IGCSEMathsEdexcelMaths AHigherRevision NotesGeometry & TrigonometryCircle TheoremsAngles in the Same Segment

Angles in the Same Segment (Edexcel IGCSE Maths A) : Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

Edexcel Maths A: HigherEdexcel Maths A (Modular): Higher Unit 2Cambridge (CIE) International Maths: ExtendedCambridge (CIE) Maths: Extended

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Circles & Segments

Circle Theorem: Angles in the same segment are equal

  • Any two angles on the circumference of a circle that are formed from the same two points on the circumference are equal

    • These two angles are in the same segment of the circle

      • To see this, add the chord PQ below to split the circle into two segments

A circle with points P and Q and the arc between them highlighted. Two chords from each point meet at another point on the circumference forming an angle y.

  • To spot this circle theorem on a diagram

    • Find two points on the circumference that meet at a third point

    • See if there are any other pairs of lines from the same two original points that meet at a different point on the circumference

  • When explaining this theorem in an exam you must use the keywords:

    • Angles in the same segment are equal

  • Look out for a bowtie shape

    • The theorem works upside down, in that the angles at P and Q are also equal

Angles in the same segment are equal (this works upside-down too)

Examiner Tips and Tricks

  • An exam question diagram may have multiple equal angles

    • Look for as many as possible by seeing how many pairs of lines start from the same two points on the circumference

Worked Example

The diagram below shows a circle with centre, O.
A, B, C, D and E are five points on the circumference on the circle.

Angle AEB = 12º.
Angle BEC = 14º.
Angle CED = 73º.
Angle EBD = θº.

Find the value of θ .

A circle with centre, O, and 5 points on the circumference, A, B, C, D and E. Angle AEB = 12º, angle BEC = 14º and angle CED = 73º. The angle EBC = θ º.

CE is a diameter
This means that triangles EAC and CED are both triangles in a semicircle

Angle EAC = 90º
Angle CED = 90º
Angle in a semicircle = 90º

Find the other angles in the triangles

Angle ECA = 64º
Angle ECD = 17º
Angles in a triangle = 180º

Label these angles on the diagram

Previous circle diagram with the angle EAC and EDC marked as right angles. Angle ACE = 64º and angle DCE = 17º are also labelled.

Angle θ  is formed by two lines coming from either end of the chord ED
Angle ECD is also formed by two lines coming from either end of the chord ED

Angle θ = angle ECD = 17º
Angles in the same segment are equal

Angle θ = 17º

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Author
Reviewer
Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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