Question

Prove that the angles of an equilateral triangle are $60{}^\circ $ each.

Answer 1

- Hint: First, start by making a diagram of an equilateral triangle. Use the property that angles opposite to equal sides are equal to get $\angle $A = $\angle $B = $\angle $C. Next use the angle sum property to get $\angle $A + $\angle $B + $\angle $C = $180{}^\circ $. Substitute the previous relation in this equation to get the final answer.

Complete step-by-step answer:
In this question, we need to prove that the angles of an equilateral triangle are $60{}^\circ $ each.
Let us first draw the diagram of an equilateral triangle.

As we can see from the diagram, all sides of an equilateral triangle are equal in length.
Assuming an equilateral triangle ABC,
Then, AB = AC = BC.
Now, we know the property that angles opposite to equal sides are equal.
Using this property, we will get the following:
$\angle $A = $\angle $B = $\angle $C
Now, we know that the angle sum property of a triangle states that the sum of all the angles of a triangle is equal to $180{}^\circ $.
Using this property, we will get the following:
$\angle $A + $\angle $B + $\angle $C = $180{}^\circ $
Substituting $\angle $A = $\angle $B = $\angle $C in the above equation, we will get the following:
$\angle $A + $\angle $A + $\angle $A = $180{}^\circ $
$\Rightarrow \angle $A = $60{}^\circ $
$\Rightarrow $$\angle $A = $\angle $B = $\angle $C = $60{}^\circ $
So, each angle of an equilateral triangle is $60{}^\circ $.
Hence proved.

Note: The angle sum property of a triangle is valid for all triangles, i.e. scalene, isosceles or equilateral. But the property that all the angles are equal is valid only for equilateral triangles. Also, in case of equilateral triangles, the angle is always fixed, i.e. ${{60}^{o}}$ . But, in case of other triangles, the angles may vary from one triangle to another.