Problems on cyclic quadrilaterals

A plan for problems

Most problems on cyclic quadrilaterals are solved in a few steps.

1. Find the pairs. Mark the opposite angles. Mark any exterior angle and the angle opposite to it.
2. Use the sum. Opposite angles add to $180^\circ$. An exterior angle equals the interior opposite angle.
3. Bring in the triangles. A diagonal cuts the figure into triangles. Angles in the same part of the circle are equal, and the angles of a triangle add to $180^\circ$.
4. Check. The four angles of the quadrilateral must add to $360^\circ$.

Example 1. $ABCD$ is a cyclic quadrilateral. The diagonal $AC$ is drawn. $\angle BAC = 35^\circ$ and $\angle ACD = 48^\circ$ and $\angle ABC = 107^\circ$. Find $\angle ADC$ and $\angle CAD$.

1. $\angle ADC$ is opposite to $\angle ABC$. So $\angle ADC = 180^\circ - 107^\circ = 73^\circ$.
2. In the triangle $ACD$, the angles add to $180^\circ$. So $\angle CAD = 180^\circ - 73^\circ - 48^\circ = 59^\circ$.

Check. In the triangle $ABC$, $\angle BCA = 180^\circ - 107^\circ - 35^\circ = 38^\circ$. Then $\angle BCD = 38 + 48 = 86^\circ$. And $\angle BAD = 35 + 59 = 94^\circ$. They add to $180^\circ$.

Trapezium, parallelogram, kite

A trapezium has one pair of parallel sides. An isosceles trapezium has equal legs.

Theorem. A trapezium is cyclic if and only if it is isosceles.

If it is isosceles, then it is cyclic. Let $ABCD$ have $AB \parallel DC$ and $AD = BC$. The base angles are equal: $\angle A = \angle B$. The angles $\angle A$ and $\angle D$ lie between parallel lines, so they add to $180^\circ$. Then $\angle B + \angle D = \angle A + \angle D = 180^\circ$. By the converse, $ABCD$ is cyclic.

If it is cyclic, then it is isosceles. Now $ABCD$ is cyclic with $AB \parallel DC$. Then $\angle A + \angle D = 180^\circ$ because of the parallel lines. Also $\angle A + \angle C = 180^\circ$ because $ABCD$ is cyclic. So $\angle C = \angle D$. A trapezium with equal base angles has equal legs, so $AD = BC$.

A trapezium whose legs are not equal is not cyclic.

Other figures:

• A rectangle is always cyclic. Its diagonals are diameters.
• A parallelogram is cyclic only if it is a rectangle.
• A rhombus is cyclic only if it is a square.
• A kite is cyclic only if its two opposite angles between unequal sides are right angles.

Practise with new numbers

Try it in Python

The program reads the corners of four shapes, measures their angles, and tests the opposite angles.

Common mistakes

• Taking every trapezium to be cyclic. Only an isosceles trapezium is.
• Taking every parallelogram to be cyclic. Only a rectangle is.
• Finding an angle but not checking it. After you finish, add the four angles. They must make $360^\circ$.
• Mixing up the diagonal angles. Equal angles on the same arc come from corners on the same side of the chord, not from opposite sides.

Remember these

• Mark opposite angles first. Then bring in the exterior angle and the triangles.
• A trapezium is cyclic exactly when its legs are equal.
• A cyclic parallelogram is a rectangle. A cyclic rhombus is a square.
• Always add the four angles at the end. The total must be $360^\circ$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. $ABCD$ is a cyclic trapezium with $AB \parallel DC$ and $\angle ABC = 81^\circ$. Find $\angle ADC$.

2. Can a parallelogram with angles $70^\circ$ and $110^\circ$ be inscribed in a circle?

3. The opposite angles $\angle A$ and $\angle C$ of a cyclic quadrilateral $ABCD$ are in the ratio $4 : 5$. Find them.

4. In a cyclic quadrilateral, one angle is $90^\circ$. What can you say about the angle opposite to it?

5. Why are the angles of a cyclic trapezium at the two ends of a parallel side equal?