Segment Theorems (Cambridge O Level Maths)

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

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

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