Segment Theorems (OCR GCSE Maths)

Revision Note

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Circles & Segments

Circle Theorem: Angles at the circumference subtended by the same arc are equal

  • This theorem is also sometimes known as the same segment theorem
  • It states that any two angles at the circumference of a circle that are formed from the same two points on the circumference are equal
    • Subtended means the equal angles are created by drawing chords from the ends of an arc
    • These chords may or may not pass through the centre
  • This is one of the more tricky circle theorems to identify
    • STEP 1
      Choose an angle on the circumference and put your index fingers on it
    • STEP 2
      Use your fingers to follow the two lines that form the angle to the point where they each meet the circumference
    • STEP 3
      See if there are any other lines from these two points that meet at another angle
    • The two angles are equal

Equal subtend arc, IGCSE & GCSE Maths revision notes

  • If giving the same segment theorem as a reason in an exam, use the key vocabulary
    • "Angles in the same segment are equal"

Exam Tip

  • The same segment theorem is a common circle theorem used in GCSE exam questions
  • Don't be afraid of it, look for as many equal angles you can find using it and fill them in as they will help you find other angles
  • If you use this theorem to help you find other angles, you should still mention the same segment theorem in your reasons

Worked example

Find the value of θ in the diagram below, giving reasons for your answers. 

Q2-Circle-Theorems-1, IGCSE & GCSE Maths revision notes

There is a diameter here, splitting the circle into two semicircles.
Identify the two triangles in each semi circle and mark in the right angles using the angle in a semicircle theorem.
Find the other angles in the triangles using the rule angles in a triangle add up to 180°

Q2-Circle-Theorems-2, IGCSE & GCSE Maths revision notes

In the diagram, notice how the angle θ is subtended from the same chord as the angle that is 17°.

The angle at B is 90° because it is the angle (at the circumference) in a semi circle
The angle at C is 17°
because the angles in a triangle add up to 180°
θ = 17°
...
... because angles in the same segment are equal

Alternate Segment Theorem

Circle theorem: Alternate Segment Theorem

  • Although one of the least obvious circle theorems to identify, this is very helpful in finding angles quickly in many questions
  • The Alternate Segment Theorem states that the angle between a chord and a tangent is equal to the angle in the alternate segment
  • You can spot this circle theorem by looking for a “cyclic triangle
    • ie. all three vertices of a triangle lie on the circumference
    • but one vertex meets a tangent – look for where 2 chords meet a tangent
  • To identify which angles are equal,
    • Find the point where the 'cyclic triangle' meets the tangent and mark the angle between them
    • Look for the vertex in the triangle that is opposite the marked angle at the point where the triangle meets the circumference
    • Mark this angle as equal to the first angle you marked
  • If using this theorem as a reason for an angle in an exam, simply state the key phrase
    • "alternate segment theorem"

Alternate Segment Theorem, IGCSE & GCSE Maths revision notes 

Exam Tip

  • Spotting equal angles using the alternate segment theorem can save a lot of time in the exam
    • Identify if there are any triangles with all three vertices on the circumference early on
    • Look to see if any of the vertices meet a tangent

Worked example

Find the value of x, stating any angle facts and circle theorems you use.Q1 Circle Theorems 4, IGCSE & GCSE Maths revision notes

Identify the triangle in the circle with all three vertices at the circumference.

One vertex of this triangle meets a tangent at the bottom, so look for the vertex inside the triangle opposite this point and mark that angle with  2x + 5.  

Q1 CT4 Working in red, IGCSE & GCSE Maths revision notes

Give reasons for your working as you go.

The top left angle is 2+ 5 because of the alternate segment theorem

This angle is also subtended by the same arc as the angle at the centre.

The angle at the centre = 2(2x + 5) because of the circle theorem
'the angle at the centre is twice the angle at the circumference'

Form an equation.

2 left parenthesis 2 x space plus space 5 right parenthesis space equals space 5 x space minus space 2

Expand the brackets and solve the equation.

table row cell 4 x space plus space 10 space end cell equals cell space 5 x space minus space 2 end cell row cell 12 space end cell equals cell space x end cell end table

bold italic x bold space bold equals bold space bold 12
Using the "alternate segment theorem" and that "angles at the centre are twice angles at the circumference"

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Amber

Author: Amber

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.