# 9.10.5. Properties of Cyclic Quadrilaterals

Determine whether the quadrilateral below cyclic or not.

• Ayes
• Bno

### Example

Determine whether the quadrilateral below cyclic or not.

### Solution

It is given that is congruent to , so is an isosceles triangle. Base angles of an isosceles triangle are congruent, so . Use the fact that the sum of the measures of the interior angles of a triangle is to find the measure of as follows:

If both pairs of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. First find the sum of the measure of and the measure of .

Since the sum of the measures of and is , the angles are supplementary. Now use the fact that the sum of the measures of the interior angles of a triangle is to find the sum of the measures of and in .

Next, use the fact that the measure of is equal to the sum of the measures of and , and the fact that the measure of is equal to the sum of the measures of and , to find the sum of the measure of and the measure of .

Since the sum of the measures of and is , the angles are supplementary. Thus, quadrilateral is cyclic.

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