Proving the Converse of the Isosceles Triangle Theorem: If a Triangle has Two Congruent Sides, its Opposite Angles are Congruent

Converse of the Isosceles Triangle Theorem

The Converse of the Isosceles Triangle Theorem states that if a triangle has two congruent sides, then the angles opposite those sides are also congruent

The Converse of the Isosceles Triangle Theorem states that if a triangle has two congruent sides, then the angles opposite those sides are also congruent.

To understand this theorem better, let’s break it down step by step.

Step 1: Triangle with two congruent sides
Consider a triangle with two sides that are congruent. This means that two of the sides of the triangle are of equal length. Let’s call these sides AB and AC.

Step 2: Converse statement
The converse of the Isosceles Triangle Theorem states that if AB and AC are congruent, then the angles opposite these sides are congruent. Let’s call these angles ∠B and ∠C.

Step 3: Using the theorem
To apply the theorem, suppose we have a triangle ABC with AB ≅ AC.

Step 4: Proof
We need to prove that ∠B ≅ ∠C.

Step 5: Constructing a line segment
To start the proof, we draw a line segment AD from the vertex A to the midpoint D of the base BC. Since the two sides AB and AC are congruent, we know that BD = DC.

Step 6: Using triangle congruence
By the Side-Side-Side (SSS) congruence criterion, triangle ABD is congruent to triangle ACD.

Step 7: Congruent angles
Since triangle ABD ≅ triangle ACD, their corresponding angles are congruent. This means that ∠BAD ≅ ∠CAD.

Step 8: Vertical angles
∠BAD and ∠CAD are vertical angles. It is a geometric property that vertical angles are congruent. Therefore, ∠B ≅ ∠C.

Step 9: Conclusion
From the above steps, we have proved that if a triangle has two congruent sides (AB ≅ AC), then the angles opposite those sides (∠B and ∠C) are also congruent.

So, the converse of the Isosceles Triangle Theorem holds true.

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