Circle Theorems summary

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

Centre angle is twice circumference.

Worked example 2 — Same segment

Two angles in the same segment on the same arc.

Working

Angles in the same segment are equal.

Worked example 3 — Cyclic quadrilateral

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

Working

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

Alternate segment theorem.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.