Angles Of Equilateral Triangle

Theorem: All the three interior angles of an equilateral triangle are equal to $60^o$

Prerequisites:
Equilateral triangle (definition)
Isosceles triangle theorem (proof)
Angle sum property of a triangle (proof)

Proof:

Let $\triangle ABC$ be an equilateral triangle. Then by the definition of an equilateral triangle, all the sides are equal.

Since AB $=$ AC, hence by using the isosceles triangle theorem,

$\qquad\quad\angle B = \angle C$

Similarly, since AB $=$ BC, therefore

$\qquad\quad\angle A = \angle C$
$\therefore\quad\;\;\;\angle A = \angle B = \angle C = \theta\qquad\qquad\qquad\qquad\cdots\text{(1)}$

Further, by the angle sum property of a triangle,

$\qquad\quad\angle A +\angle B + \angle C= 180^o$
$\Rightarrow\quad\;\; 3\theta = 180^o\qquad\qquad\qquad\qquad\qquad\qquad\;\;\text{(from $(1)$)}\\ \Rightarrow\quad\;\; \theta = 60^o$

Q.E.D.


Recommended:
Converse of Isosceles Theorem
Pythagoras Theorem
Basic Proportionality Theorem