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Name Class Date 4-5Triangle Congruence: SSS and SASGoing DeeperEssential question: How can you establish the SSS and SAS triangle congruence criteria using properties of rigid motions?
You have seen that when two triangles are congruent, the corresponding sides and
corresponding angles are congruent. Conversely, if all six pairs of corresponding sides
and corresponding angles of two triangles are congruent, then the triangles are congruent.
The proofs of the SSS and SAS congruence criteria that follow serve as proof of this converse.
In each case, the proof demonstrates a “shortcut,” in which only three pairs of congruent
corresponding parts are needed in order to conclude that the triangles are congruent.
SSS Congruence Criterion
If three sides of one triangle are congruent to
three sides of another triangle, then the
triangles are congruent.
Given: ___
AB � ___
DE , ___
BC � ___
EF , and ___
AC � ___
DF .
Prove: �ABC � �DEF
To prove the triangles are congruent,
you will find a sequence of rigid motions
that maps �ABC to �DEF. Complete the
following steps of the proof.
A Since ___
AB � ___
DE , there is a sequence of rigid motions that maps ___
AB to .
Apply this sequence of rigid motions to
�ABC to get �A′B′C′, which shares a side
with �DEF.
If C ′ lies on the same side of ___
DE as F,
reflect �A′B′C′ across ___
DE . This results
in the figure at right.
B ____
A′C ′ � ___
AC because .
It is also given that ___
AC � ___
DF .
Therefore, ____
A′C ′ � ___
DF because of the Property of Congruence.
By a similar argument, ____
B′C ′ � .
C Because ____
A′C ′ � ___
DF , D lies on the perpendicular bisector of ____
FC ′ , by the Converse
of the Perpendicular Bisector Theorem. Similarly, because ____
B′C ′ � ___
EF , E lies on the
perpendicular bisector of ____
FC ′ . So, ___
DE is the perpendicular bisector of ____
FC ′ .
By the definition of reflection, the reflection across ___
DE maps C ′ to .
The proof shows that there is a sequence of rigid motions that maps �ABC to �DEF.
Therefore, �ABC � �DEF.
P R O O F1G-CO.2.8
Chapter 4 153 Lesson 5
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REFLECT
1a. The proof uses the fact that congruence is transitive. That is, if you know
figure A � figure B, and figure B � figure C, you can conclude that
figure A � figure C. Why is this true?
You can use reflections and their properties to prove theorems about angle
bisectors. These theorems will be very useful in proofs later on.
The first proof is an indirect proof (or a proof by contradiction). To write such a
proof, you assume that what you are trying to prove is false and you show that this
assumption leads to a contradiction.
Angle Bisection Theorem
If a line bisects an angle, then each side of the angle is the
image of the other under a reflection across the line.
Given: Line m is the bisector of ∠ABC.
Prove: The image of ___
› BA under a reflection across
line m is ___
› BC .
Assume what you are trying to prove is false.
Assume that the image of ___
› BA under a reflection across
line m is not ___
› BC . In that case, let the reflection image
of ___
› BA be
___ › BA′ , which is not the same ray as
___ › BC .
Complete the following to show that this assumption
leads to a contradiction.
Let D be a point on line m in the interior of ∠ABC.
Then ∠DBC and ∠DBA′ must have different measures.
However, m∠DBA = m∠DBC since line m is
That means ∠DBA and ∠DBA′ must have different measures.
This is a contradiction because reflections preserve
Therefore, the initial assumption must be incorrect, and the image of ___
› BA under
a reflection across line m is ___
› BC .
P R O O F2
A
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G-CO.3.9
Chapter 4 154 Lesson 5
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REFLECT
2a. Explain how you can use paper folding to explain why the Angle Bisection Theorem
makes sense.
Reflected Points on an Angle Theorem
If two points of an angle are located the same distance from the vertex but on different
sides of the angle, then the points are images of each other under a reflection across the
line that bisects the angle.
Given: Line m is the bisector of ∠ABC and BA = BC.
Prove: r m (A) = C and r m (C) = A.
Complete the following proof.
It is given that line m is the bisector of ∠ABC. Therefore,
when ___
› BA is reflected across line m, its image is
___ › BC .
This is justified by
This means that r m (A) lies on ___
› BC . Let r m (A) = A′.
Since point B is on the line of reflection, r m (B) = B,
and since reflections preserve distance, BA = BA′.
However, it is given that BA = BC. By the Substitution
Property of Equality, you can conclude that
Thus, A′ and C are two points on ___
› BC that are the same
distance from point B. This means A′ = C, so r m (A) = C.
A similar argument shows that r m (C) = A.
REFLECT
3a. Using the above argument as a model, write out a similar argument that shows
that r m (C) = A.
P R O O F3G-CO.3.9
Chapter 4 155 Lesson 5
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REFLECT
3b. In the figure on the previous page, suppose you reflect point A across line m. Then you
reflect the image of point A across line m. What is the final location of the point? Why?
SAS Congruence Criterion
If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of another triangle, then
the triangles are congruent.
Given: ___
AB � ___
DE , ∠B � ∠E, and ___
BC � ___
EF .
