Proving the Isosceles Triangle Theorem | Congruence of Sides and Angles

Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that if a triangle has two sides that are congruent (of equal length), then the angles opposite those sides are also congruent (of equal measure)

The Isosceles Triangle Theorem states that if a triangle has two sides that are congruent (of equal length), then the angles opposite those sides are also congruent (of equal measure).

Let’s consider an isosceles triangle ABC. If side AB is congruent to side AC, then angle A is congruent to angle B and angle A is also congruent to angle C. In other words, the angles that are opposite the congruent sides of an isosceles triangle are equal.

To prove this theorem, we can use the fact that the sum of the angles in any triangle is equal to 180 degrees. Since we know that angle A + angle B + angle C is equal to 180 degrees, and that angle A and angle C are congruent, we can write the equation as:

angle A + angle B + angle A = 180 degrees

Combining like terms, this simplifies to:

2(angle A) + angle B = 180 degrees

Since angle A and angle B share a common side, we can say that angle A + angle B is equal to the measure of angle BAC. Therefore, we can rewrite the equation as:

2(angle BAC) = 180 degrees

Dividing both sides by 2, we get:

angle BAC = 90 degrees

This means that angle BAC is a right angle. So, in an isosceles triangle, the vertex angle (angle BAC) is always a right angle, and the base angles (angle ABC and angle ACB) are congruent to each other.

In summary, the Isosceles Triangle Theorem states that if a triangle has two sides that are congruent, then the angles opposite those sides are congruent.

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