Rukasha Maths

GCSE maths Beta

Circle theorems and angle reasons · Revision guide

Higher · Geometry · 4 pages in the PDF

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Learning objectives

  • Choose a circle theorem and give a reason for an angle calculation.
  • Use radii and tangents to solve a connected angle problem.

Key ideas

The centre angle is twice the circumference angle subtended by the same arc.

θcentre=2θcircumference\theta_{\text{centre}}=2\theta_{\text{circumference}}

Angles in the same segment are equal; an angle subtended by a diameter is a right angle.

θsemicircle=90∘\theta_{\text{semicircle}}=90^\circ

Opposite angles of a cyclic quadrilateral sum to a straight angle.

A+C=180∘A+C=180^\circ

A tangent is perpendicular to its radius. Tangents from the same external point have equal lengths.

The tangent-chord angle equals the angle subtended by that chord in the opposite segment.

Worked examples

Example 1

Tangents from T touch at A and B; O is the centre. The angle at T is 56 degrees. Find the non-reflex centre angle.

A radius meets its tangent at a right angle.

∠OAT=∠OBT=90∘\angle OAT=\angle OBT=90^\circ
∠ATB=56∘\angle ATB=56^\circ

Angles in the quadrilateral sum to a full turn.

∠AOB=360∘−90∘−90∘−56∘\angle AOB=360^\circ-90^\circ-90^\circ-56^\circ
∠AOB=124∘\angle AOB=124^\circ

Example 2

A centre angle is 130 degrees. Find the circumference angle subtended by the same arc.

θcircumference=130∘2\theta_{\text{circumference}}=\frac{130^\circ}{2}
65∘65^\circ

Watch out

Match the correct arc or chord before choosing a theorem. An angle in a semicircle is a right angle, not a straight angle.