that assumption to prove a given statement is
false, thus creating a contradiction. Hence, the assumption leads to a contradiction,
therefore showing that the assumption is false. This proves the contrapositive.
Example:
Given: Triangle DEF
Prove: A triangle cannot have more than one obtuse angle.
Given �DEF, assume that �DEF has two obtuse angles. So, assume m�D � 91°
and m�E � 91°. By the Triangle Sum Theorem, m�D � m�E � m�F � 180°.
By substitution, 91° � 91° � m�F � 180°, and by subtraction, m�F � 2°.
But it is not possible for a triangle to have a negative angle, so this is a contradiction.
This proves that a triangle cannot have more than one obtuse angle.
Using the Hinge Theorem
The Hinge Theorem states: “If two sides of one triangle are congruent to two sides
of another triangle and the included angle of the first pair is larger than the included
angle of the second pair, then the third side of the first triangle is longer than the
third side of the second triangle.”
Example:
x
P80° 75°
Q
R F
G
H
8 mm
QR � GH, so x � 8 millimeters.
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Using the Hinge Converse Theorem
The Hinge Converse Theorem states: “If two sides of one triangle are congruent to
two sides of another triangle and the third side of the first triangle is longer than the
third side of the second triangle, then the included angle of the first pair of sides is
larger than the included angle of the second pair of sides.”
Example:
xX 62°
Y
ZR
S
T
5 ft 4 ft
m�T � m�Z, so x � 62°.
6.6