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.

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

**Theorem**

The sum of the measures of the opposite angle of a cyclic quadrilateral is

**Proof 1**

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

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

If we add the measure of arc and arc , they add up to a whole circle whose angle measure is . It follows that if we add up the measures of angles and , the sum is also half the measure of which is . Now, it means that the sum of the measures of and is also . Therefore, the measure of the opposite angles of a cyclic quadrilateral is .