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 (AngleAngleSide) 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 (SideAngleSide) criterion the two triangles are congruent.
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