ABCD is a cyclic quadrilateral inscribed in a circle. If ∠A measures 75°, what is the measure of ∠C? If ∠B measures 110°, what is the measure of ∠D?

Step-by-step solution

Idea: In a cyclic quadrilateral, ∠A and ∠C are opposite, and so are ∠B and ∠D; each pair adds up to 180°.

1. ∠A + ∠C = 180° (opposite angles of a cyclic quadrilateral), so ∠C = 180° − 75° = 105°.1 mark
2. ∠B + ∠D = 180°, so ∠D = 180° − 110° = 70°.1 mark

Check: 75° + 110° + 105° + 70° = 360° ✓ (angle sum of a quadrilateral).

Answer to write in the exam

Common mistakes that cost marks

• Pairing ∠A with ∠B (adjacent angles). The supplementary pairs are A with C and B with D.
• Writing ∠C = 75° (thinking opposite angles are equal, as in a parallelogram).
• Subtracting from 360° instead of 180°.

How this can come in the exam

Try one yourself

In a cyclic quadrilateral ABCD, ∠A is twice ∠C. Find ∠A and ∠C.

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