We are given a circle with center $T$. $PL$ and $KM$ are diameters of circle $T$. We are given the measures of arcs $PN = 42^\circ$, $NM = 48^\circ$, and $JK = 32^\circ$. We want to find the measure of arc $JKL$.

1. Problem Description

We are given a circle with center $T$. $PL$ and $KM$ are diameters of circle $T$. We are given the measures of arcs $PN = 42^\circ$, $NM = 48^\circ$, and $JK = 32^\circ$. We want to find the measure of arc $JKL$.

2. Solution Steps

Since $PL$ is a diameter, the angle $PTL$ is a straight angle, and thus equals $180^\circ$.

Also, since $KM$ is a diameter, the angle $KTM$ is a straight angle, and thus equals $180^\circ$.

The measure of the arc $PL$ is $180^\circ$.

We know that the measure of arc $PN = 42^\circ$ and the measure of arc $NM = 48^\circ$.

Therefore, the measure of arc $PM = PN + NM = 42^\circ + 48^\circ = 90^\circ$.

Since $PL$ is a diameter, the measure of arc $ML$ is $180^\circ - 90^\circ = 90^\circ$.

We are given that the angle $KTL$ is a right angle, so the measure of arc $KL = 90^\circ$.

We are given that the measure of arc $JK = 32^\circ$.

The measure of arc $JKL = JK + KL = 32^\circ + 90^\circ = 122^\circ$.

3. Final Answer

122