Question

If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then prove that the two triangles are similar.

Hint:

To prove that two triangles are similar, either use the AA criterion (two corresponding angles equal) or show that the sides are proportional.

Answer 1

Let \( \triangle ABC \) and \( \triangle PQR \) be two triangles, where \( \angle A = \angle P \) and the sides including these angles are proportional. That is, \[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{PR}. \] We need to prove that the two triangles are similar. Step 1: Use the condition of proportional sides. By the condition of proportionality of the sides, we have: \[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{PR}. \] This implies that the corresponding sides of the triangles are proportional. Step 2: Apply the AA (Angle-Angle) criterion of similarity. We are also given that \( \angle A = \angle P \). Since two angles of one triangle are equal to two angles of the other triangle (the corresponding angle \( \angle A = \angle P \) and the corresponding angle \( \angle B = \angle Q \)), by the AA criterion of similarity, the two triangles are similar. Thus, \[ \triangle ABC \sim \triangle PQR. \]
Conclusion:
Since the corresponding angles are equal and the corresponding sides are proportional, by the AA criterion, the two triangles are similar.