Question

to prove that 'an equilateral triangle is equiangular' draw figure write given and to prove​

Answer 1

Need to prove:

An equilateral triangle is equiangular

Let's make a triangle ABC where all three sides are equal. i.e. the triangle is equilateral and AB = BC = CA

Proof:

Since angle opposite to equal sides of a triangle are equal:

AB = BC

∠CAB = ∠BCA . . . (1)

Likewise,

AC = BC.

∠CAB = ∠CBA . . . (2)

From (1) and (2), we get:

∠CAB = ∠CBA = ∠BCA

We've proved that all three angles are of same measure, So the triangle will be an equiangular triangle. We can go even further:

As we know, sum of all angles of a triangle is 180°

∠CAB + ∠CBA + ∠BCA = 180°

As ∠CBA and ∠BCA are same as ∠CAB, we get this equation:

∠CAB + ∠CAB + ∠CAB = 180°

3∠CAB = 180°

∠CAB = 180°/3

∠CAB = 60°

∠CAB = ∠CBA = ∠BCA

60° = ∠CBA = ∠BCA

Proved!