Consider a with A, B, and C, and the to BC through A. Let ∠1 and ∠2 be the outside △ABC formed by this and the sides AB and AC.
By the , ∠B is to ∠1 and ∠C is congruent to ∠2.
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 . Therefore, by the their measures add to
180∘.
m∠BAC+m∠1+m∠2=180∘
By the , the sum of the measures of
∠BAC, ∠B, and
∠C is equal to
180∘.
m∠BAC+m∠1+m∠2=180∘⇓m∠BAC+m∠B+m∠C=180∘
Finally, in
△ABC, ∠BAC can be named
∠A.
m∠BAC+m∠B+m∠C=180∘⇓m∠A+m∠B+m∠C=180∘