4.6

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Use Congruent Triangles

4.6 Warm-Up

Suppose that ∆XYZ ∆RST. Complete each statement.

ANSWER T

ANSWER RS

1. XY ?

2. Z ?

ANSWER Y

3. m S = m ?

4.6 Warm-Up

ANSWER 50

4. If A B, m A = (2x + 40)º, and m B = (3x – 10)º,find x.

4.6 Example 1

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

SOLUTION

∠RTQ RTS

GIVEN: 1 2,

PROVE: QT ST

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

4.6 Example 1

First, copy the diagram and mark the given information. Then add the information 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, QRT SRT. Because corresponding parts of congruent triangles are congruent, QT ST.

4.6 Guided Practice

1. Explain how you can prove thatA C.

Since BD BD by the Reflexive Property, the triangles are congruent by SSS. So, ∠A ∠C because they are corresponding parts of congruent triangles.

ANSWER

4.6 Example 2

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.

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

4.6 Example 2

SOLUTION

The vertical angles ∠KML and ∠NMP are congruent. So, MLK MPN by the ASA Congruence Postulate. Then, because corresponding parts of congruent triangles are congruent, KL NP. So, you can find the distance NP across the river by measuring KL.

Because NK NP and NK KL, ∠N and ∠K are congruent right angles. Because M is the midpoint of NK , NM KM .

4.6 Example 3

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.

4.6 Example 3

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.

4.6 Guided Practice

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

ANSWER

No, since M is the midpoint of NK, NM MK. No matter how far apart the stakes at K and M are placed, the triangles will be congruent by ASA.

4.6 Guided Practice

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

ANSWER

Since you already know that TU QP and UP PU, you need only show PT UQ to prove the triangles are congruent 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.

4.6 Example 4

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

4.6 Example 4

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. Given1. AB DE,

AC DF, BC EF

2. SSS Congruence Postulate

2. FDE CAB

3. D A 3. Corresp. parts ofare .

4.6 Guided Practice

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

AC and ABANSWER

4.6 Lesson 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.

4.6 Lesson 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 .

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