We are given a quadrilateral RNPQ. We know that angle $R = 34^\circ$ and angle $Q = 108^\circ$. Also, sides $RN$ and $NQ$ are equal, and $NP$ is parallel to $RQ$. We are asked to find the measure of angle $NPQ$.

1. Problem Description

We are given a quadrilateral RNPQ. We know that angle $R = 34^\circ$ and angle $Q = 108^\circ$. Also, sides $RN$ and $NQ$ are equal, and $NP$ is parallel to $RQ$. We are asked to find the measure of angle $NPQ$.

2. Solution Steps

First, consider triangle $RNQ$. Since $RN = NQ$, triangle $RNQ$ is an isosceles triangle. Therefore, angle $NRQ$ is equal to angle $NQR$, which is $34^\circ$.

The sum of the angles in triangle $RNQ$ is $180^\circ$, so

$RNQ + NRQ + NQR = 180^\circ$

$RNQ + 34^\circ + 34^\circ = 180^\circ$

$RNQ = 180^\circ - 68^\circ = 112^\circ$

Since $NP$ is parallel to $RQ$, angles $PNQ$ and $NQR$ are alternate interior angles, therefore they sum up to $180^\circ$. The sum of co-interior angles is $180^\circ$ i.e. $NQR + PNQ = 180^\circ$

Therefore, the angle $PNQ = 180^\circ - Q = 180^\circ - 108^\circ = 72^\circ$.

Now we know $PNQ = 72^\circ$.

Consider the quadrilateral $RNPQ$. The sum of the angles in a quadrilateral is $360^\circ$.

$R + N + P + Q = 360^\circ$

$34^\circ + RNQ + PNQ + NPQ + 108^\circ = 360^\circ$

$34^\circ + RNQ + NPQ + 108^\circ = 360^\circ - PNQ - NPQ$

We know that $RNQ = 112^\circ$ and $PNQ = 72^\circ$ from the property of parallel lines.

$34^\circ + 112^\circ + NPQ + 108^\circ = 360^\circ$

$254^\circ + NPQ = 360^\circ$

$NPQ = 360^\circ - 254^\circ = 106^\circ$

3. Final Answer

The measure of angle $NPQ$ is $106^\circ$.