We are given a circle with center T. $PL$ and $KM$ are diameters of the circle. We need to find the measure of arc $PJ$. From the diagram, we are given that $m\angle MTN = 48^\circ$, $m\angle NTP = 42^\circ$, $m\angle JTK = 32^\circ$, and $m\angle KTL = 90^\circ$.

1. Problem Description

We are given a circle with center T. $PL$ and $KM$ are diameters of the circle. We need to find the measure of arc $PJ$. From the diagram, we are given that $m\angle MTN = 48^\circ$, $m\angle NTP = 42^\circ$, $m\angle JTK = 32^\circ$, and $m\angle KTL = 90^\circ$.

2. Solution Steps

Since $PL$ is a diameter, $m\angle PTL = 180^\circ$. Also, $m\angle PTJ = m\angle PTK + m\angle JTK$.

Since the measure of an arc is equal to the measure of its central angle, we need to find the measure of the central angle $\angle PTJ$.

First, we find $m\angle MTP$.

$m\angle MTL + m\angle LTP = 180^\circ$ (Since $PL$ is a straight line).

Also, $m\angle MTN + m\angle NTK = 180^\circ - m\angle MTP$.

Since $KM$ is a diameter, $m\angle KTM = 180^\circ$.

$m\angle KTM = m\angle KTL + m\angle MTL = 90^\circ + m\angle MTL = 180^\circ$.

From this, we get $m\angle MTL = 180^\circ - 90^\circ = 90^\circ$.

So, $m\angle MTP = 180^\circ - (m\angle NTP + m\angle NTJ)$

$m\angle PTK + m\angle KTL = 180^\circ - (m\angle MTP)$

We also know that the sum of the angles around the center T is $360^\circ$.

$m\angle MTN + m\angle NTP + m\angle PTK + m\angle KTL + m\angle LTM = 360^\circ$.

$48^\circ + 42^\circ + m\angle PTK + 90^\circ + m\angle LTM = 360^\circ$.

Also, $m\angle MTL = 90^\circ$ (because KM is a diameter and angle KTL is a right angle)

Since PL is a straight line, $m\angle PTL = 180^\circ$.

Therefore $m\angle PTL = m\angle PTK + m\angle KTL = m\angle PTK + 90^\circ = 180^\circ$.

So $m\angle PTK = 180^\circ - 90^\circ = 90^\circ$.

Then, $m\angle PTJ = m\angle PTK + m\angle JTK = 90^\circ + 32^\circ = 122^\circ$.

Since the measure of an arc equals the measure of its central angle, $m\widehat{PJ} = m\angle PTJ = 122^\circ$.

3. Final Answer

122