Any angle subtended in a semicircle — from the two ends of a diameter to any point on the circumference — is always exactly $90°$.

A tangent touches a circle at exactly one point. At that point of contact, the tangent is always perpendicular to the radius drawn to it: the angle between them is $90°$.

For angles subtended by the same arc, the angle at the centre is always twice the angle at the circumference:

Angles in the same segment are equal — any two angles at the circumference subtended by the same arc (on the same side of it) are equal to each other.

A cyclic quadrilateral has all four vertices on the circle. Its opposite angles sum to $180°$ (they are supplementary).

The angle between a tangent and a chord drawn from the point of contact equals the angle in the alternate segment — the angle subtended by that same chord in the segment on the other side of it. "Alternate" here means the segment on the far side of the chord from the angle you're comparing it to — it can help to physically trace the chord with your finger and look at the region not containing the tangent angle you started with.

Look first at what the angle is formed between: two chords meeting at the circumference (same-segment or cyclic-quadrilateral rules), a radius and a tangent (always $90°$), a diameter and a circumference point (semicircle rule, always $90°$), or a tangent and a chord (alternate segment theorem). Identifying these two "ingredients" of an angle — what two lines actually form it, and where its vertex sits (centre, circumference, or point of tangency) — narrows the six theorems down to exactly one almost every time.

$A$, $B$, $C$ are points on a circle with centre $O$. Angle $AOC = 130°$ (the angle at the centre). Find angle $ABC$ (the angle at the circumference, subtended by the same arc $AC$).

1. Angle at centre is twice angle at circumference (same arc $AC$): $\text{angle } AOC = 2 \times \text{angle } ABC$.
2. $130° = 2 \times \text{angle } ABC \implies \text{angle } ABC = 65°$

$PQRS$ is a cyclic quadrilateral. Angle $PQR = 108°$. $PT$ is a tangent to the circle at $P$, and angle $TPS = 40°$. Find (a) angle $PSR$, and (b) angle $PRS$ (the angle in the alternate segment to angle $TPS$, subtended by chord $PS$).

1. (a) Opposite angles of a cyclic quadrilateral sum to $180°$: angle $PSR = 180° - 108° = 72°$.
2. (b) By the alternate segment theorem, the angle between tangent $PT$ and chord $PS$ equals the angle in the alternate segment subtended by $PS$: angle $PRS = 40°$.