Cyclic Quadrilaterals: Quick Revision

A brief note on Cyclic Quadrilaterals. Read it before a test. Try each question at the end before you open its answer.

The theorems

• A cyclic quadrilateral has all four corners on one circle.
• Opposite angles add up to $180^\circ$: $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.
• An exterior angle equals the interior opposite angle.
• The four angles of any quadrilateral add up to $360^\circ$. If one pair adds up to $180^\circ$, so does the other.

The converse

• If a pair of opposite angles of a quadrilateral adds up to $180^\circ$, the quadrilateral is cyclic.
• If an exterior angle equals the interior opposite angle, it is cyclic.
• If $C$ and $D$ are on the same side of $AB$ and $\angle ACB = \angle ADB$, the four points are on one circle.

Which figures are cyclic

• A rectangle and a square: always.
• An isosceles trapezium: yes. A trapezium with unequal legs: no.
• A parallelogram: only a rectangle. A rhombus: only a square.

A worked example

The angles of a quadrilateral $PQRS$ are $100^\circ$, $76^\circ$, $80^\circ$ and $104^\circ$, in order. Is it cyclic?

1. Pair them. $\angle P$ is opposite to $\angle R$, and $\angle Q$ is opposite to $\angle S$.
2. Add. $100 + 80 = 180$ and $76 + 104 = 180$.
3. Decide. Both pairs are supplementary, so $PQRS$ is cyclic.

Always pair the opposite angles. The neighbouring angles $100 + 76 = 176$ do not matter.

Practice with new numbers

A program to try

The program tests three sets of angles. It checks both pairs of opposite angles.

Mistakes to avoid

• Adding neighbouring angles instead of opposite ones.
• Using $360^\circ$ for a pair of opposite angles.
• Taking every trapezium or parallelogram to be cyclic.
• Pairing the exterior angle with the wrong corner. It equals the opposite interior angle.

Check yourself

1. A cyclic quadrilateral has $\angle A = 66^\circ$ and $\angle B = 109^\circ$. Find $\angle C$ and $\angle D$.

2. The side $QR$ of a cyclic quadrilateral $PQRS$ is extended to $X$, and $\angle SRX = 69^\circ$. Which interior angle is equal to it?

3. Is a quadrilateral with angles $90^\circ$, $85^\circ$, $90^\circ$ and $95^\circ$, in order, cyclic?

4. Why is a rhombus that is not a square never cyclic?