This is a key topic in circle geometry. A cyclic quadrilateral is simply a four-sided figure where all four of its vertices lie on the circumference of a single circle. This special property leads to two very important angle theorems.

1. The Two Core Theorems

Theorem 1: Opposite angles are supplementary.

• What it means: The pairs of angles opposite each other inside a cyclic quadrilateral always add up to 180°.

• In the diagram:

  • ∠A + ∠C = 180°

  • ∠B + ∠D = 180°

This is the most fundamental rule and is used in almost every problem.

Theorem 2: The exterior angle is equal to the interior opposite angle.

• What it means: If you extend one of the sides of a cyclic quadrilateral, the angle formed outside the shape (the exterior angle) is equal to the angle at the opposite corner inside the shape.

• This is a fantastic shortcut! It saves you from first finding the interior angle using the "angles on a straight line add to 180°" rule and then using Theorem 1.

Example: Instead of saying ∠DCB + ∠DAB = 180°, you can directly say that the exterior angle at C is equal to ∠DAB.

3. The Converse: Proving a Quadrilateral is Cyclic

Sometimes, a question will ask you to prove that four points lie on a circle (i.e., that a quadrilateral is cyclic). To do this, you use the theorems in reverse:

• How to Prove: Show that one pair of opposite angles adds up to 180°. If you can prove that ∠A + ∠C = 180° OR ∠B + ∠D = 180°, then you can conclude that ABCD is a cyclic quadrilateral.

4. Strategy for Solving Problems (Riders)

1. Identify the Cyclic Quadrilateral: The first step is always to find the four vertices that lie on the circle.

2. Apply the Theorems:

   • Look for opposite angles. You can write an equation: angle1 + angle2 = 180°.

   • Look for an exterior angle. You can write an equation: exterior angle = interior opposite angle.

3. Combine with Other Circle Theorems: These problems rarely use only one theorem. You will almost always need to combine it with other geometry rules, such as:

   • Angles in the same segment are equal.

   • The angle at the centre is twice the angle at the circumference.

   • Angle in a semicircle is 90°.

   • Properties of isosceles triangles (especially those formed by two radii).

   • Properties of parallel lines (alternate and corresponding angles).

Exam Tips & Common Pitfalls

• Pitfall 1: Assuming a shape is cyclic. Do not use the cyclic quadrilateral theorems unless you are told it's a cyclic quadrilateral OR you have proven it yourself.

• Pitfall 2: Confusing the interior/exterior angles. The exterior angle is equal to the interior opposite angle, not the one adjacent to it.

• Exam Tip 1: Look for Hidden Information. If a line passes through the centre of the circle, it's a diameter. This means it creates a 90° angle at the circumference (angle in a semicircle). This is often a starting point for finding other angles.

• Exam Tip 2: Chain of Logic. Geometry proofs are like a chain. Use one theorem to find an angle, then use that new angle to apply another theorem, and so on, until you reach the answer. Write down every step and the reason for it (e.g., "Opposite angles of a cyclic quad").