Learning pack · IGCSE handout

Circle Theorems

What's in this pack: printable worksheet PDF, matching answers PDF (IGCSE / Secure–Stretch practice). Circle theorems give angle relationships: angle at the centre, angles in the same segment, cyclic quadrilaterals, tangent–radius.

Resources at the top · full summary · videos with short notes below.

🎒 Resources in this pack

Jump straight to the printable files, or scroll to the summary and videos.

Scaffolded Learning Sheet — attempt the worksheet, then use the Answers for a full written solution.

📝 Open Learning Sheet ✅ Open answers 🎬 More topic videos (6)📖 Full summary

Files: 41_circle_theorems_handout.pdf · 41_circle_theorems_handout_answers.pdf

Full topic summary

Circle theorems give angle relationships: angle at the centre, angles in the same segment, cyclic quadrilaterals, tangent–radius.

Key learning points

Angle at the centre is twice the angle at the circumference on the same arc.
Angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to $180^\circ$ .
Tangent ⊥ radius; two tangents from an external point are equal.

Worked example 1 — Angle at the centre

Angle at circumference $40^\circ$ on the same arc. Angle at centre?

Working

80^\circ

Centre angle is twice circumference.

Worked example 2 — Same segment

Two angles in the same segment on the same arc.

Working

\text{equal}

Angles in the same segment are equal.

Worked example 3 — Cyclic quadrilateral

One angle $110^\circ$ . Opposite angle?

Working

70^\circ

Opposite angles sum to $180^\circ$ .

Worked example 4 — Tangent-chord

Tangent meets chord; angle between tangent and chord is $55^\circ$ . Angle in the alternate segment?

Working

55^\circ

Alternate segment theorem.

🎬 Topic videos

Angle Properties in a Circle
Eddie Woo · 6 min
Circle angle properties. CIE 0580 / Pearson 4MA1.
Angle Properties - Circle Geometry (Angle at the centre/circumference)
Eddie Woo · 7 min
Angle at centre vs circumference. CIE 0580 / Pearson 4MA1.
Angle Properties - Circle Geometry (Angle in a semicircle)
Eddie Woo · 10 min
Angle in a semicircle. CIE 0580 / Pearson 4MA1.
Tangent Properties - Circle Geometry (Tangents from an external point are equal)
Eddie Woo · 9 min
Equal tangents from an external point. CIE 0580 / Pearson 4MA1.
Tangent Properties - Circle Geometry (Tangent is perpendicular to radius)
Eddie Woo · 9 min
Tangent perpendicular to radius. CIE 0580 / Pearson 4MA1.
Circle Theorems - GCSE Higher Maths
1st Class Maths · 14 min
Lower Secondary classroom-friendly explainer. Scaffold→stretch friendly; use with mixed-ability groups.

More Lower Secondary packs

Browse the hub Learning Packs & Summary, or open the compact Circle Theorems summary.