Question

PQR and XTZ are two triangles in which PQ - XY and _PRQ = 70°, ZPQR = 50°, 2XYZ = 70", and 2YXZ = 60°. Prove that the two triangles are congruent.
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Answer 1

In a triangle, the sum of three angles is 180°.

Therefore, in PQR, ∠PRQ + ∠PQR + ∠QPR = 180°.

Therefore, 70° + 50° + ∠QPR = 180°

⟹ ∠QPR = 180° – (70° + 50°)

⟹ ∠QPR = 180° – 120°

⟹ ∠QPR = 60°.

In ∆PQR and ∆XYZ,

PQ = XZ, ∠PRQ = ∠XYZ = 70° and ∠QPR = ∠YXZ = 60°.

Therefore, by AAS (Angle-Angle-Side) criterion, the two triangles are congruent.