Cyclic quadrilaterals

Opposite angles of a cyclic quadrilateral are supplementary, and the exterior angle equals the interior opposite angle.

A cyclic quadrilateral has all four vertices on a circle.

Example 1●●

In cyclic quadrilateral $ABCD$, $\angle A = 75^\circ$. Find $\angle C$.

$\angle C = 180 - 75 = 105^\circ$ (opposite angles of a cyclic quadrilateral)

Example 2●●●

Opposite angles of a cyclic quadrilateral are $x^\circ$ and $2x^\circ$. Find $x$.

$3x = 180$, so $x = 60$.

Flashcards

Opposite angles in a cyclic quad

Add to $180^\circ$

Exterior angle of a cyclic quad

Equals the interior opposite angle

Can a rectangle be cyclic?

Yes: opposite angles $90 + 90$.

Practice

Cyclic quadrilateral, $\angle A = 75^\circ$. Find $\angle C$.
Opposite angles are $x^\circ$ and $2x^\circ$. Find $x$.
An exterior angle of a cyclic quadrilateral is $98^\circ$. The interior opposite angle?
$\angle B = 110^\circ$ in cyclic $ABCD$. Find $\angle D$.
Can a parallelogram with angles $70^\circ$ and $110^\circ$ be cyclic?
Every rectangle is cyclic.
Opposite angles are $(3x + 12)^\circ$ and $(2x - 12)^\circ$. Find $x$.
Using that $x$, find the larger of the two angles.