Geometry & measures · Circles, arcs & sectors
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Circle theorems
The theorems about angles in a circle — at the centre, in a semicircle, in the same segment, and in a cyclic quadrilateral — why each is true, and the exact wording examiners require as a reason.
Geometry & measures · Circles, arcs & sectors
Circle theorems
The theorems about angles in a circle — at the centre, in a semicircle, in the same segment, and in a cyclic quadrilateral — why each is true, and the exact wording examiners require as a reason.
Why it works
Every circle theorem below comes from one fact: all radii are equal, so any triangle with two radii as sides is isosceles.The angle at the centre is twice the angle at the circumference (on the same arc). Draw the radius from the centre to the point on the circumference. It splits the figure into two isosceles triangles; in each, the exterior angle at is twice the base angle at (exterior angle = sum of the two equal interior ones). Adding the two halves gives centre circumference.
The angle in a semicircle is 90°. This is the theorem above with the "angle at the centre" stretched into a straight line: the centre angle is , so the angle at the circumference is . Spotting it needs the diameter — if a triangle's longest side passes through the centre, the opposite angle is a right angle.
Angles in the same segment are equal. Two points on the same arc both subtend the same centre angle, and each is half of it — so they equal each other. In a figure this is the "bow-tie": two triangles standing on the same chord, with the equal angles at the two far vertices.
Opposite angles in a cyclic quadrilateral add up to 180°. Take the two centre angles standing on the two arcs cut by a diagonal pair of vertices: together they make . Each opposite angle is half of one of them, so together the two opposite angles are half of . (A useful corollary: an exterior angle of a cyclic quadrilateral equals the interior angle at the opposite vertex.)
The reasons are marked, and the wording is fixed. Write them in full:
- "the angle at the centre is twice the angle at the circumference"
- "the angle in a semicircle is 90°"
- "angles in the same segment are equal"
- "opposite angles in a cyclic quadrilateral add up to 180°"
- "triangle is isosceles because and are radii"
Check the theorem actually applies. Angles in the same segment must stand on the same chord and be on the same side of it; a cyclic quadrilateral needs all four vertices on the circle. Half the lost marks here come from quoting a theorem whose conditions aren't met.