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.

cyclic quadrilateral theorem

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.

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