Learning Objectives
  • Use all prescribed circle theorems to calculate angles.
  • Give correct geometrical reasons for every deduction.
  • Apply tangent, chord and symmetry properties.
  • Use the alternate segment theorem accurately.
  • Combine circle theorems with triangle, parallel-line and polygon facts.
Key Terms
Chord
A straight line joining two points on a circle.
Tangent
A line touching a circle at exactly one point.
Cyclic quadrilateral
A quadrilateral whose four vertices lie on a circle.
Segment
A region cut off by a chord.
Subtended angle
An angle formed by lines from the endpoints of an arc or chord.
Alternate segment theorem
The angle between a tangent and chord equals the angle in the opposite segment.
Perpendicular bisector
A line crossing a segment at 90° through its midpoint.
What This Chapter Covers
  • Angle in a semicircle
  • Tangent–radius and tangent–chord properties
  • Centre and circumference angle theorem
  • Same-segment and cyclic-quadrilateral theorems
  • Chord symmetry and equal tangents
Reasoning Is Part Of The Answer

Circle-theorem questions often award marks for correct reasons. State the theorem used, not merely “circle theorem”. Use three-letter angle notation when clarity is needed.

Property Usable reason
Angle in a semicircle is 90^\circ Angle in a semicircle
Radius is perpendicular to tangent Tangent is perpendicular to radius at point of contact
Centre angle is twice circumference angle on same arc Angle at centre is twice angle at circumference
Angles standing on same chord are equal Angles in the same segment are equal
Opposite cyclic angles sum to 180^\circ Opposite angles of a cyclic quadrilateral are supplementary
Tangent–chord angle equals angle in opposite arc Alternate segment theorem
Two tangents from one external point have equal lengths Tangents from an external point are equal
Angle At The Centre
\angle AOB=2\angle ACB

Both angles must stand on the same arc AB.

Worked Example: Centre And Circumference

Question: Angle AOB at the centre is 124^\circ. Find angle ACB at the circumference on the same arc.

  1. The centre angle is twice the circumference angle.
  2. \angle ACB=124^\circ/2.

Answer: 62^\circ.

Angles In The Same Segment

Angles at the circumference subtended by the same chord and lying in the same segment are equal. Check that the vertices lie on the same side of the chord.

Worked Example: Same Segment

Question: Angles ACB and ADB stand on chord AB in the same segment. If \angle ACB=47^\circ, find \angle ADB.

  1. Both angles are subtended by chord AB.
  2. They lie in the same segment.

Answer: 47^\circ.

Cyclic Quadrilaterals
\angle A+\angle C=180^\circ
Worked Example: Cyclic Opposites

Question: ABCD is cyclic and \angle BAD=108^\circ. Find \angle BCD.

  1. Opposite angles in a cyclic quadrilateral sum to 180^\circ.
  2. \angle BCD=180^\circ-108^\circ.

Answer: 72^\circ.

Tangent And Radius

A tangent is perpendicular to the radius drawn to the point of contact. This creates a right triangle and often connects circle geometry with Pythagoras or trigonometry.

Worked Example: Tangent Length

Question: From external point P, PT is tangent to a circle with centre O. If OP=13 cm and radius OT=5 cm, find PT.

  1. OT\perp PT, so triangle OPT is right-angled at T.
  2. PT=\sqrt{13^2-5^2}=\sqrt{144}.

Answer: 12 cm.

Alternate Segment Theorem

The angle between a tangent and chord at the point of contact equals the angle in the alternate segment subtended by that chord. Identify the chord first, then identify the angle at the circumference standing on it.

Worked Example: Tangent–Chord Angle

Question: A tangent at A makes an angle of 38^\circ with chord AB. Find angle ACB on the opposite arc AB.

  1. Both angles are associated with chord AB.
  2. By the alternate segment theorem they are equal.

Answer: 38^\circ.

Chord Symmetry Properties

Equal chords are equidistant from the centre. The perpendicular bisector of a chord passes through the centre. Conversely, a perpendicular from the centre to a chord bisects the chord. These facts often create congruent right triangles.

Worked Example: Chord Distance

Question: Two chords in the same circle have equal lengths. One is 6 cm from the centre. Find the distance of the other chord from the centre.

  1. Equal chords in the same circle are equidistant from the centre.

Answer: 6 cm.

Tangents From An External Point

If PA and PB are tangents from the same external point P, then PA=PB. With radii OA and OB, the right triangles OAP and OBP are congruent, which may also show that OP bisects the angle between the tangents.

Worked Example: Equal Tangents

Question: PA and PB are tangents from P. If PA=3x+2 and PB=5x-8, find x and the tangent length.

  1. Equal tangents give 3x+2=5x-8.
  2. Solve: 10=2x, so x=5.
  3. Substitute: PA=17.

Answer: x=5, tangent length 17.

Combining Theorems

A multi-step problem may require several facts: radius–tangent angle, triangle angle sum, alternate segment theorem, same-segment angles and cyclic quadrilateral properties. Write each reason beside the line where it is used.

Examination Guidance
  • Name the chord or arc subtending the angles.
  • Use “same segment” only when the angles are on the same side of the chord.
  • Write “opposite angles of a cyclic quadrilateral sum to 180°” rather than only “cyclic”.
  • Mark radii equal when isosceles triangles are formed.
  • Give a reason for each angle found in a proof-style question.
Common Mistakes
  • Assuming all angles at the circumference are equal.
  • Using the centre theorem for angles standing on different arcs.
  • Applying the alternate segment theorem to the angle between two chords rather than tangent and chord.
  • Forgetting that a tangent is perpendicular only to the radius at the point of contact.
  • Giving numerical answers without geometrical reasons.
Knowledge Check And Practice

1. An angle at the centre is 150^\circ. Find the angle at the circumference on the same arc.

Answer: 75^\circ.

2. One angle of a cyclic quadrilateral is 96^\circ. Find the opposite angle.

Answer: 84^\circ.

3. What is the angle between a tangent and radius at the point of contact?

Answer: 90^\circ.

4. Two tangents from P have lengths 2x+7 and 5x-8. Find x.

Answer: 5.

5. State the theorem linking a tangent–chord angle to an angle at the circumference.

Answer: Alternate segment theorem.

6. What line passes through the centre and midpoint of a chord at right angles?

Answer: The perpendicular bisector of the chord.
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