This is one of the most important geometry topics. It introduces three powerful theorems that will be the key to solving many exam problems involving circles. Your goal is to memorize these theorems and, more importantly, learn to recognize the patterns they create in diagrams.

1. The Three Key Theorems (Core Concepts)

Theorem 1: The Angle at the Centre Theorem

The angle an arc creates at the centre of a circle is twice the angle it creates at any point on the circumference.

• How to spot it: Look for an "arrowhead" shape where the point is at the centre (O) and the base is on the circumference.

• The Rule: Angle at Centre = 2 × Angle at Circumference.

• Example: If the angle at the centre ∠AOB is 120°, then the angle at the circumference ∠ACB is 60°.

• Important Note: This also works for the reflex angle. If the reflex ∠AOB is 260°, the angle on the other side of the circle ∠ADB would be 130°.

Theorem 2: Angles in the Same Segment Theorem

Angles created by the same arc at the circumference are equal.

• How to spot it: Look for a "bow-tie" or "butterfly" shape. All angles that stand on the same chord or arc are equal.

• The Rule: Angles subtended by the same arc are equal.

• Example: If ∠AXB and ∠AYB are both created from the arc AB, and ∠AXB = 40°, then ∠AYB must also be 40°.

Theorem 3: The Angle in a Semicircle Theorem

The angle in a semicircle is always a right angle (90°).

• How to spot it: Look for a triangle inside a circle where one of its sides is the diameter.

• The Rule: The angle opposite the diameter is 90°.

• Example: If AB is the diameter of a circle, any angle ∠ACB on the circumference will be 90°.

• Connection: This is just a special case of Theorem 1. A diameter is a straight line, so the angle at the centre is 180°. The angle at the circumference is half of that, which is 90°.

2. A Step-by-Step Strategy for Solving Problems

When you see a circle geometry question, don't panic. Use this checklist to find the solution.

1. SCAN FOR A DIAMETER: This is your first and easiest check. If you see a line passing through the centre, immediately find the angle opposite it and mark it as 90°. This often solves the problem instantly.

2. LOOK FOR THE CENTRE (O): If the centre is marked, look for the "arrowhead" pattern. Find the angle at the centre and either double or halve it to find the angle at the circumference using Theorem 1.

3. LOOK FOR A "BOW-TIE": If you see multiple angles on the circumference, check if they stand on the same arc. If they do, they are equal (Theorem 2). This is great for finding unknown angles in different parts of the circle.

4. USE OTHER GEOMETRY RULES: Circle theorems are almost never used alone. After applying a circle theorem, you will likely need to use one of these:

   • Isosceles Triangles: A triangle formed by two radii and a chord (like ΔAOB) is always isosceles. This means its base angles are equal (∠OAB = ∠OBA). This is a very common and important step.

   • Angles in a triangle add up to 180°.

   • Angles on a straight line add up to 180°.

3. Important Points to Remember for Exams

• State Your Reasons Clearly: You lose marks if you don't provide the correct reason. Write the full theorem in brackets after your statement.

  • (Angle at centre = 2 × angle at circumference)

  • (Angles in the same segment)

  • (Angle in a semicircle)

• Identify the Correct Arc: For Theorem 2, make sure the angles you are setting as equal come from the exact same two points on the circumference.

• **Annotate the Diagram