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Warm Up

Use Congruent Triangles. Warm Up. Lesson Presentation. Lesson Quiz. Suppose that ∆ XYZ ∆ RST . Complete each statement. ?. ?. ?. 1. XY. 3. m S = m. RS. Y. ANSWER. ANSWER. Z. 2. . T. ANSWER. Warm-Up.

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Warm Up

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  1. Use Congruent Triangles Warm Up Lesson Presentation Lesson Quiz

  2. Suppose that ∆XYZ ∆RST. Complete each statement. ? ? ? 1. XY 3. m S = m RS Y ANSWER ANSWER Z 2. T ANSWER Warm-Up

  3. 4. If A B, m A = (2x + 40)º, and m B = (3x – 10)º,findx. 50 ANSWER Warm-Up

  4. GIVEN: ∠ RTQ RTS 1 2, QTST PROVE: If you can show that QRT SRT, you will know that QT  ST. Example 1 Explain how you can use the given information to prove that the hang glider parts are congruent. SOLUTION

  5. 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. Example 1

  6. Explain how you can prove that AC. ANSWER Since BD BD by the Reflexive Property, the triangles are congruent by SSS. So, AC because they are corresponding parts of congruent triangles. Guided Practice

  7. 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 Surveying Use the following method to find the distance across a river, from point Nto point P. • Explain how this plan allows you to find the distance.

  8. Example 2 SOLUTION Because NK NPand NK KL, Nand Kare congruent right angles. Because Mis the midpoint ofNK,NM KM. The vertical angles KMLand NMP are congruent. So, MLK MPN by theASACongruence 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.

  9. 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 Use the given information to write a plan for proof. SOLUTION

  10. Plan for Proof 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 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.

  11. In Example 2, does it matter how far from point Nyou 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. Guided Practice

  12. Using the information in the diagram at the right, write a plan to prove thatPTU UQP. ANSWER Since you already know thatTUQPandUPPU, you need only showPTUQto prove the triangles are congruent bySSS. This can be done by showing right trianglesQSPand TRUare congruent byHL leading to right trianglesUSQandPRTbeing congruent byHL which gives youPTUQ. Guided Practice

  13. GIVEN: AB DE,AC DF, BC EF 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. D A PROVE: Example 4 Write a proof to verify that the construction for copying an angle is valid. SOLUTION

  14. STATEMENTS REASONS AB DE, Given AC DF, BC EF SSS Congruence Postulate FDE CAB D A Corresp. parts of are . Example 4 Plan For Proof Show that CAB  FDE, so you can conclude that the corresponding parts Aand Dare congruent. Plan in Action

  15. ANSWER AC and AB Guided Practice Look back at the construction of an angle bisector in Explore 4 on page 34. What segments can you assume are congruent?

  16. 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. Lesson Quiz

  17. 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 . Lesson Quiz

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