Question

In the given figure, lines PQ, RS and TV intersect at O. If x : y : z = 1 : 2 : 3, then find the values of x, y and z.

• A. 20°, 40°, 80°
• B. 10°, 20°, 40°
• C. 30°, 60°, 90°
• D. 50°, 70°, 90°

Answer 1

The correct option is C 30°, 60°, 90°

The sum of all the angles around a point is 360°.

∠POR = ∠SOQ = x° (Pair of vertically opposite angles are equal)

∠VOQ = ∠POT = y° (Pair of vertically opposite angles are equal)

∠TOS = ∠ROV = z° (Pair of vertically opposite angles are equal)

Therefore, ∠POT + ∠POR + ∠ROV + ∠VOQ + ∠QOS + ∠SOT = 360°

y + x + z + y + x + z = 360°

⟹ 2x + 2y + 2z = 360°

⟹ 2(x + y + z) = 360°

⟹ x + y + z = 180° ...(i)

Let the common ratio be 'a'.

Since, x : y : z is in the ratio 1 : 2 : 3, we can write, x = a, y = 2a and z = 3a.

From the equation (i), we get,

a + 2a + 3a = 180°

⟹ 6a = 180°

⟹ a = 30°

Therefore, x = a = 30°

y = 2a = 2 × 30 = 60°

z = 3a = 3 × 30 = 90°

Therefore, the measures of the angles are 30°, 60°, 90°.