The circle theorems
Four angle facts
For points A, B, C and D on a circle with centre O:
- The angle at the centre is twice the angle at the circumference when both stand on the same arc.
- The angle in a semicircle is 90°: if AB is a diameter, angle ACB = 90°. (The angle at the centre is then 180°, and half of it is 90°.)
- Angles in the same segment are equal: angles ACB and ADB, on the same side of the chord AB, are the same size.
- Opposite angles of a cyclic quadrilateral add up to 180°. A cyclic quadrilateral has all four corners on the circle.
Two more facts you will need
- Chords: the perpendicular from the centre to a chord bisects the chord. It cuts the chord into two equal halves and you can use Pythagoras with the radius.
- With tangents (next lesson): a tangent meets a radius at 90°, tangents from the same point are equal in length, and the alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
Other reasons still count: base angles of an isosceles triangle are equal, angles on a straight line add up to 180°, and the angles in a triangle add up to 180°.
Say which theorem
On a paper, "give reasons" means a reason for every angle you find, in words.
Write "the angle at the centre is twice the angle at the circumference", not "centre rule". Name the angle with three letters (angle ACB), so the examiner knows which one you mean.
Move the points
In the circle theorems simulation, choose the angle at the centre. Put A at 40°, B at 150° and C at 270°. Work out angle ACB first, then check. Now drag C onto the short arc between A and B: what happens to the angle at the centre?
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This question is in Brainlag Plus