Angles in the Same Segment (Cambridge International IGCSE Mathematics)

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 and that lie in the same segment are equal

• It is important that both angles are in the same segment

  • They are not equal if they are in different segments
  • In this case, a cyclic quadrilateral is formed

• To spot this circle theorem on a diagram

  • Find two points on the circumference that meet at a thir **d 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

Exam Hint 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 .

Answer:

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

Angle EAC = 90º Angle CDE = 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

Final answer: Angle θ = 17º