Quadrilaterals inscribed in a circle

Three quick questions from earlier lessons.

A quadrilateral has all four vertices on a circle. Opposite angles stand on arcs that together make the whole circle. What do opposite angles add to?

Move D around the circle. Add the angles at A and C.

Cyclic quadrilaterals.

A quadrilateral inscribed in a circle has opposite angles supplementary. Conversely, if opposite angles add to

180^\circ

, the quadrilateral can be inscribed in a circle.

$\angle A = 75^\circ$ , $\angle B = 110^\circ$ .

\angle C = 105^\circ

Opposite A.

\angle D = 70^\circ

Opposite B.

Your turn: $\angle P = 98^\circ$ in cyclic $PQRS$ . Find $\angle R$ .

82^\circ

180 − 98.

$ABCD$ is inscribed in a circle and $\angle A = 128^\circ$ . Find $\angle C$ (degrees).
$ABCD$ is inscribed in a circle and $\angle A = 40^\circ$ . Find $\angle C$ (degrees).

Which quadrilateral can always be inscribed in a circle?
1. A rectangle (for example 9 by 11)
2. Any parallelogram
3. Any rhombus

$PQRS$ is inscribed in a circle with $\angle Q = 113^\circ$ . Find $\angle S$ (degrees).
$PQRS$ is inscribed in a circle with $\angle Q = 42^\circ$ . Find $\angle S$ (degrees).

Three questions to check you have got it.

$ABCD$ is inscribed in a circle and $\angle A = 128^\circ$ . Find $\angle C$ (degrees).
$ABCD$ is inscribed in a circle and $\angle A = 40^\circ$ . Find $\angle C$ (degrees).
$ABCD$ is inscribed in a circle and $\angle A = 107^\circ$ . Find $\angle C$ (degrees).

Cyclic quadrilaterals

Opposite angles add to $180^\circ$ .

Watch out: supplementary, not equal.

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