Segment Theorems

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 theorem can be a bit challenging to identify.
• Steps to identify:
  1. Choose an angle on the circumference and put your index fingers on it.
  2. Use your fingers to follow the two lines that form the angle to the point where they each meet the circumference.
  3. See if there are any other lines from these two points that meet at another angle. If so, the two angles are equal.

• Remember, angles in the same segment of a circle are equal. In an exam, if you need to cite the same segment theorem, use the key phrase "Angles in the same segment are equal."

Alternate Segment Theorem

Circle Theorem: Alternate Segment Theorem

• One of the more subtle circle theorems, the Alternate Segment Theorem is a valuable tool for swiftly determining angles in various problems.
• The theorem states that the angle formed between a chord and a tangent is equal to the angle in the alternate segment.
• To recognize this theorem, look for a "cyclic triangle," where all three vertices of a triangle are on the circumference, with one vertex meeting a tangent. Focus on situations where two chords intersect a tangent.
• To identify the equal angles:
  • Locate the point where the 'cyclic triangle' intersects the tangent and determine the angle between them.
  • Identify the vertex in the triangle opposite to the marked angle at the intersection and mark this angle as equal to the initial angle.