Circle Theorem – SS2 Mathematics Lesson Note (Second Term)

Circle theorems are rules that link angles in a circle. The angle an arc subtends at the centre is twice the angle it subtends at any point on the circumference. Angles in the same segment are equal, the angle in a semicircle is a right angle, and opposite angles of a cyclic quadrilateral add up to 180°.

Behavioural objectives

By the end of the lesson, students should be able to:

• state and prove that the angle an arc subtends at the centre of a circle is twice the angle it subtends at the circumference;
• prove that the angle in a semicircle is a right angle;
• state and prove that angles in the same segment of a circle are equal;
• define a cyclic quadrilateral;
• prove that opposite angles of a cyclic quadrilateral are supplementary.

Angles at the centre of a circle

Theorem 1: angle at the centre and angle at the circumference

The angle which an arc of a circle subtends at the centre of a circle is twice that which it subtends at any point on the circumference of the circle.

Given: A circle PQR, centre O.

To prove that: ∠POQ = 2∠PRQ

Construction: Join RO, and extend to any point T.

Proof:

1. PO = RO = QO (radii)
2. a = a1 (base angles of isosceles triangle POR)
3. ∠POQ = a + a1 = 2a (end angles of triangle POR)
4. But ∠POT + ∠QOT
5. Reflex ∠POQ = 2a + 2b = 2(a + b) = 2∠PRQ
6. Angle at centre = 2 × angle at circumference of a circle.

Corollary: what is the angle in a semicircle?

The angle in a semicircle is a right angle.

Given: A semicircle PQR, centre O.

To prove that: ∠PRQ = 90°

Construction: None

Proof:

1. PQ is a diameter (given)
2. ∠POQ = 180° (angle at centre = twice angle in circumference)
3. 2∠PRQ = 180°
4. Hence ∠PRQ = 180° ÷ 2 = 90° as required.

Angles in the same segment

Theorem 2: are angles in the same segment equal?

Angles in the same segment of a circle are equal.

Given: Points on a circle ABCD, AB is an arc of the circle.

To prove that: ∠ADB = ∠ACB

Construction: Join D and C to A and B, as shown above.

Proof:

1. ∠AOB = 2(x1 + x2)
2. x1 = (∠AOB)
3. ∠ADB = ∠ACB as required.

What is a cyclic quadrilateral?

A cyclic quadrilateral is a four-sided figure whose vertices lie on the circumference of the circle. The opposite angles of a cyclic quadrilateral lie in the opposite segments of the circle.

Theorem 3: opposite angles of a cyclic quadrilateral

Opposite angles of a cyclic quadrilateral are supplementary.

Given: A cyclic quadrilateral PQRS.

To prove that: ∠PQR + ∠PSR = 180°

Construction: Join PO and RO of the circle.

Proof:

1. ∠POR = 2y (angle at centre = twice angle on the circumference)
2. 2x + 2y = 360° (angles at a point)
3. 2(x + y) = 360°
4. x + y = 180°
5. ∠PQR + ∠PSR = 180°

Evaluation

1. State the theorem that relates the angle at the centre of a circle to the angle at the circumference.
2. Prove that the angle in a semicircle is a right angle.
3. What is a cyclic quadrilateral?
4. Prove that opposite angles of a cyclic quadrilateral are supplementary.

Assignment

Prove that if a straight line touches a circle at a point, and from the point of contact a chord is drawn, then the acute angles which this chord makes with the tangent are equal to the angle in the alternate segment.