Circle Theorems — Part 1
Learn the angle at centre, angles in same segment, and cyclic quadrilateral theorems in MYP Maths Year 5. Includes exam justification tips for Standard level.
What This Topic Covers
Circle Theorems Part 1 introduces three fundamental theorems that relate angles inside and around a circle. Students learn to state each theorem precisely, identify the conditions under which it applies, and use it to find unknown angles in geometric diagrams.
Theorem 1: Angle at the Centre
The angle subtended at the centre of a circle is twice the angle subtended at the circumference by the same arc.
If angle AOB is at the centre and angle ACB is at the circumference, both standing on the same arc AB, then ∠AOB = 2 × ∠ACB.
Theorem 2: Angles in the Same Segment
Angles subtended by the same arc at the circumference are equal. If points C and D both lie on the major arc AB, then ∠ACB = ∠ADB.
This theorem allows students to identify equal angles in diagrams that initially appear to have no obvious relationship.
Theorem 3: Cyclic Quadrilateral
Opposite angles of a cyclic quadrilateral (a quadrilateral with all four vertices on a circle) add up to 180°.
∠A + ∠C = 180° and ∠B + ∠D = 180°
Students must confirm that all four vertices lie on the circle before applying this theorem.
Common Mistakes
- Applying the angle-at-centre theorem when the centre is not explicitly shown in the diagram
- Confusing the angle at the centre with the reflex angle — using the wrong one
- Assuming opposite angles sum to 180° in any quadrilateral, not just cyclic ones
- Not stating the theorem by name when justifying angle calculations
MYP Question Style
Circle theorem questions in MYP Criterion A tasks require students to find one or more angles and give a reason for each step. Writing the theorem name or a brief statement of the rule is expected — a bare numerical answer will not receive full marks on reasoning steps.
Practice Approach
Draw your own diagrams from scratch for each theorem to understand why it works, not just how to apply it. Then practise diagrams where multiple theorems are needed in sequence. Part 2 covers the remaining circle theorems — tangent-radius, alternate segment, and further applications.