Question

If a triangle is equilateral then it has 3 congruent angles proof

Answer 1

Allow ABC to be an equilateral triangle so AB = AC = BC. Let X be the midpoint of BC, AX is the median of BC, and BX = CX. Look at triangles BAX and CAX. Obviously, AX = AX since AB = AC and BX = CX, subsequently by SSS Congruence Test, we have that triangles BAX and CAX are congruent. Therefore, corresponding angles are congruent so <ABX = <ACX. Because <ABX and <ABC are the same angle, they are obviously congruent, and then <ABX = <ABC. Likewise, <ACX and <ACB are the same angles so <ACX = <ACB. Then, <ABC = <ACB. Please note that <ABC = <B and <ACB = <C. Therefore, <B = <C. 

At present, disregard AX and let Y be the midpoint of AC. Then BY is the median of AC. Then AY = CY. Evidently, BY = BY.  Because AB = BC, we see that triangles ABY and CBY are congruent by SSS. Then corresponding sames are congruent so <BAY = <BCY. Since <BAY and <BAC are the same angles then <BAY = <BAC. Similarly, <BCY = <BCA since they are the same angles. Then <BAC = <BCA. Note that <BAC = <A and <BCA = <C. Therefore, <A = <C. 

Thus, <A = <C = <B. So all the angles are congruent.