Application of Congruency of Triangles

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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.

9th Grade Math

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