Isosceles Triangle Theorem: Explained with Proof and Applications

Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that in an isosceles triangle, the two legs (sides) are congruent

The Isosceles Triangle Theorem states that in an isosceles triangle, the two legs (sides) are congruent. This means that if two sides of a triangle are equal in length, then the angles opposite those sides are also equal.

Let’s denote the triangle as ΔABC, where AB = AC. To prove the Isosceles Triangle Theorem, we can consider the following:

1. Construct the line segment AD, where D is the midpoint of BC. (This can be done by drawing the perpendicular bisector of BC from point A).

2. Since D is the midpoint of BC, we have BD = CD.

3. Now, consider the triangles ΔABD and ΔACD. We have AB = AC (given), BD = CD (midpoint property), and AD is common.

4. By applying the Side-Angle-Side (SAS) congruence criterion, we can conclude that ΔABD ≅ ΔACD.

5. Therefore, the corresponding angles are congruent, which implies that ∠BAD = ∠CAD.

Hence, we have shown that in an isosceles triangle, if two sides are equal, then the angles opposite those sides are also equal. This is known as the Isosceles Triangle Theorem.

This theorem is particularly useful when solving problems involving isosceles triangles, as it allows us to make certain assumptions about the triangle’s angles or sides based on the given information.

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