A triangle is a closed polygon having three sides, three internal angles. While studying geometry and measurement, you can across some properties of a triangle that makes the concept more interesting. Here we are giving one of the isosceles triangle theorems that states that sides opposite to the equal angles of a triangle are equal. Check the following sections to know more about the concept.

## Prove That Sides Opposite to the Equal Angles of a Triangle are Equal

Here we will prove that the sides opposite to the equal angles of a triangle are equal. We have already given the proof for the theorem Angles Opposite to Equal Sides of an Isosceles Triangle are Equal.

Proof:
Consider a triangle ABC, ∠CAB = ∠CBA
We have to prove AC = BC

Construction: Draw the bisector CD of ∠BCA so that it meets AB at a point D.

Statement Reason
In ∆ACD and ∆BCD
∠CAB = ∠CBA
CD = CD
Given
CD bisects ∠ACB
Common side
∆BCD ≅ ∆ACD AAS Criterion

Hence proved.

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### Solved Example Questions

Question 1:
If a : b : c = 2 : 3 : 4 and s = 27 inches, find the area of the triangle ABC.

Solution:
Given that,
a : b : c = 2 : 3 : 4
Let a = 2x, b = 3x, c = 4x
a + b + c = 2x + 3x + 4x = 9x
Therefore, 9x = 2s
9x = 2 x 27
x = 6
Therefore, the lengths of three sides are 2 x 6 = 12 inches, 3 x 6 = 18 inches, 4 x 6 = 24 inches.
Area of the triangle ABC = √(s(s – a)(s – b)(s – c))
= √(27(27 – 12)(27 – 18)(27 – 24))
= √(27 x 15 x 9 x 3)
= 27√15 sq inches

Question 2:
Find x° from the below figures.

Solution:
In ∆XYZ, XY = XZ.
angles opposite to equal sides of an isosceles triangle are equal
Therefore, ∠XYZ = ∠XZY = x°
∠YXZ + ∠XYZ + XZY = 180°
84° + x° + x° = 180°
2x° = 180° – 84° = 96
x° = 48°

### FAQ’s on Isosceles Triangle Theorem

1. When two sides of the triangle are equal two of its angles are also?

In an isosceles triangle, if two sides of a triangle are equal, then two of its angles are also equal.

2. When the sides of one triangle are equal to the side of another triangle?

Two triangles are congruent if all the sides of one triangle are equivalent to the corresponding sides of another triangle?

3. What is the meaning of angle opposite to equal sides?

In an isosceles triangle, the angles opposite to equal sides are equal.