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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Cyclic Quadrilaterals”.

1. Which among the following is a cyclic quadrilateral?
a) b) c) d) Explanation: All the angles of a cyclic quadrilateral lie on a circle (circumscribed circle) and sum of either pair of opposite angles of cyclic quadrilateral is 180˚. Among the given figures, only the answer figure satisfies the angle sum property of the quadrilateral and the conditions of cyclic quadrilateral.

2. Find the value of x if ABCD is a cyclic quadrilateral if ∠1 : ∠2 = 3 : 6. a) 90°
b) 45°
c) 60°
d) 20°

Explanation: Since ABCD is a cyclic quadrilateral, Sum of either pair of opposite angles of cyclic quadrilateral is 180°
⇒ ∠1 + ∠2 = 180°
⇒ 3k + 6k = 180° ⇒ k = 20°
Now, x = ∠1 [Exterior angle formed when one side of cyclic quadrilateral is produced is equal to the interior opposite angle]
⇒ x = 3k ⇒ x = 60°.

3. Find the value of ∠PTR if PQRS is a cyclic quadrilateral. a) 90°
b) 65°
c) 130°
d) 115°

Explanation: From figure, ∠POQ = 2∠PQR  [Angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference]
⇒ ∠PQR = 65°
Now, Since PQRS is a cyclic quadrilateral, Sum of either pair of opposite angles of cyclic quadrilateral is 180°. ⇒ ∠PQR + ∠PSR = 180° ⇒ ∠PSR = 115°
As angles subtended by an arc in the same segment are equal ⇒ ∠PTR = ∠PSR = 115°.

4. If PQRS is a cyclic quadrilateral and PQ is diameter, find the value of ∠PQS. a) 45°
b) 110°
c) 20°
d) 80°

Explanation: Since PQRS is a cyclic quadrilateral, Sum of either pair of opposite angles of cyclic quadrilateral is 180°. ⇒ ∠QRS+ ∠QPS = 180° ⇒ ∠QPS = 70°  ————–(i)
Also, ∠PSQ = 90°  [Angle in a semicircle] ———-(ii)
In ΔPSQ, ∠PSQ + ∠SPQ + ∠SQP = 180°  [Angle sum property of triangle]
⇒ 90° + 70° + ∠PQS = 180°  [from equation i and ii]
⇒ ∠PQS = 20°.

5. Find the value of x and y if ABCD is cyclic quadrilateral. a) 60°, 60°
b) 50°, 60°
c) 45°, 45°
d) 80°, 90°

Explanation: Since PQRS is a cyclic quadrilateral, Sum of either pair of opposite angles of cyclic quadrilateral is 180°.
From figure, ∠BAD + ∠BCD = 180° ⇒ x + 2y = 180°  ————-(i)
and ∠ADC + ∠CBA = 180° ⇒ (x + y) + (2x – y) = 180°  ————-(i)
Solving equation i and ii, x = 60° and y = 60°.

6. What is the value of ∠PQR if PQRS is cyclic quadrilateral and PS = SR? a) 90°
b) 70°
c) 40°
d) 30°

Explanation: In ΔPSR, PS = PR ⇒ ∠PRS = ∠SPR = 20°  [Angles opposite to equal sides are equal]
Now, ∠PSR + ∠SPR + ∠SRP = 180°  [Angle sum property of triangle]
⇒ ∠PSR + 20° + 20° = 180°
⇒ ∠PSR = 140°
Since PQRS is a cyclic quadrilateral, Sum of either pair of opposite angles of cyclic quadrilateral is 180°. ⇒ ∠PQR + ∠PSR = 180° ⇒ ∠PQR = 40°.

7. What is the value of ∠PRQ if ∠PSR : ∠PQR = 1 : 2? a) 50°
b) 10°
c) 90°
d) 45°

Explanation: Since PQRS is a cyclic quadrilateral, Sum of either pair of opposite angles of cyclic quadrilateral is 180°. ⇒ ∠PSR + ∠PQR = 180° ⇒ k + 2k = 180° ⇒ k = 60°
Hence, ∠PSR = 60° and ∠PQR = 120°.
In ΔPQR, ∠PQR + ∠PRQ + ∠RPQ = 180°  [Angle sum property of triangle]
⇒ ∠PRQ + 120° + 50° = 180°
⇒ ∠PRQ = 10°.

8. Find the value of ∠PQR if PS || RQ and PQRS is cyclic quadrilateral. a) 45°
b) 50°
c) 80°
d) 90°

Explanation: Since PQRS is a cyclic quadrilateral, Sum of either pair of opposite angles of cyclic quadrilateral is 180°. ⇒ ∠P + ∠R = 180° ⇒ ∠P = 100° ————-(i)
Now, PS || RQ ⇒ ∠SPQ + ∠PQR = 180°  [Sum of interior angles]
⇒ ∠PQR + 100° = 180°
⇒ ∠PQR = 80°.

9. What is the value of x if ∠AOC if ABCD is cyclic quadrilateral? a) 140°
b) 110°
c) 70°
d) 45°

Explanation: If one of the side of cyclic quadrilateral is produced, then the exterior angle is equal to the opposite interior angle. ⇒ ∠CBE = ∠ADC = 70°
As angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference, ∠AOC = 2∠ADC = 2 x 70° = 140°.

10. From the figure given below, a quadrilateral ABCD is cyclic. a) True
b) False

Explanation: A quadrilateral is said to be cyclic if four vertices of it lies on a circle.
We can see that four vertices A, B, C and D lies on a circle and hence given quadrilateral is cyclic.

11. From the figure given below, ∠B + ∠D = __________ a) 90°
b) 270°
c) 180°
d) 145°

Explanation: According to theorem 10.11, the sum of either part of opposite angles of a cyclic quadrilateral is 180°.
We can see that ∠B and ∠D are opposite pair of angles.
Therefore, ∠B + ∠D = 180°
Similarly, ∠A + ∠C = 180°

12. From the figure given below, ∠PAB = __________ a) 90°
b) 110°
c) 95°
d) 75°

Explanation: ∠APB = 90° and ∠PQA = 45° (Given)
We know that angles in the same segment are equal.
Hence, ∠APB = ∠AQB = 60°
Now, ∠PQB = ∠PQA + ∠AQB = 45° + 60°
∠PQB = 105°
∠PQB and ∠PAB are opposite pairs of angles of a quadrilateral.
Therefore, ∠PQB + ∠PAB = 180°
∠PAB = 180° – ∠PQB
= 180° – 105°
∠PAB = 75°

Sanfoundry Global Education & Learning Series – Mathematics – Class 9.

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