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

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.

good luck on your studies! : )

david edward.

step-by-step explanation:

this can be solved first with pythagorean theorem, which tells us to square both legs of the triangle and sum it. this will be equal to the square of the hypotenuse.

the hypotenuse is the side "opposite" the 90 degree angle, which is x.

so we can write using pythagorean theorem,

simplifying:

to further simplify this radical, we need to use the property  

thus