# Theorem, Proof of Equal Sides of an Isosceles Triangle are Produced, the Exterior Angles are equal

An isosceles triangle is a special kind of triangle that has two sides of equal length, no matter in what direction the peak of the triangle points. The important properties of an isosceles triangle are it has two equal sides, two equal angles (base angles). According to the isosceles triangle theorem if equal sides of an isosceles triangle are produced, then the exterior angles are equal. The detailed proof for the statement is given here.

## Equal Sides of an Isosceles Triangle are Produced, Exterior Angles are Equal

Here is the proof of the statement of equal sides of an isosceles triangle are produced, the exterior angles are equal and obtuse.

Proof:
Let us take an isosceles triangle ABC. The equal sides AB, AC are produced to D and E.
To prove: ∠CBD = ∠BCE. Statement Reason
∠ABC = ∠ACB Angles opposite to equal sides are equal
∠CBD = 180 – ∠ABC Linear pair
∠BCE = 180 – ∠ACB Linear Pair
∠BCE = 180 – ∠ABC Angles opposite to equal sides are equal
Therefore, ∠CBD = ∠BCE From 2 & 4 statements

Hence, proved.

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### Problems on Isosceles Triangle

Problem 1:
If the two equal sides of an isosceles triangle are 3x – 1, 2x + 2. The third side is 2x. Find the value of x and the perimeter of the triangle.

Solution:
Given that,
Equal sides of an isosceles triangle are 3x – 1, 2x + 2. So, equate them.
3x – 1 = 2x + 2
3x – 2x = 2 + 1
x = 3
Third side of an isosceles triangle is 2x
Third side = 2 x 3 = 6
First two sides are 3x – 1, 2x + 2
= 3(3) – 1, 2(3) + 2
= 9 – 1, 6 + 2
= 8, 8
Perimeter of the triangle = 8 + 8 + 3
= 19.
Therefore, the value of x is 3 units, perimeter of the triangle is 19 units.

Problem 2:
The base BC of triangle ABC has produced both ways and the measure of exterior angles formed are 94°, 126°. Find ∠BAC.

Solution:
Given that,
Base = BC
∠ACD = 126°, ∠ABE = 94°
We know that,
∠ABE + ∠ABC = 180°
∠ABC = 180° – ∠ABE = 180° – 94°
= 86°
Again, ∠ACB + ∠ACD = 180°
∠ACB = 180° – ∠ACD = 180° – 126°
= 54°
In ΔABC
∠ABC + ∠ACB + ∠BAC = 180°
86° + 54° + ∠BAC = 180°
∠BAC = 180° – 140° = 40°

1. What are the rules of an isosceles triangle?

In an isosceles triangle, two sides are congruent, the third side is called the base. Two angles opposite to equal sides are congruent.

2. What two angles are equal in an isosceles triangle?

Two sides are equal in the measure in an isosceles triangle called legs. The angles opposite to legs are also equal.

3. What happens when equal sides of an isosceles triangle are produced?

If the equal sides of an isosceles triangle are produced, then the exterior angles are equal and obtuse.

4. What are the 3 properties of an isosceles triangle?

• It has two sides of equal length.
• The angles opposite to equal sides are equal in measure.
• The altitude from one vertex to the base is the perpendicular bisector of the base.