EXAMPLE 1

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

2. 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 QTST. 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, RTRT .

3. Explain how you can prove that AC. AB BC Given AD DC Given BD BD Reflexive property ANSWER Thus the triangle by SSS ABDBCD for Example 1 GUIDED PRACTICE SOLUTION

4. Use the following method to find the distance across a river, from point Nto point P. • 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 Use congruent triangles for measurement Surveying

5. Because NK NPand NK KL, Nand Kare congruent right angles. Because Mis the midpoint of NK, NMKM. The vertical angles KMLand NMP are congruent. So, MLK MPNby the ASA Congruence Postulate. Then, because corresponding parts of congruent triangles are congruent, KL NP. So, you can find the distance NPacross the river by measuring KL. EXAMPLE 2 Use congruent triangles for measurement • Explain how this plan allows you to find the distance. SOLUTION

6. 1 2,3 4 GIVEN BCEDCE 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 Plan a proof involving pairs of triangles Use the given information to write a plan for proof. SOLUTION

7. 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 Plan a proof involving pairs of triangles 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. Plan for Proof

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

9. Using the information in the diagram at the right, write a plan to prove thatPTU UQP. for Examples 2 and 3 GUIDED PRACTICE No matter how far apart the strikes at K and M are placed the triangles will be congruent by ASA. SOLUTION

10. STATEMENTS REASONS TU PQ Given PT QU Given Reflexive property PU PU PTUUQP SSS PTUUQP 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. TU PQ for Examples 2 and 3 GUIDED PRACTICE

11. GIVEN AB DE, AC DF, BC EF DA PROVE 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. EXAMPLE 4 Prove a construction Write a proof to verify that the construction for copying an angle is valid. SOLUTION

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

13. AC and AB for Example 4 GUIDED PRACTICE Look back at the construction of an angle bisector in Explore 4 on page 34. What segments can you assume are congruent? SOLUTION