IGCSE Maths / Circle theorems

Angles in the same segment

Edexcel International GCSE Maths (4MA1), chapter 17: Circle theorems, specification point 4.6c. Free lesson.

Read the revision notes · IGCSE Maths past papers

Angles standing on the same chord from the same side are equal. Identify the chord and check which side the points are on, so you choose the right circle theorem and give its full reason.

What a segment is

A chord is a straight line joining two points on a circle. It cuts the circle into two pieces, called segments: a small segment on one side and a larger segment on the other. An angle stands on a chord when its two arms meet the chord's endpoints.

When angles in the same segment are equal

If two angles stand on the same chord and their points are on the same side of that chord, the angles are equal. The angle's point can move around the same arc without changing its size. Find the chord both angles stand on. Check that both angle points are on the same side of the chord. Copy the given angle to the other angle. No calculation is needed.

How to tell the two rules apart

The picture can look almost the same in both rules, so check where the angle points sit in relation to the chord. If they are on opposite sides, the angles add to 180°, rather than being equal. Same side of the chord: the angles are equal. The reason is angles in the same segment are equal. Opposite sides of the chord: the angles add to 180°. The reason is opposite angles of a cyclic quadrilateral add up to 180°.

Worked examples

Use the chord first to decide which rule applies. Then write the angle and its reason.

The common mistake and the exam reason

Do not subtract from 180° when the two points are on the same side of the chord. For example, if P and Q are both above chord AB and one angle is 41°, the other is 41°, not 180° − 41° = 139°. Subtraction is for the opposite-side picture. Write the full reason: angles in the same segment are equal. Writing only “same segment” or saying the angles look the same does not give the required reason mark.

More lessons in Circle theorems

• Parts of a circle and cyclic quadrilaterals
• Tangent and radius, and two tangents from a point
• The angle at the centre is twice the angle at the circumference
• The angle in a semicircle
• Opposite angles of a cyclic quadrilateral
• The alternate segment theorem
• Using the alternate segment theorem in a reasoning chain
• Reasoning chains with circle theorems
• Multi-step circle theorem problems
• Intersecting chords inside a circle
• Intersecting chords from a point outside the circle