Based on this diagram, the following relation holds true.
m∠A+m∠B+m∠C=180∘
This theorem is also known as the Triangle Angle Sum Theorem.
Proof
Consider a triangle with verticesA,B, and C, and the parallel line to BC through A. Let ∠1 and ∠2 be the angles outside △ABC formed by this line and the sides AB and AC.
By the definition of congruent angles, ∠1 and ∠B have the same measure. For the same reason, ∠2 and ∠C also have the same measure.
∠B≅∠1⇕m∠B=m∠1∠C≅∠2⇕m∠C=m∠2
Furthermore, in the diagram it can be seen that ∠BAC,∠1, and ∠2 form a straight angle. Therefore, by the Angle Addition Postulate their measures add to 180∘.
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