Prove: �ABC � �DEF
To prove the triangles are congruent, you will find a sequence of rigid motions
that maps �ABC to �DEF . Complete the following steps of the proof.
A The first step is the same as the first step in the proof
of the SSS Congruence Criterion. In particular, the fact
that ___
AB � ___
DE means there is a sequence of rigid motions
that results in the figure at right.
B Rigid motions preserve distance, so ____
B′C ′ � ___
BC . Also, it is given that
___ BC �
___
EF .
So, because congruence is transitive.
It is given that ∠DEF � ∠B. Also, ∠B � because rigid motions
preserve angle measure.
Therefore, ∠DEF � because congruence is transitive. You can use
this to conclude that ___
DE is the bisector of ∠FEC ′.
C Now consider the reflection across ___
DE .
Under this reflection, the image of C ′ is by the Reflected Points on
an Angle Theorem.
The proof shows that there is a sequence of rigid motions that maps �ABC to �DEF.
Therefore, �ABC � �DEF.
P R O O F4G-CO.2.8
Chapter 4 156 Lesson 5
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REFLECT
4a. Explain how the Reflected Points on an Angle Theorem lets you conclude that the
image of C ′ under a reflection across ___
DE is F.
Using the SSS Congruence Criterion
Complete the proof.
Given: M is the midpoint of ___
RT ; ___
SR � ___
ST
Prove: �RSM � �TSM
A Use a colored pen or pencil to mark the figure using the
given information.
B Write a statement in each cell to complete the proof.
The reason for each statement is provided.
REFLECT
5a. What piece of additional given information in the above example would allow you to
use the SAS Congruence Criterion to prove that �RSM � �TSM ?
5b. Suppose the given information had been that M is the midpoint of ___
RT and ∠R � ∠T.
Would it have been possible to prove �RSM � �TSM? Explain.
E X AM P L E5
Definition of
midpoint
SSS Congruence
Criterion
Given
Reflexive Property
of Congruence
Given
G-SRT.2.5
Chapter 4 157 Lesson 5
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190 ft
190 ft210 ft
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K
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P R A C T I C E
1. Given: ___
AB � ___
CD , ___
AD � ___
CB Prove: �ABD � �CBD
2. Given: ____
GH ‖ __
JK , ____
GH � __
JK Prove: �HGJ � �KJG
3. To find the distance JK across a large rock formation, you
locate points as shown in the figure. Explain how to use this
information to find JK.
4. To find the distance RS across a lake, you locate points as
shown in the figure. Can you use this information to find RS ?
Explain.
5. �DEF � �GHJ, DF = 3x + 2, GJ = 6x - 13, and HJ = 5x.
Find HJ.
6. In the figure, ‹
___
› MC is the perpendicular bisector of
___ AB . Is it possible
to prove that �AMC � �BMC? Why or why not?
Complete the two-column proof.
Statements Reasons
1. ___
AB � ___
CD 1.
2. ___
AD � ___
CB 2.
3. 3.
4. �ABD � �CBD 4.
Statements Reasons
1. ___
GH ‖ ___
JK 1.
2. ∠HGJ � ∠KJG 2.
3. 3. Given
4. ___
GJ � ___
GJ 4.
5. 5.
Chapter 4 158 Lesson 5
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Name ________________________________________ Date __________________ Class__________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
yrtemoeG laguoDcM tloH
Practice
Triangle Congruence: SSS and SAS
Write whether SSS or SAS, if either, can be used to prove the triangles
congruent. If no triangles can be proved congruent, write neither.
1. _________________________ 2. _________________________
3. _________________________ 4. _________________________
Find the value of x so that the triangles are congruent.
5. x = _________________________ 6. x = _________________________
The Hatfield and McCoy families are feuding over some land. Neither family will
be satisfied unless the two triangular fields are exactly the same size. You know
that C is the midpoint of each of the intersecting segments. Write a two-column
proof that will settle the dispute.
7. Given: C is the midpoint of AD and .BE
Prove: ABC ≅ DEC
Proof:
25
LESSON
4-5
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4-5Name Class Date
Additional Practice
Chapter 4 159 Lesson 5
Page 8
Use the diagram for Exercises 1 and 2. A shed door appears to be divided into congruent right triangles.
1. Suppose .AB CD≅ Use SAS to show ABD ≅ DCA.
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2. J is the midpoint of AB and .AK BK≅ Use SSS to explain why AKJ ≅ BKJ.
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3. A balalaika is a Russian stringed instrument. Show that the triangular parts of the two balalaikas are congruent for x = 6.
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A quilt pattern of a dog is shown. Choose the best answer.
5. P is the midpoint of TS and TR = SR =
4. ML = MP = MN = MQ = 1 inch. 1.4 inches. What can you conclude Which statement is correct? about ΔTRP and ΔSRP? A LMN ≅ QMP by SAS. F TRP ≅ SRP by SAS. B LMN ≅ QMP by SSS. G TRP ≅ SRP by SSS. C LMN ≅ MQP by SAS. H TRP ≅ SPR by SAS. D LMN ≅ MQP by SSS. J TRP ≅ SPR by SSS.
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Problem Solving
Chapter 4 160 Lesson 5
