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MathidiaMathematics Academy
Circle theorems
Lesson 42 of 75
Easy practice

Semicircles and same segment

Objective. Apply right-angle and same-segment theorems.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Semicircles and same segment theorems cover two circle facts: the angle in a semicircle is always 90°, and angles subtended by the same arc in the same segment are equal to each other. For the right-angle rule, any triangle drawn from the two ends of a diameter to a point on the circumference has a right angle at that point. For the same-segment rule, if two angles both stand on chord AB from the same side of the circle, they are equal, so if one is 42° the other is also 42°.

Worked example

Points A, B, C and D lie on a circle with AB as diameter. Angle ACB and angle ADB both relate to semicircles and same segment facts. If angle ACD = 25° and angle BCD = 90°, find angle ACB and explain why.

Diagram of a circle
  1. 1Since AB is the diameter, angle ACB is the angle in a semicircle
  2. 2The angle in a semicircle is always 90°
  3. 3So angle ACB = 90°
  4. 4This is confirmed since angle ACD + angle DCB = 25° + ... but the key rule used is the semicircle theorem directly giving angle ACB = 90°

Answer: Angle ACB = 90° (angle in a semicircle)

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Common mistake. A common error is applying the same-segment equal-angles rule to angles on opposite sides of the chord, where the angles are actually not equal (they're supplementary in a cyclic quadrilateral case instead). Always check both angles are on the same side of the chord before equating them.

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