Question

pl and km are diameters of ⊙t. find m\\(\overarc{ml}\\).

Explanation:

Step1: Recognize straight angle sum

A diameter forms a $180^\circ$ angle at center.
For diameter $\overline{KM}$: $\angle PTN + \angle NTM + \angle MTL = 180^\circ$, but first find $\angle JTP$ using $\overline{PL}$:
$\angle JTP = 180^\circ - \angle PTN - \angle NTK$
$\angle JTP = 180^\circ - 42^\circ - 48^\circ - 32^\circ = 58^\circ$

Step2: Use vertical angles property

Vertical angles are equal, so $\angle MTL = \angle JTP = 58^\circ$.

Step3: Arc equals central angle

The measure of arc $\overset{\frown}{ML}$ equals its central angle $\angle MTL$.

Answer:

$58^\circ$