#### CPCTC

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. It is statement developed from the definition of congruent triangles. It allows us to prove things about the remaining unproven parts of the triangles that we have just proven congruent. It allows us to state correctly after two triangles are congruent, then corresponding parts that were not previously known to be congruent are now allowed to be considered congruent.

Example

Since BO ≅ MA and BW ≅ MN and OW ≅ AN, we can conclude that ΔBOW ≅ ΔMAN because of SSS.

Now we can say ∠B ≅ ∠M, ∠O ≅ ∠A, and ∠W ≅ ∠N because of CPCTC. Since the two triangle were proven congruent, we can now correctly assume that corresponding parts that we knew nothing about, are now congruent.

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Another example

Since BO ≅ MA and OW ≅ AN and ∠O ≅ ∠A, then ΔBOW ≅ ΔMAN by SAS.

Now we can say BW ≅ MN, ∠B ≅ ∠M, and ∠W ≅ ∠N because of CPCTC.

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Another example

Since BO ≅ MA and ∠B ≅ ∠M and ∠O ≅ ∠A, we can conclude that ΔBOW ≅ ΔMAN because of ASA.

Now we can say ∠W ≅ ∠N, BW ≅ MN and OW ≅ AN because of CPCTC.

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Another example

Since BO ≅ MA and ∠O ≅ ∠A and ∠W ≅ ∠N, we can conclude that ΔBOW ≅ ΔMAN because of AAS.

Now we can say ∠B ≅ ∠M, BW ≅ MN and OW ≅ AN because of CPCTC.

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Another example

Since ΔBOW and ΔMAN are right triangle, BO ≅ MA and OW ≅ AN , then ΔBOW ≅ ΔMAN by HL.

Now we can say ∠B ≅ ∠M, ∠O ≅ ∠A, and BW ≅ MN because of CPCTC.

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Remember, we use CPCTC to prove parts are congruent after we have proven triangles congruent. CPCTC is used after SSS, or SAS, or ASA, or AAS, or HL, never before. First we prove that two triangles are congruent. Then if we haven’t already proven that a desired pair of corresponding sides or angles are congruent, we can now do so using CPCTC.

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