Circle theorems connect angles, chords, tangents, and arcs. They are essential for geometry proofs and exam reasoning.

Key circle theorems

• Angle at center is twice angle at circumference on same arc.
• Angles in the same segment are equal.
• Angle in a semicircle is \(90^\circ\).
• Opposite angles of cyclic quadrilateral sum to \(180^\circ\).
• Tangent is perpendicular to radius at point of contact.
• Two tangents from same external point are equal in length.
• Alternate segment theorem: angle between tangent and chord equals angle in opposite arc.

Using theorems in proofs

1. Mark known equal angles/lengths clearly.
2. Name theorem used at each step.
3. Chain results to target statement.

Cyclic quadrilaterals

A quadrilateral is cyclic if all vertices lie on one circle.

Useful tests:

• Opposite angles supplementary.
• Exterior angle equals opposite interior angle.

Tangent-chord applications

When a tangent touches a chord endpoint, use alternate segment theorem to connect tangent angle with an inscribed angle subtending that chord.

Exam strategy

• Label all equal angles immediately.
• Write theorem names, not just numbers.
• Avoid assuming symmetry unless given or proved.
• Finish with a clear conclusion sentence.