Tangents and the Tangent–Chord Theorem

Tangents and the Tangent–Chord Theorem

The tangent-chord theorem says the angle between a tangent and a chord equals the angle in the alternate segment subtended by that chord. Pair the same chord endpoints before applying it. Together with the radius-tangent right angle, this theorem turns a contact-point angle into an angle on the far side of a circle.

Worked reasoning

If the angle between tangent at A and chord AB is $48^\circ$, then any angle ACB in the alternate segment is $48^\circ$.

Exam method

1. Identify contact point and chord. 2. Find the angle on the opposite arc using the same chord endpoints. 3. State ‘tangent-chord theorem’ as the reason.

The alternate segment is specific

Name the chord used by the tangent angle and then locate the angle that stands on that same chord from the other side. If the tangent is at A and the chord is AB, the matching circumference angle must have arms to A and B. This letter-tracing method is more reliable than visual guessing. Combine it with the right angle between a radius and tangent only when the radius at the contact point is present or constructed.

One-minute retrieval: Tangents and the Tangent–Chord Theorem

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.