Cyclic quadrilaterals · 2 marks

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.

Answer: Yes. The angles add up to 360°, and the opposite angles are supplementary: ∠A + ∠C = 80° + 100° = 180° and ∠B + ∠D = 110° + 70° = 180°. A quadrilateral whose opposite angles add up to 180° is cyclic.

Step-by-step solution

Idea: Two checks: any quadrilateral needs angle sum 360°, and a cyclic one needs each pair of opposite angles to add up to 180°. If opposite angles add up to 180°, the quadrilateral is cyclic (the converse also holds).

Angle sum: 80° + 110° + 100° + 70° = 360° ✓, so a quadrilateral with these angles exists.½ mark
Opposite angles: ∠A + ∠C = 80° + 100° = 180° and ∠B + ∠D = 110° + 70° = 180° ✓.1 mark
Opposite angles of a cyclic quadrilateral add up to 180°, and conversely a quadrilateral whose opposite angles add up to 180° has its vertices on a circle. So yes, such a cyclic quadrilateral can be drawn.½ mark
One way to draw it: on a circle with centre O, mark A, B, C, D in order so that ∠AOB = 60°, ∠BOC = 80°, ∠COD = 80° and ∠DOA = 140° (total 360°). Each angle of ABCD is half the angle at O of the arc it does not touch: ∠A = ½(80° + 80°) = 80°, ∠B = ½(80° + 140°) = 110°, ∠C = ½(140° + 60°) = 100°, ∠D = ½(60° + 80°) = 70° ✓. (Many other choices work too.)

Yes. The angles total 360° and each pair of opposite angles adds up to 180° (80° + 100° and 110° + 70°), which is exactly the condition for a quadrilateral to be cyclic.

Answer to write in the exam

∠A + ∠B + ∠C + ∠D = 80° + 110° + 100° + 70° = 360°

∠A + ∠C = 80° + 100° = 180°

∠B + ∠D = 110° + 70° = 180°

Opposite angles supplementary ⇒ the quadrilateral is cyclic (converse of the cyclic quadrilateral property)

∴ Yes, such a cyclic quadrilateral can be drawn.

Common mistakes that cost marks

Pairing adjacent angles (∠A + ∠B = 190°) and concluding “no”. The condition is about opposite angles: A with C, B with D.
Checking only the angle sum (360°). Every quadrilateral has that; the cyclic condition is the 180° pairs.
Thinking opposite angles must be equal. They must be supplementary.

How this can come in the exam

MCQ (1 mark)

Which set of angles, taken in order ∠A, ∠B, ∠C, ∠D, can belong to a cyclic quadrilateral?

90°, 80°, 100°, 90°
70°, 120°, 110°, 60°
85°, 95°, 85°, 95°
60°, 100°, 140°, 60°

Show answer

(B) 70°, 120°, 110°, 60°
Only (B) has both opposite pairs adding to 180°: 70° + 110° = 180° and 120° + 60° = 180°. In (A) 90° + 100° = 190°; in (C) 85° + 85° = 170°; in (D) 60° + 140° = 200°.

Try one yourself

A quadrilateral has angles ∠P = 75°, ∠Q = 95°, ∠R = 105° and ∠S = 85°. Is PQRS cyclic?