Cyclic Quadrilaterals — O/L Mathematics

What must I remember about Cyclic Quadrilaterals?

• Cyclic Quadrilateral Definition: A quadrilateral whose all four vertices lie on the circumference of a single circle.
• Opposite Angles Theorem: The opposite interior angles of a cyclic quadrilateral are SUPPLEMENTARY (sum to 180°): ∠A + ∠C = 180° and ∠B + ∠D = 180°.
• Exterior Angle Theorem: The exterior angle formed by producing one side of a cyclic quadrilateral is EQUAL to the interior opposite angle.
• Converse Theorems for Proving Cyclic: 1) If a pair of opposite angles of a quadrilateral are supplementary, it is cyclic; 2) If the exterior angle equals the interior opposite angle, it is cyclic; 3) If a line segment subtends equal angles at two points on the same side, the four points are concyclic.

Test yourself (1 minute)

1. Tangent-Chord (Alternate Segment) interplay

   In Paper 2 riders, cyclic quadrilateral angle theorems are frequently combined with the alternate segment theorem (angle between tangent and chord equals angle in alternate segment).

2. Opposite vs Adjacent angles

   Only OPPOSITE angles add to 180°; adjacent angles generally DO NOT add to 180° unless the cyclic quadrilateral is also an isosceles trapezium or rectangle.

3. Angle in a Semicircle

   If a cyclic quadrilateral diagonal passes through circle center (diameter), the angle subtended on the circumference is 90°.

Practise Cyclic Quadrilaterals questions today

Source: NIE G.C.E. (O/L) Mathematics Syllabus (Grade 11, Unit 21)

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What comes next?

Next in the syllabus

Where does Cyclic Quadrilaterals fit in the syllabus?

Grade 11 · Term 3