Quadrilateral PQRS is inscribed in a circle. If ∠P = (2x + 10)° and ∠R = (3x − 20)°, find the value of x and the measures of ∠P and ∠R.

Step-by-step solution

Idea: ∠P and ∠R are opposite angles of a cyclic quadrilateral, so they add up to 180°. That gives a linear equation in x.

1. ∠P + ∠R = 180° (opposite angles of a cyclic quadrilateral): (2x + 10) + (3x − 20) = 180.1 mark
2. 5x − 10 = 180 ⇒ 5x = 190 ⇒ x = 38.1 mark
3. ∠P = 2(38) + 10 = 86°; ∠R = 3(38) − 20 = 94°.1 mark

Check: 86° + 94° = 180° ✓.

Answer to write in the exam

Common mistakes that cost marks

• Setting ∠P = ∠R (giving x = 30). Opposite angles of a cyclic quadrilateral are supplementary, not equal.
• Sign slip: 10 − 20 = −10, so 5x = 190, not 170.
• Giving only x and forgetting to find the two angles.

How this can come in the exam

Try one yourself

PQRS is cyclic with ∠Q = (4y − 6)° and ∠S = (2y + 6)°. Find y, ∠Q and ∠S.

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