4.7 Use Congruent Triangles

• You will use congruent triangles to prove corresponding parts congruent.

• Essential Question: How can you use congruent triangles to prove angles or sides congruent?

You will learn how to answer this question by using corresponding parts of congruent triangles.

Warm-Up ExercisesEXAMPLE 1 Use congruent triangles

Explain how you can use the given information to prove that the hanglider parts are congruent.

SOLUTION

GIVEN 1 2,∠RTQ RTS

PROVE QT ST

If you can show that QRT SRT, you will know that QT ST.First, copy the diagram and mark the giveninformation.

Warm-Up ExercisesEXAMPLE 1 Use congruent triangles

Then add the information that you can deduce. In this case, RQT and RST are supplementary to congruent angles, so∠RQT RST. Also, RT RT .

Mark given information. Add deduced information.

Two angle pairs and a non-included side are congruent, so by the AAS Congruence Theorem, . Because corresponding parts of congruent triangles are congruent,

QRT SRT

QT ST.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

1. Explain how you can prove thatA C.

SOLUTION

GivenAB BCGivenAD DC

Reflexive propertyBD BD

ABD BCDThus the triangle by SSS

ANSWER

Warm-Up ExercisesEXAMPLE 2 Use congruent triangles for measurement

Surveying

Use the following method to find the distance across a river, from point N to point P.

• Place a stake at K on thenear side so that NK NP

• Find M, the midpoint of NK .

• Locate the point L so that NK KL and L, P, and Mare collinear.

Warm-Up ExercisesEXAMPLE 2 Use congruent triangles for measurement

• Explain how this plan allows you to find the distance.

SOLUTION

Because NK NP and NK KL , N and K are congruent right angles.

Then, because corresponding parts of congruent triangles are congruent, KL NP . So, you can find the distance NP across the river by measuring KL .

MLK MPN by the ASA Congruence Postulate.

Because M is the midpoint of NK , NM KM . The vertical angles KML and NMP are congruent. So,

Warm-Up ExercisesEXAMPLE 3 Plan a proof involving pairs of triangles

Use the given information to write a plan for proof.

SOLUTION

GIVEN 1 2, 3 4

PROVE BCD DCE

In BCE and DCE, you know 1 2 and CE CE . If you can show that CB CD , you can use the SAS Congruence Postulate.

Warm-Up ExercisesEXAMPLE 3 Plan a proof involving pairs of triangles

CBA CDA. You are given 1 2 and 3 4. CA CA by the Reflexive Property. You can use the ASA Congruence Postulate to prove that CBA CDA.

To prove that CB CD , you can first prove that

Plan for ProofUse the ASA Congruence Postulate to prove that CBA CDA. Then state that CB CD . Use the SAS Congruence Postulate to prove that BCE DCE.

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

2. In Example 2, does it matter how far from point N you place a stake at point K ? Explain.

SOLUTIONNo, it does not matter how far from point N you place a stake at point K . Because M is the midpoint of NK

GivenNM MKDefinition of right triangle

MNP MKL areboth right triangles

Vertical angleKLM NMPASA congruence MKL MNP

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

No matter how far apart the strikes at K and M are placed the triangles will be congruent by ASA.

3. Using the information in the diagram at the right, write a plan to prove that PTU UQP.

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

Given TU PQ

Given PT QU

Reflexive property PU PU

This can be done by showing right triangles QSP and TRU are congruent by HL leading to right triangles USQ and PRT being congruent by HL which gives you PT UQ

STATEMENTS REASONS

SSSPTU UQP

PTU UQP By SSS

Warm-Up ExercisesEXAMPLE 4 Prove a construction

Write a proof to verify that the construction for copying an angle is valid.

SOLUTION

Add BC and EF to the diagram. In the construction, AB , DE , AC , and DF are all determined by the same compass setting, as are BC and EF . So, you can assume the following as given statements.

GIVEN AB DE, AC DF, BC EF

PROVE D A

Warm-Up ExercisesEXAMPLE 4 Prove a construction

STATEMENTS REASONS

Plan in Action

Plan For Proof

Show that CAB FDE, so you can conclude that the corresponding parts A and D are congruent.

1. AB DE 1. GivenAC DF, BC EF

2. 2. SSS Congruence Postulate

FDE CAB

3. 3. Corresp. parts ofD Aare .

Warm-Up ExercisesGUIDED PRACTICE for Example 4

4. Look back at the construction of an angle bisector in Explore 4 on page 34. What segments can you assume are congruent?

SOLUTION

AC and AB

Warm-Up ExercisesDaily Homework Quiz

Tell which triangles you can show are congruent in order to prove AE = DE. What postulate or theorem would you use?

1.

ANSWER AEC DEB by the AAS cong. Thm. or by the ASA cong. Post.

Warm-Up ExercisesDaily Homework Quiz

Write a plan to prove 1 2.2.

ANSWER Show LM LM by the Refl. Prop.Of Segs. Hence OLM NML by the SAS cong. Post. This gives NLM OML, since Corr. Parts of are . So 1 2 by the Vert. Thm. and properties of .

s

s s

• You will use congruent triangles to prove corresponding parts congruent.

• Essential Question: How can you use congruent triangles to prove angles or sides congruent?

• If triangles are congruent, theircorresponding parts arecongruent.

Use the fact that Corr. Parts of congruent triangles are congruent.