Cyclic quadrilaterals

Objective. Use opposite angles summing to 180 degrees.

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Learn: the key idea

A cyclic quadrilateral has all four vertices lying on a circle, and its key property is that opposite angles always sum to 180 degrees. Identify the two pairs of opposite angles, then use the fact that each pair adds to 180° to find an unknown angle by subtraction. If one angle in a cyclic quadrilateral is 110°, the opposite angle must be 180° − 110° = 70°.

Worked example

In cyclic quadrilateral PQRS, angle P = 95° and angle Q = 80°. Find angle R, the angle opposite P.

1 $Recall that opposite angles in a cyclic quadrilateral sum to 180°$
2 $Angle R is opposite angle P$
3 $Set up the equation: angle P + angle R = 180°$
4 $Solve: angle R = 180° - 95° = 85°$

Answer: 85°

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Common mistake. Students sometimes add adjacent angles instead of opposite ones, forgetting that only the vertices diagonally across from each other sum to 180°. Always identify which two vertices are diagonally opposite before applying the rule.

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