4.2.3 Triangle Congruence: CPCTC Key Objectives • Using CPCTC to prove parts of triangles are congruent. Key Terms • CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent,” which can be used as a justification in a proof after two triangles have been shown to be congruent. Remember, congruent triangles are triangles with corresponding parts (angles and sides) that are congruent. The SSS, SAS, ASA, AAS, and HL theorems and postulates are used to prove that two triangles are congruent by using some combination of their corresponding parts. In other words, once some combination of corresponding parts has been shown to be congruent, then SSS, SAS, ASA, AAS, or HL can be used to justify that the two triangles are congruent. CPCTC works in the opposite way. CPCTC is used to prove that corresponding parts of two triangles are congruent by using the congruence of two triangles. In other words, once two triangles are shown to be congruent, then CPCTC can be used to justify that any of their corresponding parts are congruent. CPCTC is typically used in a proof after SSS, SAS, ASA, AAS, or HL has been used to show that the two triangles are congruent. Example 1 Measuring Application
Indirect measurement is used in this example to show how the distance from M to N could be found using only measurements from Manda’s side of the fence. Manda constructed the two triangles so that ∠ M and ∠ Q are right angles and MP = PQ = 20 ft. Use this given information to explain how she could find MN without actually measuring that distance. First, prove that △ NMP ≅ △ RQP by using ASA since two angles ( ∠ M and ∠ P ) and their included side in △ NMP are congruent to two angles ( ∠ Q and ∠ P ) and their included side in △ RQP. Then, since the two triangles are congruent, CPCTC can be used to show that any pair of corresponding parts are congruent. So, use CPCTC to show that by finding QR , MN can also be found since QR and MN are the lengths of corresponding sides.
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