Suppose that ∆XYZ ∆RST. Complete each statement. ? ANSWER RS 2. Z ? ANSWER T

Suppose that ∆XYZ ∆RST. Complete each statement. 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

EXAMPLE 1 Use congruent triangles Explain how you can use the given information to prove that the hanglider 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 BCD Thus the triangle by SSS ANSWER

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,

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 BCD 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 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

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

GUIDED PRACTICE for Examples 2 and 3 STATEMENTS REASONS TU PQ PT QU Given TU PQ Given PT QU Reflexive property PU PU SSS PTU UQP PTU UQP 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

EXAMPLE 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

SSS Congruence Postulate FDE CAB EXAMPLE 4 Prove a construction Plan For Proof Show that CAB FDE, so you can conclude that the corresponding parts A and D are congruent. STATEMENTS REASONS AB DE Given AC DF, BC EF Plan in Action SSS Congruence Postulate FDE CAB Corresp. parts of D A are .

GUIDED PRACTICE for Example 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

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