Cyclic Quadrilateral

What is Cyclic Quadrilateral?

Cyclic quadrilateral is defined as a four-sided figure whose vertices lie on the circumference of a circle. A cyclic quadrilateral is a quadrilateral inscribed in a circle. Remember that not all quadrilaterals inside a circle are cyclic as its vertices must lie on the circle.

The above example is not a cyclic quadrilateral even though the four-sided polygon is inscribed in a circle. One of its vertices does not lie around the circumference.

A rectangle inscribed in a circle (above) is an example of a cyclic quadrilateral. All four vertices lie in the circumference of a circle.

Properties of Cyclic Quadrilaterals

1.The sum of the opposite angles of a cyclic quadrilateral is 180 degrees. The formula to get the measure of the opposite angle are:

∠A + ∠C = 180
∠B + ∠D =180

Example: Based on the properties of quadrilaterals, solve for the measure of all angles.

For angle C:
∠A + ∠C = 180
80 + ∠C = 180
∠C = 180 – 80
∠C =100

For angle B:
∠B + ∠D =180
∠B + 95 = 180
∠B = 180 – 95
∠B = 85

The measure of all the angles of a quadrilateral are: ∠A = 80, ∠ B = 85, ∠C = 100, and ∠D = 95

2. In a cyclic quadrilateral, the exterior angle is congruent to the interior opposite angles.

Based on the example above, ∠CDE is congruent with ∠B. If ∠B = 85, then ∠CDE = 85.