A theorem is a statement that can be proved using known mathematical facts and reasoning.

An angle can be subtended by a line segment or curve. The legs of the subtended angle meet the endpoints of the line segment or curve.

Two segments are created when dividing a circle into two parts using a chord.

A tangent of a circle is a line that intersects the circle exactly once.

The alternate segment theorem

Maths

Checking and securing understanding of the parts of a circle

The angle at the centre of the circle is twice the angle at any point on the circumference

The angle in a semicircle is a right angle

The perpendicular from the centre of a circle to a chord bisects the chord

The tangent at any point on a circle is perpendicular to the radius at that point

The angles in the same segment are equal

The opposite angles of a cyclic quadrilateral sum to 180°

The tangents from an external point are equal in length

Identifying which circle theorem to use

Problem solving with circle theorems

In the prior unit, pupils applied their understanding of angles to solve increasingly complex problems. In this unit, pupils will use their developed mathematical reasoning skills to derive and use various circle theorems.

In this unit, pupils will use their developed mathematical reasoning skills to derive and use various circle theorems.

Circle theorems

Find the size of angle $$a$$° (in degrees).

Find the size of angle $$b$$° (in degrees).

Find the size of angle $$c$$° (in degrees).

The size of the angle labelled $$d$$° is {{}}°.

∠ATC = 46°
∠ABT is subtended by the chord AT.
Find the size of angle $$e$$° (in degrees).

∠PEF is subtended by the chord PF.
∠PFE is subtended by the chord PE.
Find ∠EPD (in degrees).

Both ∠GHF and ∠GIF are angles at the circumference subtended by the chord GF. 

The size of the angle labelled $$p$$° is {{}}°.

Both ∠JMK and ∠JLK are angles at the circumference subtended by the arc JK. 

The size of the angle labelled $$t$$° is {{}}°. 

The size of the angle labelled $$a$$° is {{}}°.

The alternate segment theorem

Lesson outcome