Corresponding parts of congruent triangles are congruent. Tell what other parts are congruent by CPCTC. What does “CPCTC” stand for? Use the diagram for Exercises 2 and 3. So the officer must stand perpendicular to the ground, and the ground must be level. What conditions must hold for that to be true? DEG and DEF are the angles the officer makes with the ground. GEOMETRY LESSON 4-4 Real-World Connection The Given states that DEG and DEF are right angles. That leaves two congruence statements remaining that also can be proved: CLS CRS CL CR Quick Check 4-4 GEOMETRY LESSON 4-4 (continued) SL SR Given SC SC Reflexive Property of Congruence CL CR Other congruence statement Sides: 1 2 Given 3 4Ĝorresponding Parts of Congruent Triangles CLS CRS Other congruence statement Angles: In the proof, three congruence statements are used, and one congruence statement is proven. Umbrella Frames In an umbrella frame, the stretchers are congruent and they open to angles of equal measure. When two triangles are congruent, you can form congruence statements about three pairs of corresponding angles and three pairs of corresponding sides. 3 4 by corresponding parts of congruent triangles are congruent. 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. 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. B E (CPCTC)ġ4 Then, how do you prove triangles congruent? (SSS, SAS, ASA, AAS) bis) LM NM (Given) PM PM (Ref) PMN PML (SAS) LP NP (CPCTC) QED Given: MP bisects LMN and LM NM Prove: LP NPġ2 Given: AB DC, AD BC Prove: A C Statements Reasons A Bģ. By definition, A is the midpoint of segment MT. MA AT (CPCTC) Plan: Show the triangles are congruent using AAS, then MA =AT. s) S T MS TR (Given) Given: MS || TR and MS TR SAM RAT (AAS) Prove: A is the midpoint of MT. HJL KLJ (Alt Int s) LJ LJ (Reflexive) HLJ KJL (Alt Int s) JHL LKJ (ASA) H K (CPCTC) QEDġ0 Example 2 Since MS || TR, M T (Alt. Helpful Hint 4-4Ħ Example A B C L J K Is ABC JKL? YES What’s the reason? SASħ What other angles are congruent? B K and C LĮxample 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 KLĨ Why? CPCTC ABC JKL What other angles are congruent?Įxample 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 KLĩ Example Given: HJ || LK and JK || HL Prove: H K H J K L Plan: Show JHL LKJ by ASA, then use CPCTC. Then look for triangles that contain these angles. To show that ED || GF, look for a pair of angles that are congruent. Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 Work backward when planning a proof. Remember! 4-4ĥ Then look for triangles that contain these angles. CPCTC uses congruent triangles to prove corresponding parts congruent. GEOMETRY LESSON 4-4 SSS, SAS, ASA, AAS, (and HL) use corresponding parts to prove triangles congruent. The Basic Idea: Given Information SSS SAS ASA AAS Prove Triangles Congruent Show CorrespondingParts Congruent 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. RF S and U GHI PQR AAS not possible ABX ACX AAS 4-3 If you can, state a triangle congruence and the postulate or theorem you used. Which angles in STU include US? Tell whether you can prove the triangles congruent by ASA or AAS. Which side is included between R and F in FTR? 2.
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