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.
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 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.