In each set, one thing changes and everything else stays the same. Work them out in order and look for the pattern — the last line tells you what to notice.
①Angle at the centre is twice the angle at the circumference – the given angle changeschanging: the angle at the circumference
In each circle, chord AB subtends the given angle at C (on the circumference) and the angle $x$ at the centre O. Work out $x$.
angle at C = 25°
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angle at C = 40°
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angle at C = 58°
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The angle at the centre is always double the angle at the circumference on the same arc. What is the given angle when $x=90°$?
②The angle in a semicircle is 90° – the given base angle changeschanging: the angle at A
AB is a diameter, so the angle at C is 90°. For each given angle at A, work out the angle $x$ at B.
angle at A = 30°
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angle at A = 48°
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angle at A = 64°
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The three angles of the triangle add to 180°, and one of them is always 90°. So what do the other two always add to?
③Opposite angles of a cyclic quadrilateral add to 180° – the given angle changeschanging: angle A
ABCD is a cyclic quadrilateral (all four vertices lie on the circle). For each value of angle A, work out the opposite angle C.
angle A = 70°
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angle A = 85°
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angle A = 100°
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angle A = 115°
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Opposite angles always add to 180°. What is the largest angle A could be while angle C stays positive?
Answers · Circle Theorems
Variation practice
① Angle at the centre is twice the angle at the circumference – the given angle changes
angle at C = 25°: x = 50°angle at C = 40°: x = 80°angle at C = 58°: x = 116°
② The angle in a semicircle is 90° – the given base angle changes
angle at A = 30°: x = 60°angle at A = 48°: x = 42°angle at A = 64°: x = 26°
③ Opposite angles of a cyclic quadrilateral add to 180° – the given angle changes
angle A = 70°: angle C = 110°angle A = 85°: angle C = 95°angle A = 100°: angle C = 80°angle A = 115°: angle C = 65°
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