Year 10
Apply the 8 main circle theorems (angles in the same segment, angle at centre, angle in a semicircle, opposite angles in a cyclic quadrilateral, tangent-radius, alternate segment, two tangents from a point, and intersecting chords) with clear geometric reasoning.
Geometry - properties of circles
Circle theorems are fundamental results about angles, lines and circles. The main theorems are: (1) Angles in the same segment are equal — two angles inscribed in a circle that subtend the same arc are equal. (2) The angle at the centre is twice the angle at the circumference — an angle at the centre of a circle is twice the angle at the circumference subtending the same arc. (3) The angle in a semicircle is a right angle — any angle inscribed in a semicircle is 90°. (4) Opposite angles in a cyclic quadrilateral sum to 180° — if all four vertices of a quadrilateral lie on a circle, opposite angles are supplementary. (5) The angle between a tangent and a radius is 90° — a tangent to a circle is perpendicular to the radius at the point of contact. (6) The alternate segment theorem — the angle between a tangent and a chord equals the angle in the alternate segment. (7) Two tangents from the same external point are equal in length, and the angle between them relates to the angle at the centre by angle AOB = 180° − angle APB (P the external point, O the centre). (8) When two chords intersect inside a circle, the angle between them can be found using angle sum in the triangle the chords form with the circle. These theorems allow us to find unknown angles and prove geometric properties.
Angles in the same segment
In a circle, two angles both subtend the same arc. What can we say about these angles?
Angles inscribed in a circle that subtend the same arc are equal. This is true regardless of where the vertex is on the circle (as long as it's on the same side of the chord).
Angle at the centre is twice the angle at the circumference
An angle at the centre of a circle is 120°. What is the angle at the circumference that subtends the same arc?
The angle at the centre is always twice the angle at the circumference subtending the same arc.
Angle in a semicircle is a right angle
A triangle is inscribed in a circle with its hypotenuse as a diameter. What is the angle opposite the hypotenuse?
Any angle inscribed in a semicircle (subtending a diameter) is automatically a right angle. This is a special case of Thales' theorem.
Opposite angles in a cyclic quadrilateral
A quadrilateral ABCD is inscribed in a circle. If angle A = 80°, what is angle C?
In a cyclic quadrilateral (all vertices on the circle), opposite angles are supplementary (sum to 180°).
Tangent meets radius at a right angle
A tangent touches a circle at point P. OP is a radius. What is the angle between the tangent and the radius at P?
A tangent is always perpendicular to the radius drawn to the point of contact. In the right-angled triangle formed with an external point E, the other two angles still sum to 90° (angles in a triangle sum to 180°, minus the 90° at P).
Alternate segment theorem
A tangent touches a circle at A. Chord AB makes a 65° angle with the tangent. What is the angle in the alternate segment (angle ACB)?
The angle between a tangent and a chord equals the angle subtended by that chord in the alternate segment — the segment on the other side of the chord from the tangent-chord angle.
Two tangents from an external point
PA and PB are tangents to a circle from an external point P, and O is the centre. If angle APB = 70°, what is angle AOB?
PA and PB are equal in length, and OA/OB are radii meeting each tangent at 90°. Quadrilateral OAPB's four angles sum to 360°, so with two 90° angles already used, angle AOB and angle APB are supplementary (sum to 180°).
Intersecting chords
Chords AC and BD intersect at point E inside a circle. Angle BAC = 40° and angle ABD = 35°. What is angle AEB?
E, A and B form a triangle, since E sits on both chords. Angle EAB = angle BAC (E is on line AC) and angle EBA = angle ABD (E is on line BD), so angle AEB follows from angle sum in a triangle: 180° minus the other two angles.
In a circle, a chord subtends an angle of 55° at a point on the major arc. What angle does the same chord subtend at the centre of the circle?
For each question, identify which circle theorem applies, state the theorem clearly, and then use it to find the unknown angle. Show your reasoning.
Two angles in the same segment of a circle subtend the same arc. If one angle is 45°, what is the other angle?
💡 Angles in the same segment are equal. This is the first circle theorem.
An angle at the circumference of a circle is 40°. What is the angle subtended at the centre by the same arc?
💡 The angle at the centre is twice the angle at the circumference subtending the same arc.
A triangle has one side as a diameter of a circle, and the third vertex lies on the circle. What is the angle at the third vertex?
💡 Any angle inscribed in a semicircle (subtending a diameter) is a right angle.
In cyclic quadrilateral ABCD, angle A = 75° and angle B = 100°. What is angle C?
💡 Opposite angles in a cyclic quadrilateral sum to 180°. So angle C = 180° − angle A = 180° − 75° = 105°.
A tangent to a circle meets the radius at the point of contact. What angle is formed between the tangent and the radius?
💡 A tangent is always perpendicular to the radius at the point of contact.
Watch for these when working through the lesson.
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