Circle Theorems
Angles in circles - the standard circle theorems and using them to find angles and build reasons.
Learn
Circle Theorems, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Use: the angle at the centre is twice the angle at the circumference
When two angles stand on the same arc, the one at the centre is double the one at the circumference. Spot the arrowhead shape: centre angle 140° means 70° at the edge. Check both angles really do stand on the SAME arc before using it.
Use: angles in the same segment are equal, and the angle in a semicircle is 90°
Angles at the circumference standing on the same arc are equal - they look like a bowtie. And any triangle drawn from a diameter has a right angle at the circumference, because the "centre angle" is the straight line, 180°, and half of that is 90°.
Use: opposite angles of a cyclic quadrilateral add to 180°
A cyclic quadrilateral has all four corners on the circle, and each pair of OPPOSITE angles sums to 180°: if one angle is 95°, the angle across from it is 85°. All four vertices must touch the circle for the theorem to apply.
Use tangent properties: tangent ⊥ radius, and tangents from a point are equal
A tangent meets its radius at exactly 90° - drawing that radius in usually unlocks the question. Two tangents drawn from the same outside point are equal in length, creating an isosceles triangle with the equal base angles that follow.
Use the alternate segment theorem
The angle between a tangent and a chord equals the angle in the ALTERNATE segment - the inscribed angle on the other side of the chord. It is the hardest to spot: look for a triangle with one vertex at the point of tangency.
Give clear reasons in circle-theorem proofs, naming each theorem you use
Every step needs its theorem quoted in full - "opposite angles of a cyclic quadrilateral sum to 180°", not "circle theorems". Chase angles around the diagram, writing each new angle on it, and give one reason per step.
Practice
Try a Circle Theorems question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
ABCD is a cyclic quadrilateral - its four vertices all lie on a circle. Angle ABC = 91°. Work out angle ADC.
- 89°
- 33°
- 58°
- 91°
Show the answer and the working
Answer: 89°
Opposite angles of a cyclic quadrilateral add up to 180°.
- ABC and ADC are opposite angles of the cyclic quadrilateral ABCD.
- Opposite angles of a cyclic quadrilateral add up to 180°.
- So angle ADC = 180 − 91 = 89°.