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E4.7 Circle theorems I

Syllabus
0580–2028–2029
Topic
E4.7
Level
Extended

Learning objectives

E4.7.1Calculate unknown angles and give explanations using the following• Calculate unknown angles and give explanations using the following geometrical properties of circles: • angle in a semicircle = 90° • angle between tangent and radius = 90° • angle at the centre is twice the angle at the circumference • angles in the same segment are equal • opposite angles of a cyclic quadrilateral sum to 180° (supplementary) • alternate segment theorem. Candidates are expected to use the geometrical properties listed in the syllabus when giving reasons for answers.

Choose and chain circle theorems

A circle-theorem solution starts by matching an angle to an exact circle configuration, then names that theorem as the reason before moving to the next angle.

Configuration to recognise Angle fact Reason to state
angle subtended by a diameter at the circumference 90∘90^\circ angle in a semicircle
radius and tangent at the point of contact 90∘90^\circ tangent is perpendicular to the radius
centre angle and circumference angle standing on the same arc centre angle =2×=2\times circumference angle angle at the centre is twice the angle at the circumference
two circumference angles standing on the same chord and in the same segment equal angles in the same segment are equal
four vertices on one circle opposite angles sum to 180∘180^\circ opposite angles of a cyclic quadrilateral are supplementary
tangent and chord at the contact point, compared with the angle subtended by that chord in the opposite segment equal alternate segment theorem

Mark the centre, radii, diameter, tangent and the endpoints of the relevant chord. Decide which two rays form the required angle, identify the arc or chord it stands on, apply one theorem, and write its reason. Then use ordinary angle facts—triangle sum, straight line, isosceles radii or angles at a point—as separate justified steps.

Let ABAB be a diameter, let CC lie on the circle, and let a tangent touch the circle at AA. If ∠BAC=32∘\angle BAC=32^\circ, then ∠ACB=90∘\angle ACB=90^\circ because it is an angle in a semicircle. Hence ∠ABC=180∘−90∘−32∘=58∘\angle ABC=180^\circ-90^\circ-32^\circ=58^\circ. The angle between the tangent at AA and chord ACAC is also 58∘58^\circ by the alternate segment theorem.

The same chord or arc endpoints must be used when comparing centre, circumference or same-segment angles. A tangent is perpendicular only to the radius drawn to its contact point, and a quadrilateral is cyclic only when all four vertices lie on the circle. Do not choose a theorem from the picture's appearance alone.