How many pairs of opposite angles are there in a cyclic quadrilateral?

**Two pairs** — and each pair adds to 180°.

What is the sum of all four angles in any quadrilateral?

**360°** — but in a cyclic quadrilateral, each pair of opposite angles adds to 180°, so both pairs sum to 360° total.

Can a rectangle be cyclic?

**Yes** — a rectangle can always be inscribed in a circle (all four vertices on the circle). Opposite angles are 90° each, and 90° + 90° = 180°.

What is a cyclic quadrilateral?

A quadrilateral with **all four vertices** on the circumference of a circle.

Does this theorem work for any quadrilateral?

**No** — only for cyclic quadrilaterals (all four vertices on the circle).

If angle A = 70° in a cyclic quadrilateral, what is the opposite angle?

**110°** — because opposite angles add to 180°, so 70° + 110° = 180°.

How do you spot a cyclic quadrilateral in a diagram?

Look for a quadrilateral with **all four points on the circumference** of the circle.

Angles in Cyclic Quadrilaterals | Pearson Edexcel IGCSE Maths

Q: What do opposite angles in a cyclic quadrilateral add up to?

**180°** — opposite angles in a cyclic quadrilateral add up to 180°.

Q: What is the exam phrase for this theorem?

**Opposite angles in a cyclic quadrilateral add up to 180°**.

A cyclic quadrilateral is a four-sided shape with all four vertices on the circumference of a circle. The key theorem states that opposite angles in a cyclic quadrilateral add up to 180°. This property only applies to cyclic quadrilaterals — quadrilaterals that can be inscribed in a circle — and is essential for solving angle problems in circle geometry.

A cyclic quadrilateral is a quadrilateral formed by four points on the circumference of a circle.

All four vertices lie on the circle.

Quadrilateral ABCD is cyclic — all vertices lie on the circle.

Circle Theorem: Opposite angles in a cyclic quadrilateral add up to 180°.

For a cyclic quadrilateral, angle A + angle C = 180° and angle B + angle D = 180°.

To spot: look for quadrilaterals with all four points on the circumference.

Exam phrase: Opposite angles in a cyclic quadrilateral add up to 180°.

The theorem only works for cyclic quadrilaterals.

Not every quadrilateral inscribed in a diagram is cyclic — all four vertices must lie on the same circle.

A common exam trap: a quadrilateral with only three vertices on the circle is NOT cyclic.

Problem: Find x. Identify the cyclic quadrilateral and the radius perpendicular to the chord.

The radius bisects the chord, creating two congruent triangles. Use this to find 72° (equal angles) and 18° (angles in triangle = 180°).

Opposite angles in cyclic quadrilateral: 2x + 4 + 20 + 18 = 180 → 2x = 138 → x = 69°.

A cyclic quadrilateral is a quadrilateral formed by four points on the circumference of a circle.

All four vertices lie on the circle.

Quadrilateral ABCD is cyclic — all vertices lie on the circle.

Circle Theorem: Opposite angles in a cyclic quadrilateral add up to 180°.

For a cyclic quadrilateral, angle A + angle C = 180° and angle B + angle D = 180°.

To spot: look for quadrilaterals with all four points on the circumference.

Exam phrase: Opposite angles in a cyclic quadrilateral add up to 180°.

The theorem only works for cyclic quadrilaterals.

Not every quadrilateral inscribed in a diagram is cyclic — all four vertices must lie on the same circle.

A common exam trap: a quadrilateral with only three vertices on the circle is NOT cyclic.

Problem: Find x. Identify the cyclic quadrilateral and the radius perpendicular to the chord.

The radius bisects the chord, creating two congruent triangles. Use this to find 72° (equal angles) and 18° (angles in triangle = 180°).

Opposite angles in cyclic quadrilateral: 2x + 4 + 20 + 18 = 180 → 2x = 138 → x = 69°.

Cyclic quadrilaterals are often easy to spot in busy diagrams.

Mark on their angles even if you think you do not need them — they may help later.

Combine with other theorems (e.g. radius bisecting chord) when solving multi-step problems.

Cyclic quadrilaterals are often easy to spot in busy diagrams.

Mark on their angles even if you think you do not need them — they may help later.

Combine with other theorems (e.g. radius bisecting chord) when solving multi-step problems.

Angles in Cyclic Quadrilaterals

Summary

1. Definition of a Cyclic Quadrilateral

2. The Theorem

3. Important Caveat

4. Worked Example

5. Examiner Tips