How to Solve: Circle Theorems

How to Solve: Circle Theorems

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Master circle theorems, and you’ll crack 80% of geometry exam questions—from finding missing angles to proving shapes are cyclic. Let’s get you full marks."

What You Need To Know First

1. Basic angle properties – Sum of angles in a triangle (180°), straight line (180°), quadrilateral (360°).
2. Parts of a circle – Radius, diameter, chord, tangent, arc, sector, segment.
3. Isosceles triangles – Two sides equal → two angles equal.

(If you’re shaky on these, pause and review first.)

Key Vocabulary

Formulas To Know

(All these are NOT given on exam sheets—memorise them!)

Step-by-Step Method

Step 1: Identify the circle and key points

• Label the centre (O) if given.
• Mark all given angles and points on the circle (A, B, C, etc.).
• Highlight tangents, chords, or diameters (diameter = longest chord).

Step 2: Look for right angles (90°)

• If a triangle has a diameter as one side, the angle opposite is 90° (angle in a semicircle).
• If a tangent meets a radius, the angle is 90° (tangent-radius theorem).

Step 3: Check for isosceles triangles

• Any triangle with two radii as sides is isosceles (e.g., △OAB where OA = OB).
• Mark equal angles (e.g., ∠OAB = ∠OBA).

Step 4: Apply the angle at centre vs. circumference rule

• If you see an angle at the centre (∠AOB) and an angle at the circumference (∠ACB), use: ∠AOB = 2 × ∠ACB

Step 5: Use cyclic quadrilateral properties

• If a quadrilateral is cyclic (all vertices on circle), opposite angles sum to 180°: ∠A + ∠C = 180° and ∠B + ∠D = 180°

Step 6: Apply the alternate segment theorem (if tangent is involved)

• The angle between a tangent and chord (∠TAC) equals the angle in the alternate segment (∠ABC).

Step 7: Solve for missing angles

• Use angle sums (triangle = 180°, straight line = 180°) to find unknowns.
• Write every step with reasons (e.g., "Angle in semicircle = 90°").

Step 8: Check for hidden isosceles triangles

• If two sides are radii, the angles opposite them are equal.

Worked Examples

Example 1 – Basic (Angle in a semicircle)

Question: In the circle with centre O, AB is a diameter. Point C lies on the circle. Find ∠ACB.

Solution: 1. Identify key points: AB is diameter, O is centre, C is on circle. 2. Apply angle in semicircle: ∠ACB = 90° (angle opposite diameter is 90°). 3. Answer: ∠ACB = 90°

What we did and why: - Recognised AB as a diameter → used the angle in a semicircle theorem. - No calculations needed—just recall the rule.

Example 2 – Medium (Angle at centre vs. circumference)

Question: In the circle with centre O, ∠AOB = 100°. Find ∠ACB.

Solution: 1. Identify key points: O is centre, A/B/C on circle. 2. Apply angle at centre rule: ∠AOB = 2 × ∠ACB 3. Substitute: 100° = 2 × ∠ACB 4. Solve: ∠ACB = 100° ÷ 2 = 50°

What we did and why: - Saw angle at centre (100°) and angle at circumference (∠ACB) → used angle at centre = 2 × angle at circumference. - Divided by 2 to find the smaller angle.

Example 3 – Exam Style (Cyclic quadrilateral + tangent)

Question: In the diagram, ABCD is a cyclic quadrilateral. TA is a tangent at A. ∠TAD = 50° and ∠BCD = 80°. Find ∠BAD.

Solution: 1. Identify key points: ABCD is cyclic, TA is tangent at A. 2. Apply alternate segment theorem: ∠TAD = ∠ABD = 50° 3. Use cyclic quadrilateral property: ∠BCD + ∠BAD = 180° (opposite angles sum to 180°) 4. Substitute: 80° + ∠BAD = 180° 5. Solve: ∠BAD = 180° – 80° = 100° 6. But wait! ∠BAD includes ∠BAT (from tangent). We need to subtract ∠ABD. 7. Find ∠BAT: ∠BAD = ∠BAT + ∠ABD
100° = ∠BAT + 50°
∠BAT = 50° 8. Final answer: ∠BAD = 100° (but the question asks for ∠BAD, which is already 100°—no further steps needed).

What we did and why: - Used alternate segment theorem to find ∠ABD. - Applied cyclic quadrilateral rule to relate ∠BCD and ∠BAD. - Exam trap: Some students forget to check if the angle is split—always verify!

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Circle theorems are all about angles and rules—here’s the cheat sheet:

1. Angle in a semicircle = 90° – If one side is a diameter, the opposite angle is right.
2. Angle at centre = 2 × angle at circumference – Double the small angle to get the big one.
3. Cyclic quadrilateral? Opposite angles add to 180° – No exceptions.
4. Tangent meets radius at 90° – Always mark this first.
5. Alternate segment theorem – Angle between tangent and chord = angle in the opposite segment.

Before the exam: - Draw diagrams for every theorem—visuals stick better. - Practice 3 problems (one basic, one with a tangent, one cyclic quadrilateral). - Check for isosceles triangles—radii are equal, so angles opposite them are too.

You’ve got this. Go smash that exam!