Suppose that ∆XYZ ∆RST. Complete each statement. ? ANSWER RS 2. Z ? ANSWER T 3. m S = m ? ANSWER Y 4. If A B, m A = (2x + 40)º, and m B = (3x – 10)º, find x. ANSWER 50 Homework Check 4.6 – With your textbook closed, copy your solutions from your homework for # # #

Proving triangles congruent. Target Proving triangles congruent. GOAL: 4.7 Use congruent triangles to prove corresponding parts congruent.

EXAMPLE 1 Use congruent triangles Explain how you can use the given information to prove that the hand glider parts are congruent. GIVEN 1 2, ∠ RTQ RTS PROVE QT ST SOLUTION If you can show that QRT SRT, you will know that QT ST. First, copy the diagram and mark the given information.

Use congruent triangles 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, 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.

GUIDED PRACTICE for Example 1 Explain how you can prove that A C. SOLUTION Given AB BC Given AD DC Reflexive property BD BD ABD CBD SSS Congruence A C CPCTC

EXAMPLE 2 Use congruent triangles for measurement Use the following method to find the distance across a river, from point N to point P. Surveying Place a stake at K on the near side so that NK NP Find M, the midpoint of NK . Locate the point L so that NK KL and L, P, and M are collinear.

EXAMPLE 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,

GUIDED PRACTICE for Examples 2 and 3 In Example 2, does it matter how far from point N you place a stake at point K ? Explain. SOLUTION No, it does not matter how far from point N you place a stake at point K . Because M is the midpoint of NK Given NM MK Definition of right triangle MNP MKL are both right triangles Vertical angle KLM NMP ASA congruence MKL MNP

EXAMPLE 3 Plan a proof involving pairs of triangles Use the given information to write a plan for proof. GIVEN 1 2, 3 4 PROVE BCE DCE SOLUTION 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.

EXAMPLE 3 Plan a proof involving pairs of triangles To prove that CB CD , you can first prove that 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. Plan for Proof Use the ASA Congruence Postulate to prove that CBA CDA. Then state that CB CD . Use the SAS Congruence Postulate to prove that BCE DCE.

GUIDED PRACTICE for Examples 2 and 3 Using the information in the diagram at the right, write a plan to prove that PTU UQP.

Daily 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.

Daily 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