We are given a circle with center $T$. $NL$ and $MK$ are diameters of the circle. We are given that $\angle NTJ = 88^\circ$, $\angle JTK = 44^\circ$, and $\angle KTL = 48^\circ$. We need to identify whether arc $MNL$ is a major arc, a minor arc, or a semicircle, and find its measure.

1. Problem Description

We are given a circle with center $T$. $NL$ and $MK$ are diameters of the circle. We are given that $\angle NTJ = 88^\circ$, $\angle JTK = 44^\circ$, and $\angle KTL = 48^\circ$. We need to identify whether arc $MNL$ is a major arc, a minor arc, or a semicircle, and find its measure.

2. Solution Steps

First, let's find the measure of $\angle NTL$. Since $NL$ is a diameter, $\angle NTL = 180^\circ$.

Since $\angle NTK + \angle KTL = \angle NTL$, we have $\angle NTK + 48^\circ = 180^\circ$.

Thus, $\angle NTK = 180^\circ - 48^\circ = 132^\circ$.

Also, $\angle NTK = \angle NTJ + \angle JTK = 88^\circ + 44^\circ = 132^\circ$. This confirms the given information.

Since $MK$ is a diameter, $\angle MTK = 180^\circ$. We have $\angle MTK = \angle MTN + \angle NTJ + \angle JTK$. Also, $\angle MTN + \angle NTJ + \angle JTK = 180^\circ$.

We know that $\angle NTJ = 88^\circ$ and $\angle JTK = 44^\circ$, so $\angle MTN + 88^\circ + 44^\circ = 180^\circ$.

$\angle MTN = 180^\circ - 88^\circ - 44^\circ = 180^\circ - 132^\circ = 48^\circ$.

Now we want to find the measure of arc $MNL$, which corresponds to the central angle $\angle MTL$.

We have $\angle MTL = \angle MTN + \angle NTK + \angle KTL$, or $\angle MTL = \angle MTN + \angle NTL$. Since $\angle NTL = 180^\circ$,

$\angle MTL = 48^\circ + 180^\circ = 228^\circ$.

Since $228^\circ > 180^\circ$, the arc $MNL$ is a major arc.

The measure of arc $MNL$ is equal to the measure of $\angle MTL$, which is $228^\circ$.

3. Final Answer

Major arc

228