Know and use the circle theorems (angle at the centre, angle in a semicircle, angles in the same segment, cyclic quadrilateral, tangent properties and alternate segment) to find angles and construct reasoned proofs (Higher tier).

What this dot point is asking

OCR reference G10 at Higher tier covers the circle theorems: the relationships between angles, chords, tangents and radii in a circle, used both to find angles and to write reasoned proofs. Circle theorems are a signature Higher-tier topic that tests AO2 reasoning heavily, because every step needs a named justification. They build directly on the basic angle facts and are a reliable source of multi-mark questions.

Angles at the centre and circumference

The most-used theorem relates a central angle to an inscribed one.

So if a central angle is $80^\circ$, the inscribed angle on the same arc is $40^\circ$. The semicircle case is a frequent shortcut: any triangle with the diameter as one side has a right angle opposite it. Identifying which arc both angles stand on is the decisive step.

Same segment and cyclic quadrilaterals

Two theorems concern angles inside the circle.

So two inscribed angles standing on the same chord are equal, no matter where on the arc their vertices sit. In a cyclic quadrilateral $ABCD$, angle $A$ + angle $C$ = $180^\circ$ and angle $B$ + angle $D$ = $180^\circ$. These let you chain through a diagram, finding one angle and using it to find the next.

Tangents and the alternate segment

Tangents introduce right angles and a subtle equal-angle rule.

The tangent-radius right angle often combines with Pythagoras to find a length. The equal-tangents fact creates isosceles triangles, which then give equal base angles. The alternate segment theorem is the hardest to spot, so look for a tangent, a chord from the point of contact, and an inscribed angle in the far segment.

Why circle theorems matter

Circle theorems are the clearest test of geometric reasoning at GCSE, and OCR awards marks specifically for the named justification at each step, in line with the $30$ percent AO2 weighting. They reward a systematic approach: label what you know, find an angle you can justify, then use it to unlock the next. Quoting the theorem precisely ("angle at the centre is twice the angle at the circumference") is as important as the arithmetic.