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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-Up1Exercises EXAMPLE Use congruent triangles Explain how you can use the given information to prove that the hanglider parts are congruent. GIVEN PROVE 1 QT 2, RTQ RTS ST SOLUTION If you can show that QRT SRT, you will know that QT ST. First, copy the diagram and mark the given information. Warm-Up1Exercises EXAMPLE 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, QRT SRT so by the AAS Congruence Theorem, . Because corresponding parts of congruent triangles are ST. congruent, QT Warm-Up Exercises GUIDED PRACTICE 1. for Example 1 Explain how you can prove that A C. SOLUTION AB BC AD DC Given Given BD BD Reflexive property ANSWER Thus the triangle ABD BCD by SSS Warm-Up2Exercises EXAMPLE 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 the near side so that NK NP • Find M, the midpoint of NK . • Locate the point L so that NK are collinear. KL and L, P, and M Warm-Up2Exercises EXAMPLE 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. Because M is the midpoint of NK , NM KM . 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 . Warm-Up3Exercises EXAMPLE Plan a proof involving pairs of triangles Use the given information to write a plan for proof. GIVEN 1 PROVE 2, BCD 3 4 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. Warm-Up3Exercises EXAMPLE 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. Warm-Up Exercises GUIDED PRACTICE 2. 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 NM MK MNP MKL are both right triangles KLM NMP MKL MNP Given Definition of right triangle Vertical angle ASA congruence Warm-Up Exercises 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. 3. Using the information in the diagram at the right, write a plan to prove that PTU UQP. Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 STATEMENTS TU PQ Given PT QU Given PU PU Reflexive property PTU PTU REASONS UQP SSS 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 Warm-Up4Exercises EXAMPLE 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 PROVE AB D DE, AC A DF, BC EF Warm-Up4Exercises EXAMPLE Prove a construction Show that CAB FDE, so you can conclude that the corresponding parts and D are congruent. Plan For Proof STATEMENTS 1. Plan in Action 2. 3. AB AC REASONS DE DF, BC FDE D A CAB A 1. Given EF 2. SSS Congruence Postulate 3. Corresp. parts of are . Warm-Up Exercises GUIDED PRACTICE 4. 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 Warm-Up Exercises 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 Warm-Up Exercises 2. Write a plan to prove ANSWER 1 2. Show LM LM by the Refl. Prop.Of Segs. Hence OLM NML by the SAS cong. Post. This gives NLM OML, since s are Corr. Parts of . So 1 2 by the s. Vert. s Thm. and properties of • 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.