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4.7 Use Congruent Triangles

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- 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.

GIVEN

∠ RTQ RTS

1 2,

QTST

PROVE

If you can show that QRT SRT, you will know that QT ST.

First, copy the diagram and mark the given

information.

EXAMPLE 1

Use congruent triangles

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

SOLUTION

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

QTST.

EXAMPLE 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,

RTRT .

Explain how you can prove that

AC.

AB BC

Given

AD DC

Given

BD BD

Reflexive property

ANSWER

Thus the triangle by SSS

ABDBCD

for Example 1

GUIDED PRACTICE

SOLUTION

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

- Place a stake at Kon the

near side so that NK NP

- Find M,the midpoint of NK .

- Locate the point Lso that NKKLand L, P,and M

are collinear.

EXAMPLE 2

Use congruent triangles for measurement

Surveying

Because NK NPand NK KL, Nand Kare congruent right angles.

Because Mis the midpoint of NK, NMKM. The vertical angles KMLand NMP are congruent. So,

MLK MPNby the ASA Congruence Postulate.

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

EXAMPLE 2

Use congruent triangles for measurement

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

SOLUTION

1 2,3 4

GIVEN

BCDDCE

PROVE

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

EXAMPLE 3

Plan a proof involving pairs of triangles

Use the given information to write a plan for proof.

SOLUTION

Use the ASA Congruence Postulate to prove that

CBACDA.Then state that CB CD. Use the SAS Congruence Postulate to prove that BCE DCE.

EXAMPLE 3

Plan a proof involving pairs of triangles

To prove that CBCD, you can first prove that

CBACDA. You are given 12 and 34. CACAby the Reflexive Property. You can use the ASA Congruence Postulate to prove that CBACDA.

Plan for Proof

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

No, it does not matter how far from point Nyou place a stake at point K . Because M is the midpoint of NK

NM MK

Given

MNPMKL are

Definition of right triangle

both right triangles

KLMNMP

Vertical angle

ASA congruence

MKLMNP

for Examples 2 and 3

GUIDED PRACTICE

SOLUTION

Using the information in the diagram at the right, write a plan to prove thatPTU UQP.

for Examples 2 and 3

GUIDED PRACTICE

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

STATEMENTS

REASONS

TU PQ

Given

PT QU

Given

Reflexive property

PU PU

PTUUQP

SSS

PTUUQP

By SSS

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

for Examples 2 and 3

GUIDED PRACTICE

GIVEN

AB DE,AC DF, BC EF

DA

PROVE

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

EXAMPLE 4

Prove a construction

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

SOLUTION

Show that CAB FDE, so you can conclude that the corresponding parts Aand Dare congruent.

Plan For Proof

STATEMENTS

REASONS

AB DE

Given

D A

Corresp. parts of

AC DF, BC EF

are .

SSS Congruence Postulate

FDE CAB

EXAMPLE 4

Prove a construction

Plan in Action

AC and AB

for Example 4

GUIDED PRACTICE

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

SOLUTION

1.

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

ANSWER

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

Daily Homework Quiz

2.

Write a plan to prove 1 2.

s

s

s

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 .

Daily Homework Quiz

- 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, their

corresponding parts are

congruent.

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