Practice Maths

L22 — Circle Theorems

Key Terms

Arc
A portion of the circumference of a circle.
Chord
A line segment joining two points on a circle. A diameter is the longest chord, passing through the centre.
Central angle
An angle with its vertex at the centre of the circle; equals twice the inscribed angle that subtends the same arc.
Inscribed angle
An angle with its vertex on the circumference, subtended by an arc. All inscribed angles subtending the same arc are equal.
Cyclic quadrilateral
A quadrilateral with all four vertices on a circle. Opposite angles sum to 180°.
Tangent
A line touching the circle at exactly one point; always perpendicular to the radius at the point of contact.

The Six Core Circle Theorems

#TheoremKey phrase
1 The angle at the centre is twice the angle at the circumference (same arc). Centre = 2 × circumference
2 The angle in a semicircle is 90°. Diameter ⇒ right angle
3 Angles in the same segment are equal. Same arc ⇒ equal angles
4 Opposite angles of a cyclic quadrilateral sum to 180°. Cyclic quad: opp. angles supplementary
5 The angle between a tangent and a chord equals the inscribed angle in the alternate segment. Alternate segment theorem
6 A radius to a tangent point is perpendicular to the tangent. Radius ⊥ tangent
2x x O A B P Theorem 1: angle AOB = 2 × angle APB
Hot Tip: Always name the theorem before applying it — it shows clear reasoning and prevents using the wrong rule. The most common error: applying Theorem 1 (centre = 2 × circumference) when the vertex is on the circumference, not at the centre.

Theorem 1: Angle at the Centre

The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference by the same arc.

Worked Example 1

The central angle AOB = 130°. Find the inscribed angle APB.

APB = ½ × 130° = 65°.

Theorem 2: Angle in a Semicircle

If AB is a diameter, then any angle ACB inscribed in the circle (C on the circumference) is exactly 90°. This is a special case of Theorem 1 (central angle = 180°, so inscribed = 90°).

Theorem 3: Angles in the Same Segment

All inscribed angles that subtend the same arc are equal to each other.

Worked Example 2

Points P, Q, R, S lie on a circle. Angle PAQ = 38° where A is any point on the major arc. Find angle PBQ where B is another point on the same major arc.

Angle PBQ = 38° (same arc, same segment).

Theorem 4: Cyclic Quadrilateral

A cyclic quadrilateral has all four vertices on a circle. Its opposite angles add to 180°.

Worked Example 3

ABCD is a cyclic quadrilateral. Angle A = 75°, angle B = 110°. Find angles C and D.

C = 180 − 75 = 105°. D = 180 − 110 = 70°.

Theorem 5: Alternate Segment Theorem

The angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle on the opposite side of the chord.

Proving Your Answer

Always state the theorem you use. For example: “∠APB = 65° (angle at centre is twice angle at circumference, same arc AB).”

  1. Angle at the centre. Fluency

    O is the centre of the circle in each part. Find the unknown angle, giving the theorem used.

    • (a) Central angle AOB = 80°. Find the inscribed angle APB.
    • (b) Inscribed angle AQB = 55°. Find the central angle AOB.
    • (c) Central angle AOB = 150°. Find inscribed angle ACB on the major arc.
    • (d) Central angle reflex AOB = 240°. Find the inscribed angle APB (where P is on the major arc).

  2. Angle in a semicircle. Fluency

    • (a) AB is a diameter. C is on the circle. Angle CAB = 35°. Find angle CBA.
    • (b) AB is a diameter. C is on the circle. Angle ACB = 90°. Find AC if AB = 10 cm and angle ABC = 40°.
    • (c) AB is a diameter. D is on the circle. Angle DAB = 52°. Find angle ADB and angle ABD.
    • (d) A triangle inscribed in a circle has one side as a diameter (length 13 cm). The other two sides are 5 cm and 12 cm. Show this is consistent with the theorem.

