Here we will learn how to prove different types of problems on congruency of triangles.

**1.** PQR and XYZ are two triangles in which PQ = XY and ∠PRQ
= 70°, ∠PQR = 50°, ∠XYZ = 70°, and ∠YXZ = 60°. Prove that the two triangles are
congruent.

**Solution:**

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.

**2.** In the given figures, prove that two triangles are
congruent.

**Solution:**

In ∆ABC, ∠BAC + ∠ABC + ∠BCA = 180°

⟹ 65° + ∠ABC +55° = 180°

⟹ ∠ABC = 60°.

In ∆ABC and ∆XYZ,

AB = XZ = 4 cm, BC = YZ = 5 cm and ∠ABC = ∠XZY = 60°.

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

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