Angles in the Same Segment (Cambridge (CIE) IGCSE International Maths)

Revision Note

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Maths

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

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

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

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 .

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

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

the (exam) results speak for themselves:

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