Circle theorems I.

Circle theorems connect angles to particular chords, arcs and tangents. They are not rules for every angle drawn inside a circle. Begin by identifying the centre, circumference points and any tangent.

Use a finger or coloured pencil to trace the two endpoints of the relevant chord or arc. Then locate the angles standing on those same endpoints. This careful matching makes the theorem meaningful and prevents selecting a rule merely because a familiar-looking circle is present.

An angle standing on a chord is formed by joining its two endpoints to a vertex elsewhere on the circle. Moving that vertex around the same segment preserves the angle.

Trace the two rays of each angle back to their circumference endpoints. Angles in the same segment use the same chord and lie on the same side of it. If one angle is at the centre instead, use the appropriate centre-to-circumference relationship rather than treating them as equal.

A circle problem often needs an ordinary triangle or straight-line fact between two theorem steps. There may be no single rule that takes you directly from the given angle to the requested one.

Label intermediate angles as soon as they are known. Work through a chain: identify a circle theorem, calculate a related angle, then use a triangle sum or supplementary-angle relation. Give the reason at every link so the final answer can be followed and checked.

For each theorem, check its required configuration before using its numerical relationship. Mark the relevant chord, tangent or centre clearly.

Write a reason beside each new angle and retain intermediate labels on your sketch. A plausible final number is not enough if it comes from a theorem applied to the wrong arc or segment.