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Triangle Congruence: SSS and SAS - Dolfanescobar's Weblog · 2017-11-28 · The proofs of the SSS...

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A F D E B C A B C E D F © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company Name Class Date 4-5 Triangle Congruence: SSS and SAS Going Deeper Essential 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 ABC, which shares a side with DEF. If C lies on the same side of ___ DE as F, reflect ABC across ___ DE . This results in the figure at right. B ____ AC ___ AC because . It is also given that ___ AC ___ DF . Therefore, ____ AC ___ DF because of the Property of Congruence. By a similar argument, ____ BC . C Because ____ AC ___ DF , D lies on the perpendicular bisector of ____ FC , by the Converse of the Perpendicular Bisector Theorem. Similarly, because ____ BC ___ 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. PROOF 1 G-CO.2.8 Chapter 4 153 Lesson 5
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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

B

G-CO.3.9

Chapter 4 154 Lesson 5

C

A

B

m

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A

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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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CD

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JK

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190 ft

190 ft210 ft

210 ftM

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

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

CS10_G_MEPS710006_C04PWBL05.indd 25 4/21/11 5:59:57 PM

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4-5Name Class Date

Additional Practice

Chapter 4 159 Lesson 5

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.

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

2. J is the midpoint of AB and .AK BK≅ Use SSS to explain why AKJ ≅ BKJ.

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

3. A balalaika is a Russian stringed instrument. Show that the triangular parts of the two balalaikas are congruent for x = 6.

_________________________________________

_________________________________________

_________________________________________

_________________________________________

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


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