# Triangle Congruence by ASA and AAS

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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. RF S and U GHI PQR AAS not possible ABX ACX AAS 4-3

Using Congruent Triangles: CPCTC
GEOMETRY LESSON 4-4 CPCTC is 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

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

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

Then look for triangles that contain these angles.
Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 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. Helpful Hint 4-4

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

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

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

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

Example 2 Since MS || TR, M  T (Alt. Int. s) M R A SAM  RAT (Vert. s) S T MS  TR (Given) Given: MS || TR and MS  TR SAM  RAT (AAS) Prove: A is the midpoint of MT. MA  AT (CPCTC) 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 3 MP bis. LMN (Given) P N L M NMP  LMP (def.  bis) LM  NM (Given) PM  PM (Ref) PMN  PML (SAS) LP  NP (CPCTC) QED Given: MP bisects LMN and LM  NM Prove: LP  NP

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

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

Then, how do you prove triangles congruent? (SSS, SAS, ASA, AAS)
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.

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? SC SC by the Reflexive Property of Congruence, and LSC RSC 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

Using Congruent Triangles: CPCTC
GEOMETRY LESSON 4-4 (continued) SL SR Given SC SC Reflexive Property of Congruence CL CR Other congruence statement Sides: 1 2 Given 3 4 Corresponding Parts of Congruent Triangles CLS CRS Other congruence statement Angles: 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 CL CR Quick Check 4-4

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

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 RUQ TUS. 5. Tell what other parts are congruent by CPCTC. Corresponding parts of congruent triangles are congruent. You are given two pairs of s, and AM AM by the Reflexive Prop., so ABM ACM by ASA. AB AC, BM CM, B C You are given a pair of s and a pair of sides, and RUQ TUS because vertical angles are , so RUQ TUS by AAS. RQ TS, UQ US, R T 4-4

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