Revision notes
When all four corners sit on a circle, opposite angles add up to 180° — and the exterior angle equals the opposite interior angle.
Ngozi marks four points on the rim of her mother's round tray and joins them to make a quadrilateral. Whatever four points she picks, two of its angles always add up to exactly 180°. Let's find out which two — and why.
A cyclic quadrilateral has all four vertices on a circle.
Its opposite angles are supplementary (add up to 180°): ∠A + ∠C = 180° and ∠B + ∠D = 180°.
Why? ∠A and ∠C stand on the two arcs that make up the whole circle. Their angles at the centre are 2∠A and 2∠C, which add up to 360°, so ∠A + ∠C = 180°.
🔑 Key idea: Opposite angles of a cyclic quadrilateral add up to 180°.
Extend side AB beyond B to a point E. The exterior angle ∠CBE equals the interior opposite angle ∠ADC.
Why? ∠CBE + ∠ABC = 180° (straight line) and ∠ADC + ∠ABC = 180° (opposite angles). Both equal 180° − ∠ABC.
So if ∠CBE = 75°, then ∠ADC = 75° too.
🔑 Key idea: Exterior angle = interior opposite angle.
Be careful to pair opposite angles, not next-door ones. In PQRS, ∠P is opposite ∠R and ∠Q is opposite ∠S.
If the angles are given in algebra, e.g. ∠A = (2y + 10)° and ∠C = (3y − 5)°, write 2y + 10 + 3y − 5 = 180, so 5y = 175 and y = 35.
Rectangles and squares are always cyclic. A parallelogram that is not a rectangle is never cyclic.
🔑 Key idea: Pair the opposite angles, set their sum to 180°, and solve.
ABCD is a cyclic quadrilateral. ∠BCD = 95°. Find ∠BAD.
Step 1: Opposite angles are supplementary
∠BAD = 180° − 95° = 85°
Why? ∠BCD and ∠BAD are opposite corners of the quadrilateral.
Answer: 85°
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