# Application of Congruency of Triangles

Here we will prove some Application of congruency of triangles.

1. PQRS is a rectangle and POQ an equilateral triangle. Prove that SRO is an isosceles triangle.

Solution:

Given:

PQRS is a rectangle. POQ is an equilateral triangle to prove ∆SOR is an isosceles triangle.

Proof:

 Statement Reason 1. ∠SPQ = 90° 1. Each angle of a rectangle is 90° 2. ∠OPQ = 60° 2. Each angle of an equilateral triangle is 60° 3. ∠SPO = ∠SPQ - ∠OPQ = 90° - 60° = 30° 3. Using statements 1 and 2. 4. Similarly, ∠RQO = 30° 4. Proceeding as above. 5. In ∆POS and ∆QOR, (i) PO = QO (ii) PS = QR(iii) ∠SPO = ∠RQO = 30° 5. (i) Sides of an equilateral triangle are equal.(ii) Opposite sides of a rectangle are equal.(iii) From statements 3 and 4. 6. ∆POS ≅ ∆QOR 6. By SAS criterion of congruency. 7. SO = RO 7. CPCTC. 8. ∆SOR is an isosceles triangle. (Proved) 8. From statement 7.

2. In the given figure, triangle XYZ is a right angled at Y. XMNZ and YOPZ are squares. Prove that XP = YN.

Solution:

Given:

In ∆XYZ, ∠Y = 90°, XMNZ and YOPZ are squares.

To prove: XP = YN

Proof:

 Statement Reason 1. ∠XZN = 90° 1. Angle of square XMNZ. 2. ∠YZN = ∠YZX  + ∠XZN = x° + 90° 2. Using statement 1. 3. ∠YZP = 90° 3. Angle of square YOPZ. 4.  ∠XZP = ∠XZY + ∠YZP = x° + 90° 4. Using statement 3. 5. In ∆XZP and ∆YZN,(i) ∠XZP = ∠YZN(ii) ZP = YZ(iii) XZ = ZN 5.(i) Using statements 2 and 4.(ii) Sides of square YOPZ.(iii) Sides of square XMNZ. 6.  ∆XZP ≅ ∆YZN 6. By SAS criterion of congruency. 7. XP = YN. (Proved) 7. CPCTC.