  3. Angles in the same segment. Fluency

    • (a) A, B, C, D all lie on a circle. Angle ADB = 42°. Find angle ACB.
    • (b) P, Q, R, S lie on a circle. Angle PRQ = 67°. Find angle PSQ.
    • (c) In a circle, two inscribed angles both subtend arc MN. One angle is (2x + 10)° and the other is (4x − 20)°. Find x and the angle size.
    • (d) A chord AB divides a circle into two segments. Angle ACB = 38° where C is in the major segment. Find the angle ADB where D is in the minor segment.

  4. Cyclic quadrilaterals. Fluency

    • (a) ABCD is cyclic. Angle A = 82°. Find angle C.
    • (b) PQRS is cyclic. Angle P = 110°, angle Q = 75°. Find angles R and S.
    • (c) ABCD is cyclic. Angle A = (3x + 5)°, angle C = (2x − 10)°. Find x.
    • (d) A rectangle is inscribed in a circle. Explain why this is consistent with the cyclic quadrilateral theorem.

  5. Find angles from the diagram. Understanding

    The diagram shows a circle with centre O. Points A, B, C, D lie on the circle. AB is a diameter. Angle ACD = 35° and angle CBD = 25°.

    O 35° 25° A B C D
    • (a) Find angle ACB. State the theorem used.
    • (b) Find angle ADB. State the theorem used.
    • (c) Find angle CAD. (Use the triangle ACD with known angles.)
    • (d) Find angle ABD.

  6. Alternate segment theorem. Understanding

    A tangent is drawn to a circle at point T. A chord TB is drawn, and C is a point on the arc on the other side of TB.

    • (a) The angle between the tangent and chord TB is 48° (on the left of TB). Find angle TCB.
    • (b) Angle TCB = 63°. Find the angle between the tangent and chord TB.
    • (c) A chord PQ is drawn. The tangent at P makes an angle of 55° with PQ. R lies on the major arc. Find angle PRQ.
    • (d) In the alternate segment theorem, if the chord is a diameter, what does the theorem say about the tangent angle? Explain.

  7. Combining theorems. Understanding

    O is the centre of the circle in each part.

    • (a) Angle at centre AOB = 100°. C is on the major arc, D is on the minor arc. Find angles ACB and ADB.
    • (b) ABCD is a cyclic quadrilateral. Diagonal AC bisects angle BAD (angle BAC = angle CAD = 28°). Find angle BCD.
    • (c) Isosceles triangle ABC is inscribed in a circle (AB = AC). OA bisects angle BAC. Show that OA is also perpendicular to BC.
    • (d) In the diagram, AB is a diameter, angle BAC = 40°, and angle DBA = 30°. Find angle ACD, where D is on the circle.

  8. Cyclic quadrilateral with algebra. Understanding

    • (a) ABCD is cyclic. Angle A = 5x°, angle C = (x + 12)°. Find x and both angles.
    • (b) PQRS is cyclic. Angle P = (3y + 20)°, angle R = (y + 40)°. Find y.
    • (c) A cyclic quadrilateral has angles in the ratio 2:3:4:5. Find all four angles.
    • (d) Is a square always a cyclic quadrilateral? Justify your answer.

  9. Angle chains. Problem Solving

    O is the centre. A, B, C, D lie on the circle. Angle BAC = 32°, angle ABD = 28°, and BD is a diameter.

    • (a) Find angle BDA.
    • (b) Find angle BAD.
    • (c) Find angle BCD (using the cyclic quadrilateral theorem with ABCD).
    • (d) Find the central angle BOD. (Hint: BD is a diameter, so you know the central angle already. Use this to check your work.)

  10. Circle theorem proof. Problem Solving

    Prove the angle-at-the-centre theorem for the case where O (centre) lies inside triangle APB (P on circumference, arc AB not containing P).

    • (a) Draw radius OP and extend it to meet the circle at Q. Write the relationship between angles AOP and angles OAP, OPA (using isosceles triangle OAP, since OA=OP=r).
    • (b) Similarly write the relationship for triangle OBP.
    • (c) Central angle AOB = angle AOQ + angle QOB. Express this in terms of the angles at P.
    • (d) Simplify to show angle AOB = 2 × angle APB.
See Answers ➔