State the theorem: the angle in a semicircle is…
…a right angle (\( 90^\circ \)).
Circle Theorems
8 exam-style questions on the main circle theorems used at GCSE.
2 Easy 4 Medium 2 Hard
A tangent meets a radius at the point of contact. What is the angle between them?
\( 90^\circ \) — the tangent is perpendicular to the radius.
Angle at the centre is twice the angle at the circumference (standing on the same arc). If the angle at the circumference is \( 35^\circ \), what is the angle at the centre?
\( 70^\circ \).
Opposite angles in a cyclic quadrilateral sum to…? If one angle is \( 110^\circ \), find the opposite angle.
\( 180^\circ \). Opposite angle \( = 70^\circ \).
Two tangents from an external point to a circle are equal in length. Tangents from P touch at A and B, and \( PA = 8 \) cm. What is \( PB \)?
\( PB = 8 \) cm.
The angle between a tangent and a chord equals the angle in the alternate segment. A tangent meets chord AB at A; the angle between tangent and chord is \( 40^\circ \). What is the angle in the alternate segment?
\( 40^\circ \).
O is the centre of a circle. Angle at the circumference subtended by arc AB is \( 28^\circ \). Find the reflex angle AOB at the centre.
Angle at centre (minor) \( = 56^\circ \). Reflex angle AOB \( = 360^\circ - 56^\circ = 304^\circ \).
A cyclic quadrilateral has angles \( x \), \( 2x \), \( 80^\circ \) and \( 100^\circ \) in order around the shape. Find \( x \).
Opposite angles sum to \( 180^\circ \): \( x + 80 = 180 \) ⇒ \( x = 100 \), but then \( 2x + 100 = 300 eq 180 \). Correct pairing: \( x + 2x \) wait — angles in order: opposite pairs are (x, 80) and (2x, 100)? Actually consecutive. Standard: opposite sum 180. If angles in cyclic order A,B,C,D then A+C=180, B+D=180. Suppose A=x, B=2x, C=80, D=100. Then x+80=180 ⇒ x=100; 2x+100=300≠180 — inconsistent. Try A=x, B=80, C=2x, D=100: x+2x=180 ⇒ x=60; 80+100=180. Yes. \( x = 60 \).