Circle Theorems: Cyclic Quadrilaterals

What Is a Cyclic Quadrilateral?

A cyclic quadrilateral is a four-sided shape where all the vertices lie on the circumference of a circle.

If a quadrilateral is cyclic, then a special rule applies:

$\text{Opposite angles add up to } 180\degree$

This is one of the most useful theorems when working with circle geometry.

Key Facts

Opposite angles in a cyclic quadrilateral always add up to $180\degree$.

Only works when all corners lie on the circle.

This property does not apply if one corner is inside or outside the circle.

How to Spot a Cyclic Quadrilateral

Look for a four-sided shape where every corner is on the edge of the circle. These shapes often look like a kite, trapezium or square that fits perfectly inside the circle.

If you're not sure, check the points — are they all on the circumference?

Example

Question:

In a cyclic quadrilateral, one angle is $72\degree$. What is the size of the opposite angle?

 
Answer:

$\text{Opposite angles in a cyclic quadrilateral add to } 180\degree$

$180\degree - 72\degree = 108\degree$

So the opposite angle is: $\boxed{108\degree}$

Worked Example

Question:

In the diagram below, points $A, B, C, D$ lie on the circumference of a circle. The quadrilateral $ABCD$ is cyclic.

You're told:

$\angle DAB = 2x + 4$

$\angle BCD = 20\degree$

$\angle DCB = 18\degree$

Find the value of $x$.

 
Step 1: Use Triangle Rules

In triangle DCBDCB, angles must add up to $180\degree$. You’re given:

$\angle BCD = 20\degree$

$\angle DCB = 18\degree$

$\angle CBD = 180\degree - 20\degree - 18\degree = 142\degree$

But since this isn't part of the cyclic pair, we look to the quadrilateral.

Step 2: Use the Cyclic Quadrilateral Theorem

$\angle DAB + \angle BCD = 180\degree$

Substitute the given values: $(2x + 4) + (20 + 18) = 180$

$2x + 4 + 38 = 180 \to \quad 2x + 42 = 180 \to \quad 2x = 138 \Rightarrow x = 69$

Final Answer:

$\boxed{x = 69\degree}$