Mathematics Questions and Answers – Cyclic Quadrilaterals

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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)

View Answer

Answer: b
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.
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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°
View Answer

Answer: d
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°
View Answer

Answer: d
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°.
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4. If PQRS is a cyclic quadrilateral and PQ is diameter, find the value of ∠PQS.

a) 45°
b) 110°
c) 20°
d) 80°
View Answer

Answer: c
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°
View Answer

Answer: a
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°.
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6. What is the value of ∠PQR if PQRS is cyclic quadrilateral and PS = SR?

a) 90°
b) 70°
c) 40°
d) 30°
View Answer

Answer: c
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°
View Answer

Answer: b
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°.
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8. Find the value of ∠PQR if PS || RQ and PQRS is cyclic quadrilateral.

a) 45°
b) 50°
c) 80°
d) 90°
View Answer

Answer: c
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°
View Answer

Answer: a
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°.
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10. From the figure given below, a quadrilateral ABCD is cyclic.

a) True
b) False
View Answer

Answer: a
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°
View Answer

Answer: c
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°
View Answer

Answer: d
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°

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter