Question

Given: Tangent TA ∥ KM Prove: KS ≅ MS

Answer 1

If the tangent-line $TA$ is parallel to chord $KM$, that means segment $KS$ and $MS$ are congruent, which allow us to deduct that angles $SKM$ and $SMK$ are equal.

Then, by the theorem of an inscribed-angle, we have

$\angle SKM = \frac{1}{2}m(MS) \\ \angle SMK = \frac{1}{2}m(KS)$

We know, $\angle SKM \cong \angle SMK$, so

$\frac{1}{2}m(MS) = \frac{1}{2}m(KS)$

$\therefore m(MS)=m(KS)$