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