Let’s assume that we have a triangle ABC, where angle A = angle B. We want to prove that side AC = side BC.
To prove this, we can use the congruence of triangles. If we can show that triangle ABC is congruent to triangle BAC, then we can conclude that side AC is equal to side BC.
To prove the congruence of triangles, we need to show that the corresponding sides and angles of the two triangles are equal.
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Corresponding angles: We are given that angle A = angle B. Since angles A and B are corresponding angles of the two triangles, we can conclude that angle BAC = angle ABC.
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Corresponding sides: We want to show that side AC = side BC. To do this, we can use the fact that the sum of the angles in a triangle is 180 degrees. Since angle A = angle B, we can write the equation: angle A + angle B + angle C = 180 degrees. Substituting angle A = angle B, we get: angle B + angle B + angle C = 180 degrees. Simplifying, we have: 2 * angle B + angle C = 180 degrees.
Now, let’s consider triangle BAC. We know that angle BAC = angle ABC (from step 1). Therefore, we can rewrite the equation as: 2 * angle B + angle BAC = 180 degrees.
Since the sum of the angles in a triangle is 180 degrees, we can conclude that angle BAC = angle C. Therefore, we have: 2 * angle B + angle C = 180 degrees and 2 * angle B + angle BAC = 180 degrees. This implies that angle C = angle BAC.
Now, we have corresponding angles angle BAC = angle ABC and angle C = angle BAC. Therefore, we can conclude that triangle ABC is congruent to triangle BAC by the angle-angle-side congruence criterion.
Since the corresponding sides of congruent triangles are equal, we can conclude that side AC = side BC.
Therefore, if two angles of a triangle are equal, then the sides opposite those angles are also equal.