Cyclic quadrilaterals: Class 9 practice
In a cyclic quadrilateral (all four vertices on a circle) opposite angles add up to 180°. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Angle sum of a quadrilateral
The idea behind cyclic quadrilaterals
In a cyclic quadrilateral (all four vertices on a circle) opposite angles add up to 180°. The exterior angle equals the interior opposite angle.
Step-by-step method
- Check the quadrilateral is cyclic.
- Opposite angles sum to 180°.
- Use the exterior-angle fact if an exterior angle is given.
Worked example 1 (Basic)
ABCD is cyclic with ∠A = 123° and ∠B = 117°. Find ∠C and ∠D.
- ∠C = 180° − 123° = 57°
- ∠D = 180° − 117° = 63°
- Answer: ∠C = 57°, ∠D = 63°
Worked example 2 (Standard)
ABCD is cyclic; side DC is produced to E and ∠BCE = 107°. Find ∠A.
- Exterior angle = interior opposite angle
- ∠A = 107°
- Answer: 107°
Worked example 3 (Challenge)
ABCD is cyclic and ∠C is 34° more than ∠A. Find ∠A and ∠C.
- ∠A + (∠A + 34°) = 180°
- 2∠A = 146° → ∠A = 73°, ∠C = 107°
- Answer: ∠A = 73°, ∠C = 107°
Common mistakes and misconceptions
- Using adjacent angles instead of opposite
- Applying it to any quadrilateral
Practice questions
- (Basic) ABCD is cyclic with ∠A = 73° and ∠B = 114°. Find ∠C and ∠D.
- (Basic) ABCD is cyclic with ∠A = 61° and ∠B = 115°. Find ∠C and ∠D.
- (Standard) ABCD is cyclic; side DC is produced to E and ∠BCE = 115°. Find ∠A.
- (Standard) ABCD is cyclic; side DC is produced to E and ∠BCE = 82°. Find ∠A.
- (Challenge) ABCD is cyclic and ∠C is 64° more than ∠A. Find ∠A and ∠C.
- (Challenge) ABCD is cyclic and ∠C is 30° more than ∠A. Find ∠A and ∠C.
Answer key with full working
- Q1: ∠C = 180° − 73° = 107° → ∠D = 180° − 114° = 66° → Answer: ∠C = 107°, ∠D = 66°
- Q2: ∠C = 180° − 61° = 119° → ∠D = 180° − 115° = 65° → Answer: ∠C = 119°, ∠D = 65°
- Q3: Exterior angle = interior opposite angle → ∠A = 115° → Answer: 115°
- Q4: Exterior angle = interior opposite angle → ∠A = 82° → Answer: 82°
- Q5: ∠A + (∠A + 64°) = 180° → 2∠A = 116° → ∠A = 58°, ∠C = 122° → Answer: ∠A = 58°, ∠C = 122°
- Q6: ∠A + (∠A + 30°) = 180° → 2∠A = 150° → ∠A = 75°, ∠C = 105° → Answer: ∠A = 75°, ∠C = 105°
Syllabus note
This skill is listed under "Circles" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is cyclic quadrilaterals in?
"Circles", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.