The Proof of the Cyclic Quadrilateral Theorem

A cyclic quadrilateral is a quadrilateral inscribed in a circle. A polygon that is inscribed in a circle is a polygon whose vertices are on the circle. Some of the inscribed polygons are shown in the next figure.

Examples of polygons inscribed in a circle.

In this post, we are going to show a special property of one inscribed polygon which is the cyclic quadrilateral theorem about angles. We are going to prove that its opposite angles add up to $180 ^\circ$

Theorem

The sum of the measures of the opposite angle of a cyclic quadrilateral is $180 ^\circ$

Proof 1

In the figure below, $ABCD$ is a cyclic quadrilateral inscribed in a circle with center $O$.

Now, $\angle C$ is an inscribed angle that intercepts arc $DAB$. The measure of $\angle C$ is half the measure of its intercepted arc (follows from the Inscribed Angle Theorem). This is also the same with $\angle C$. Its measure is half the measure arc $DCB$.

If we add the measure of arc $DCB$ and arc $DAB$, they add up to a whole circle whose angle measure is $360 ^\circ$. It follows that if we add up the measures of angles $C$ and $A$, the sum is also half the measure of $360 ^\circ$ which is $180^\circ$. Now, it means that the sum of the measures of $\angle B$ and $\angle D$ is also $180 ^\circ$.  Therefore, the measure of the opposite angles of a cyclic quadrilateral is $180 ^\circ$.