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Alternate segment theorem with a short proof

2026-08-12Plane GeometryMs Ho

The angle between a tangent and a chord equals the angle in the alternate segment.

Key formulas

∠ between tangent and chord = angle in alternate segment

Statement

If a tangent at A meets chord AB, then the angle between the tangent and AB equals the angle subtended by AB in the alternate segment of the circle.

Worked numerical case

Tangent at A, chord AB, and point C on the circumference in the alternate segment. If the angle between the tangent and AB is 34°, then ∠ACB = 34°. If another point D lies in the same alternate segment, ∠ADB = 34° as well.

How to write it

Name the tangent, name the chord, identify the alternate segment, then quote the theorem. Add a diagram mark so the examiner sees which angle matches which.

Worked example

Worked example

Problem. Tangent at T touches a circle; chord TA is drawn. The angle between the tangent and chord is 55°. Find the angle in the alternate segment.

Solution. By the alternate segment theorem, the angle in the alternate segment equals the tangent–chord angle. So that inscribed angle is also 55°.

Combined with isosceles radius

If OA is a radius to the contact point, OA ⟂ tangent. That right angle often supplies the 34° needed for the chord-tangent angle.

Review quiz

Tangent–chord angle is 38°. Angle in the alternate segment is:

Practice questions

  1. Tangent at A meets chord AB at 41°. Point C is in the alternate segment. Find ∠ACB.
  2. Sketch and label the alternate segment for a tangent at A and chord AC.

Study notes

What does “alternate” mean here?

The segment on the other side of the chord from the tangent angle.

Is this the same as the tangent-radius theorem?

No. They are often used together, but they are different facts.

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