Cyclic Quadrilateral Opposite Angles (Motivate) MCQs Quiz | Class 9
This quiz is designed for Class IX Mathematics (Code 041), Unit IV: Geometry. It focuses on the topic of Cyclic Quadrilaterals, specifically covering the property that the sum of opposite angles is 180° and its converse. Complete the 10 questions below, submit your answers, and download the PDF solution sheet for your records.
Study Notes: Cyclic Quadrilaterals
A Cyclic Quadrilateral is a quadrilateral whose vertices all lie on a single circle. This topic is a key part of Unit IV: Geometry for Class 9 Mathematics.
Key Theorems & Properties
- Main Property: The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees (supplementary).
If ABCD is a cyclic quadrilateral, then ∠A + ∠C = 180° and ∠B + ∠D = 180°. - Converse Property: If the sum of a pair of opposite angles of a quadrilateral is 180 degrees, the quadrilateral is cyclic. This means the four vertices are concyclic (lie on the same circle).
- Exterior Angle Property: If one side of a cyclic quadrilateral is produced, the exterior angle so formed is equal to the interior opposite angle.
- Special Cases:
- A cyclic parallelogram is always a rectangle.
- A cyclic trapezium is always an isosceles trapezium (non-parallel sides are equal).
Quick Revision Summary
| Condition | Inference |
|---|---|
| Opposite angles sum to 180° | Quadrilateral is Cyclic |
| Vertices on a circle | Opposite angles are supplementary |
| Cyclic Parallelogram | Must be a Rectangle |
Extra Practice Questions
- If ABCD is a cyclic quadrilateral and ∠A = 100°, find ∠C. (Ans: 80°)
- PQRS is cyclic. If ∠P = 3x and ∠R = x, find x. (Ans: 45)
- In a cyclic quadrilateral, the exterior angle at vertex B is 95°. Find the interior angle D. (Ans: 95°)
- Can a rhombus be a cyclic quadrilateral? (Ans: Yes, only if it is a square).
- If diagonals of a cyclic quadrilateral are diameters of the circle, what kind of quadrilateral is it? (Ans: Rectangle).
