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Madhyamik Mathematics – Cyclic Quadrilateral Related Theorems (Chapter 10) | Questions and Answers
15 Sept 2026 0 views
Cyclic Quadrilateral Related Theorems is an important topic in WBBSE Class 10 Mathematics. This chapter focuses on the properties of cyclic quadrilaterals, supplementary opposite angles, exterior angles, cyclic trapeziums and proof-based geometry problems.
The practice set below is organised into MCQ, True or False, Fill in the Blanks, Short Answer and Long Answer sections so that students can revise the chapter in a structured way.
Cyclic Quadrilateral Related Theorems – Chapter 10
Core Idea: A quadrilateral whose four vertices lie on the same circle
is called a cyclic quadrilateral. The chapter mainly applies the
angle properties of such quadrilaterals to solve numerical and proof-based problems.
Important Properties to Remember
- The opposite angles of a cyclic quadrilateral are supplementary.
- If a side of a cyclic quadrilateral is extended, the exterior angle is equal to the opposite interior angle.
- A cyclic parallelogram is a rectangle.
- A cyclic trapezium is an isosceles trapezium.
- The vertices of a square are concyclic.
- The diagonal of a square inscribed in a circle is the diameter of that circle.
Multiple Choice Questions – MCQ
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1. ABCD is a cyclic quadrilateral and BC is extended to P.
If ∠BAD : ∠BCD = 2 : 3, what is the value of ∠BCD?
(a) 54°
(b) 72°
(c) 36°
(d) 108°
Answer: (c) 36°
2. In a circle with centre O, ABCD is a cyclic quadrilateral.
If the given central-angle condition is satisfied, determine the ratio
∠ADC : ∠ABC.
(a) 5 : 6
(b) 1 : 2
(c) 5 : 13
(d) 3 : 13
Answer: (b) 1 : 2
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True or False
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1. If any side of a cyclic quadrilateral is extended,
the exterior angle formed is equal to the opposite interior angle.
Answer: True
2. If the opposite sides of a quadrilateral are parallel,
then the other two sides will be equal and its diagonals will also be equal.
Answer: True
3. The sum of the two angles adjacent to any one side of
a cyclic quadrilateral is always 180°.
Answer: False
4. The opposite angles of a cyclic quadrilateral are
complementary angles.
Answer: False
5. A cyclic parallelogram is a rectangle.
Answer: True
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Fill in the Blanks
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| No. | Statement | Answer |
|---|---|---|
| 1 | The diagonal of a square inscribed in a circle is the ______ of the circle. | Diameter |
| 2 | If the opposite angles of a quadrilateral are supplementary, its vertices are ______. | Concyclic |
| 3 | The diagonal of an inscribed square is the ______ of the circle. | Diameter |
| 4 | The vertices of a square are ______. | Concyclic |
| 5 | The vertices of a square are ______. | Concyclic |
| 6 | A cyclic trapezium is an ______ trapezium. | Isosceles |
Short Answer Questions
```
1. ABCD is a cyclic quadrilateral with AB ∥ DC and centre O.
If ∠BOC = 80° and ∠AOC = 10°, determine the value of ∠BAD.
Focus: This problem requires the use of central angles,
parallel-line properties and the angle relationships of a cyclic quadrilateral.
2. ABCD is a cyclic trapezium in which ∠BAD = 60°.
Determine the remaining angles of the trapezium.
Key concept: Use the properties of a cyclic quadrilateral
together with the properties of an isosceles trapezium.
```
Long Answer Questions
```
Problem 1 – Prove that O, R, A and S are Concyclic
O is the centre of a circle. AP and AQ are two chords whose midpoints are
R and S respectively. Prove that the four points O, R, A and S are concyclic.
Proof
Given: O is the centre of the circle and R and S are the
midpoints of chords AP and AQ respectively.
Since the line joining the centre of a circle to the midpoint of a chord
is perpendicular to the chord, OR ⟂ AP.
Therefore, ∠ARO = 90°.
Similarly, OS is perpendicular to AQ, so ∠ASO = 90°.
Hence,
∠ARO + ∠ASO = 90° + 90° = 180°.
Thus, the pair of opposite angles of quadrilateral AROS are supplementary.
Therefore, AROS is a cyclic quadrilateral.
Hence proved: O, R, A and S are concyclic.
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Problem 2 – Prove that P, Q, C and D are Concyclic
ABCD is a parallelogram. A circle passing through A and B intersects
AD and BC at P and Q respectively. Prove that P, Q, C and D are concyclic.
Proof
Since ABCD is a parallelogram,
AD ∥ BC.
Therefore, using AB as a transversal,
∠DAB + ∠ABC = 180°.
Since A, B, P and Q lie on the same circle, ABQP is a cyclic quadrilateral.
Therefore,
∠PQC = ∠PAB.
Also, because ABCD is a parallelogram,
∠PDC = ∠ADC = ∠ABC.
Hence,
∠PQC + ∠PDC = ∠PAB + ∠ABC = 180°.
Thus, one pair of opposite angles of quadrilateral PQCD are supplementary.
Therefore, PQCD is a cyclic quadrilateral.
Hence proved: P, Q, C and D are concyclic.
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Chapter 10 Quick Revision
| Concept | What to Remember |
|---|---|
| Cyclic Quadrilateral | All four vertices lie on the same circle. |
| Opposite Angles | The opposite angles of a cyclic quadrilateral are supplementary. |
| Exterior Angle | An exterior angle is equal to the opposite interior angle. |
| Cyclic Parallelogram | A cyclic parallelogram is a rectangle. |
| Cyclic Trapezium | A cyclic trapezium is isosceles. |
| Inscribed Square | The diagonal of an inscribed square is the diameter of its circle. |
| Concyclicity Test | If a pair of opposite angles of a quadrilateral are supplementary, its vertices are concyclic. |
Mathematics Revision Resources
For broader WBBSE Class 10 preparation, students can also explore the Madhyamik Suggestion 2027 – All Subjects resource on Cademy.
```The Cademy resource brings together subject-wise revision materials, including Mathematics, for WBBSE Class 10 students.
```
Study Tip: For this chapter, do not rely only on memorising
theorems. Practise identifying which angle relationship or cyclic-quadrilateral
property is required in each problem. This makes proof-based questions much easier
to approach.