Circle theorems (apply and prove)
WJEC Higher Unit 1 always carries a circle-theorem question worth 5–8 marks. There are six standard theorems plus the alternate segment theorem.
The six core theorems
- Angle at centre = 2 × angle at circumference when both subtended by the same arc.
- Angle in a semicircle = 90° (special case of theorem 1).
- Angles in the same segment are equal — angles subtended by the same arc, on the same side of the chord, are equal.
- Opposite angles of a cyclic quadrilateral sum to 180° (cyclic quad).
- Tangent perpendicular to radius at the point of contact.
- Two tangents from an external point are equal in length, and the line from external point to centre bisects the angle between them.
Alternate segment theorem (Higher only)
The angle between a tangent and a chord equals the angle in the alternate segment.
Strategy for "find the angle x" problems
- Look for diameters → right angle in any inscribed triangle (semicircle).
- Look for cyclic quadrilaterals → opposite-angle sum = 180°.
- Look for tangent + radius → 90°.
- Identify which arc subtends each angle.
- Quote the theorem name as the reason for the M/A1.
Proving theorems
WJEC Higher proofs are usually theorem 1 (angle at centre = 2 × angle at circumference). Standard proof:
- Draw the centre O and join radius to the apex of the inscribed angle.
- Mark equal radii → form two isosceles triangles.
- Use exterior angle of a triangle = sum of two opposite interior angles.
- Add the two centre-angle pieces and the two inscribed-angle pieces; result is the 2:1 ratio.
WJEC exam tip
Reasons MUST be quoted using mark-scheme phrasing: e.g. "angle in a semicircle is 90°", "opposite angles of a cyclic quadrilateral sum to 180°". Bullet-list the reasons next to each line of the calculation. WJEC reserves an A1 reasoning mark for every numerical answer accompanied by the named theorem.
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