Find x in the figure.

Step-by-step solution

Idea: Use “angle at the centre = 2 × angle at the circle” twice. The arc ABC (the side away from D) and the arc ADC (the side away from B) together make the full 360° at the centre. This is why opposite angles of a cyclic quadrilateral add up to 180°.

1. D is outside arc ABC, so ∠ADC = ½ × (angle subtended by arc ABC at the centre O). So arc ABC subtends 2 × 100° = 200° at O (a reflex angle).½ mark
2. The other arc, ADC, subtends 360° − 200° = 160° at O.½ mark
3. B is outside arc ADC, so x = ∠ABC = ½ × 160° = 80°. (Quick check: ∠ADC + ∠ABC = 100° + 80° = 180°, as opposite angles of a cyclic quadrilateral should.)1 mark

Check: 100° + 80° = 180° ✓ (opposite angles of a cyclic quadrilateral are supplementary).

Answer to write in the exam

Common mistakes that cost marks

• Writing x = 100° because the angles “look opposite”. Opposite angles of a cyclic quadrilateral are supplementary, not equal.
• Writing x = 50° (half of 100°). Halving applies to the angle at the centre, not to another angle on the circle.
• Doubling 100° and stopping at 200°, which is the angle at the centre, not x.

How this can come in the exam

Try one yourself

ABCD is a cyclic quadrilateral with centre O and ∠ADC = 128°. Find ∠ABC and the angle AOC on the side of D.

More questions like this

• If a line segment AB joining two points A, B subtends equal angles at two other points C, D that lie on the same side of AB, then the four points lie on a circle.
• The sum of two opposite angles of a cyclic quadrilateral is 180°.
• A cyclic quadrilateral has angles measuring ∠A = 80°, ∠B = 110°, ∠C = 100°, and ∠D = 70°. Can such a quadrilateral be drawn? Explain why or why not.
• If two opposite angles of a quadrilateral add up to 180°, then the vertices of the quadrilateral lie on a circle, i.e., they are concyclic.
• In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord?

All Circles questions · All maths questions