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The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)° ∠C = (x + y)°, ∠D = (3y - 10)° Find x and y, and hence the values of the four angles

Solution:

Given,

∠A = (6x + 10)°

∠B = (5x)°

∠C = (x + y)°

∠D = (3y - 10)°

The property of cyclic quadrilateral says that,

Sum of opposite angles = 180°

∠A + ∠C = (6x + 10)° + (x + y)° = 180°

7x + y = 170----------(1)

∠B + ∠D = (5x)° + (3y - 10)° = 180°

5x + 3y = 190----------(2)

Let us solve the linear equations (1) and (2)

Multiplying (1) by 3 and then subtracting (2), we get,

3 x (7x + y) - (5x + 3y) = 510° - 190°

21x + 3y - 5x - 3y = 320°

16x = 320°

x = 20°.

Substitute x = 20° in (1),we get,

7 x 20 + y = 170°

y = 170° - 140°

y = 30°.

∠A = (6x + 10)° = 6 x 20° + 10° = 120° + 10° = 130°

∠B = (5x)° = 5 x 20° = 100°

∠C = (x + y)° = 20° + 30° = 50°

∠D = (3y - 10)° = 3 x 30° - 10°= 90° - 10° = 80°.

∠A = 130°

∠B = 100°

∠C = 50°

∠D = 80°

Therefore, the required values of x and y are 20° and 30°, respectively and the values of the four angles i.e., ∠A, ∠B, ∠C, and ∠D are 130°, 100°, 50° and 80°, respectively.

✦ Try This: The angles of a cyclic quadrilateral ABCD are ∠A = (5x + 10)°, ∠B = (6x)° ∠C = (x + y)°, ∠D = (3y - 20)° Find x and y, and hence the values of the four angles

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 3

NCERT Exemplar Class 10 Maths Exercise 3.3 Problem 22

The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)° ∠C = (x + y)°, ∠D = (3y - 10)° Find x and y, and hence the values of the four angles

Summary:

The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)° ∠C = (x + y)°, ∠D = (3y - 10)°, the required values of x and y are 20° and 30°, respectively and the values of the four angles i.e., ∠A, ∠B, ∠C, and ∠D are 130°, 100°, 50° and 80°, respectively

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