Circle Theorems [H]
Higher Tier — This topic is assessed at Higher tier only (grades 4–9).
Circle Theorems
Circle theorems are results about angles, radii, chords and tangents that follow from the properties of a circle. Each theorem must be stated precisely as a reason when used in an exam proof or angle calculation.
The Seven Standard Theorems
| # | Theorem | Exact reason to state |
|---|---|---|
| 1 | The angle at the centre is twice the angle at the circumference subtended by the same arc. | "angle at the centre = twice angle at circumference" |
| 2 | The angle in a semicircle is 90° (angle in a semicircle subtended by a diameter). | "angle in a semicircle" |
| 3 | Angles in the same segment are equal (subtended by the same chord). | "angles in the same segment" |
| 4 | Opposite angles of a cyclic quadrilateral sum to 180°. | "opposite angles in a cyclic quadrilateral" |
| 5 | The tangent to a circle is perpendicular to the radius at the point of contact. | "tangent perpendicular to radius" |
| 6 | Two tangents from an external point are equal in length. | "tangents from an external point are equal" |
| 7 | The angle between a tangent and a chord equals the angle in the alternate segment (tangent–chord angle). | "alternate segment theorem" |
Useful Supporting Facts
- The perpendicular from the centre to a chord bisects the chord.
- Equal chords are equidistant from the centre.
- Radii of the same circle are equal — use this to identify isosceles triangles formed by two radii.
Worked Examples
\( O \) is the centre. Angle \( AOB = 140° \). Find angle \( ACB \) where \( C \) is on the major arc.
Angle at centre = 2 × angle at circumference (same arc).
\[ \angle ACB = 140° \div 2 = 70° \]Angle \( ACB = 35° \). Find the reflex angle \( AOB \).
Angle at centre = 2 × angle at circumference, but here \( C \) is on the minor arc so the relevant angle at centre is reflex \( AOB \):
\[ \text{reflex } AOB = 2 \times 35 = 70°. \quad \text{Non-reflex }AOB = 360-70=290°. \]Wait — if \( C \) is on the minor arc, the centre angle on the major side is the reflex: reflex \( AOB = 2\times35=70° \). Check: non-reflex \( AOB=290° \). ✓
Cyclic quadrilateral \( ABCD \). \( \angle DAB = 105° \). Find \( \angle BCD \). Also \( \angle ABC = 78° \), find \( \angle ADC \).
\[ \angle BCD = 180-105 = 75° \quad \text{(opposite angles, cyclic quadrilateral)} \] \[ \angle ADC = 180-78 = 102° \quad \text{(opposite angles, cyclic quadrilateral)} \]Tangent \( TP \) touches circle at \( P \). Chord \( PQ \) makes angle 52° with the tangent. Find the angle in the alternate segment.
By the alternate segment theorem, the angle between the tangent and the chord equals the angle in the alternate segment.
\[ \angle \text{ in alternate segment} = 52° \]Key Takeaways
- Theorem 1: central angle = 2 × inscribed angle (same arc). Theorem 2: angle in semicircle = 90°.
- Theorem 3: angles in same segment are equal. Theorem 4: opposite angles in cyclic quad sum to 180°.
- Theorem 5: tangent ⊥ radius. Theorem 6: two tangents from external point are equal.
- Theorem 7 (alternate segment): tangent–chord angle = angle in alternate segment.
- Always name the theorem explicitly as a reason — never just write "circle theorem".