The Alternate Segment Theorem

The tangent touches the circle at T, and the chord TA runs from there across the circle. That chord splits the circle into two segments.

∠ATP — between the tangent and the chord — is equal to ∠ACT, the angle made by any point C in the segment on the other side of the chord. That segment is the alternate one, and naming it is most of the theorem.

How to Use

Interactive Exploration

Drag C along its arc. Its angle does not change at all — it is the “angles in the same segment” result again, which is why any point in that segment will do.

Drag T or A to open or close the chord. Now both angles change, and they change together.

Push the chord wide and the tangent-chord angle becomes obtuse. The angle in the alternate segment follows it up past 90° — the case students most often get wrong, because the obvious-looking angle at T is now the other one.

Solving Problems

New positions sets a fresh question: one angle keeps its value and the other is marked x. Tap the x on the diagram, or click ∠ATP or ∠ACT, to check.

The Style tab changes the colours and the background paper.

Teaching Notes

• T and A snap to 2° steps, so the arc between them is even and both angles — each half of it — come out whole.
• Why it works: both angles are half the arc TA that the chord cuts off. The tangent-chord angle is half that arc directly; the angle at C is the inscribed angle standing on the same arc.
• The hard part is not the arithmetic, it is deciding which pair goes together. There are two tangent-chord angles at T and they add to 180°; the one that matches is the one standing on the arc away from C.
• Worth pairing with angles in the same segment first, so students already accept that moving C changes nothing.