Circle Theorems: Angles at the Centre and in a Semicircle
Use the fact that the angle at the centre is twice the angle at the circumference on the same arc, and that the angle in a semicircle is a right angle.
What a learner can do afterwards
- Find a missing angle from the centre and circumference relationship
- Recognise the semicircle result as the case where the centre angle is 180 degrees
- Name the theorem as the reason for each step of the working
1 · Read
The angle at the centre is twice the angle at the edge on the same chord. With a centre angle of 80 degrees, the edge angle is 40. Always check both angles stand on the same chord, then halve or double.
A diameter makes a centre angle of 180 degrees, so the edge angle is half of that: 90 degrees. Every triangle in a semicircle holds a right angle. With one acute angle of 35 degrees, the third angle is 55, since angles total 180.
Name the theorem beside each step, since reasons earn marks. The diameter case has its own name: angle in a semicircle. When the reflex angle is 260 degrees, the minor angle is 100, so the edge angle is 50. Say which angle you halved.
Halve the centre angle for the edge, spot the right angle in a semicircle, and name the theorem each time.
More to read
Circle theorems | Lesson
The headline theorem links the centre to the edge: the angle at the centre is twice the angle at the circumference standing on the same chord. If the centre angle is 80 degrees, the circumference angle on the same arc is 40 degrees.
Always check both angles stand on the same chord before doubling or halving. Mark the chord, mark the two angles, then apply the rule. Naming the theorem beside each step is what earns the reason marks.
Challenge problems: Inscribed angles
Inscribed angles standing on the same chord are equal when they sit in the same segment. This follows from the centre rule, since both equal half of one centre angle. Spotting the shared chord is the whole skill.
Harder problems hide the chord inside overlapping triangles. Trace each angle back to its endpoints on the circle, then compare chords. Equal chords in the same segment mean equal angles, and that unlocks the next step.
Inscribed angle theorem proof
The semicircle result drops out of the centre rule: a diameter makes a centre angle of 180 degrees, so the circumference angle is half of that, 90 degrees. Every triangle drawn in a semicircle is right angled at the circle.
Proofs chain small facts into big ones, so state the centre rule first, then substitute 180 degrees. This same style, rule plus substitution, proves most circle facts. Learn the chain once and reuse it everywhere.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.