Triangle Congruence by ASA and AAS

1. RF GHI PQRAAS ABX ACXAAS Triangle Congruence by ASA and AAS GEOMETRY LESSON 4-3 1. Which side is included between R and F in FTR? 2. Which angles in STU include US? Tell whether you can prove the triangles congruent by ASA or AAS. If you can, state a triangle congruence and the postulate or theorem you used. If not, write not possible. 3. 4. 5. S and U not possible 4-3

2. Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 CPCTCis an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent because by definition, corresponding parts of congruent triangles are congruent. 4-4

3. The Basic Idea: Given Information • SSS • SAS • ASA • AAS Prove Triangles Congruent Show CorrespondingParts Congruent CPCTC

4. Remember! SSS, SAS, ASA, AAS, (and HL) use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 4-4

5. Helpful Hint Work backward when planning a proof. To show that ED || GF, look for a pair of angles that are congruent. Then look for triangles that contain these angles. Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 4-4

6. B K A C J L Example Is ABC  JKL? YES What’s the reason? SAS

7. B K A C J L BC  KL Example continued ABC  JKL What other angles are congruent? B  K and C  L What other side is congruent?

8. B K A C J L BC  KL Example continued ABC  JKL Why? CPCTC What other angles are congruent? B  K and C  L What other side is congruent?

9. J H K L LJ  LJ Example Given: HJ || LK and JK || HL Prove: H  K Plan: Show JHL  LKJ by ASA, then use CPCTC. HJL  KLJ (Alt Int s) (Reflexive) HLJ  KJL (Alt Int s) JHL  LKJ (ASA) H  K (CPCTC) QED

10. Since MS || TR, M  T (Alt. Int. s) MS  TR (Given) Given: MS || TR and MS  TR MA  AT (CPCTC) Prove: A is the midpoint of MT. Plan: Show the triangles are congruent using AAS, then MA =AT. By definition, A is the midpoint of segment MT. A is the midpoint of MT (Def. midpoint) Example 2 M R A SAM  RAT (Vert. s) S T SAM  RAT (AAS)

11. MP bis. LMN (Given) LM  NM (Given) PM  PM (Ref) P L LP  NP (CPCTC) N Given: MP bisects LMN and LM  NM M Prove: LP  NP Example 3 NMP  LMP (def.  bis) PMN  PML (SAS) QED

12. Given: AB  DC, AD  BC Prove: A  C Statements Reasons A B 1. AB  DC 1. Given 2. AD  BC 2. Given 3. BD  BD 3. Reflexive D C 4. ABD  CDB 4. SSS 5. A C 5. CPCTC

13. 1. AC  DC A E C B D Show B  E (given) (given) 2. A  D (vert s) 3. ACB  DCE (ASA) 4. ACB  DCE 5. B  E (CPCTC)

14. Proofs • Ask: to show angles or segments congruent, what triangles must be congruent? • Then, how do you prove triangles congruent? (SSS, SAS, ASA, AAS) • Prove triangles congruent, then use CPCTC.

15. Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 Real-World Connection What other congruence statements can you prove from the diagram, in which SL SR, and 1 2 are given? SCSC by the Reflexive Property of Congruence, and LSCRSC by SAS. 3 4 by corresponding parts of congruent triangles are congruent. When two triangles are congruent, you can form congruence statements about three pairs of corresponding angles and three pairs of corresponding sides. List the congruence statements. Umbrella Frames In an umbrella frame, the stretchers are congruent and they open to angles of equal measure. 4-4

16. Sides: SLSR Given SCSC Reflexive Property of Congruence CLCR Other congruence statement Angles: 1 2 Given 3 4 Corresponding Parts of Congruent Triangles CLS CRS Other congruence statement In the proof, three congruence statements are used, and one congruence statement is proven. That leaves two congruence statements remaining that also can be proved: CLSCRS CLCR Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 (continued) Quick Check 4-4

17. Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 Real-World Connection The Given states that DEG and DEF are right angles. What conditions must hold for that to be true? DEG and DEF are the angles the officer makes with the ground. So the officer must stand perpendicular to the ground, and the ground must be level. Quick Check 4-4

18. You are given two pairs of s, and AMAM by the Reflexive Prop., so ABM ACM by ASA. AB AC, BM CM, You are given a pair of s and a pair of sides, and RUQ TUS because vertical angles are , so RUQTUS by AAS. RQ TS, UQ US, R T B C Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 1. What does “CPCTC” stand for? Use the diagram for Exercises 2 and 3. 2. Tell how you would show ABM ACM. 3. Tell what other parts are congruent by CPCTC. Use the diagram for Exercises 4 and 5. 4. Tell how you would show RUQTUS. 5. Tell what other parts are congruent by CPCTC. Corresponding parts of congruent triangles are congruent. 4-4

19. JR  HV, RC  VG, and JC  HG TI  LO, IC  OK, and CT KL Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 (For help, go to Lesson 4-1.) In the diagram, JRCHVG. 1. List the congruent corresponding angles. 2. List the congruent corresponding sides. You are given that TICLOK. 3. List the congruent corresponding angles. 4. List the congruent corresponding sides. J H, R  V, and C G T L, I O, and C K Check Skills You’ll Need 4-4