Question 13

We will draw extra parallel lines to find the unknown angle by breaking it into smaller parts.

Step 1 — Drawing parallel lines

First, we see arrows on lines LM and PQ. This means line LM is parallel to line PQ. Let us draw a line through point N. Let us call this line RS. We draw RS so it is parallel to line LM. Let us draw another line through point O. Let us call this line TU. We draw TU so it is parallel to line PQ. Since LM is parallel to PQ, and RS is parallel to LM, and TU is parallel to PQ, it means RS is also parallel to TU.

Step 2 — Finding the first part of angle MNO

Look at line LM and line RS. They are parallel lines. Line MN cuts across them. This is a transversal line. The angle $\angle\text{LMN}$ and the angle $\angle\text{MNS}$ are alternate interior angles. Alternate interior angles are equal. So, $\angle\text{MNS}$ is equal to $\angle\text{LMN}$. $\angle\text{LMN}$ is given as $\mathbf{40^\circ}$. Let us call $\angle\text{MNS}$ as $w^\circ$.

$w^\circ = 40^\circ$

$\boxed{w^\circ = 40^\circ}$

Step 3 — Finding the second part of angle MNO

We know the whole angle $\angle\text{MNO}$ is $\mathbf{96^\circ}$. This angle is made of two parts: $\angle\text{MNS}$ and $\angle\text{SNO}$. We just found $\angle\text{MNS}$ is $w^\circ = 40^\circ$. Let us call $\angle\text{SNO}$ as $x^\circ$. So, $w^\circ + x^\circ = 96^\circ$.

$40^\circ + x^\circ = 96^\circ$

To find $x^\circ$, we subtract $40^\circ$ from $96^\circ$.

$x^\circ = 96^\circ - 40^\circ$

$x^\circ = 56^\circ$

$\boxed{x^\circ = 56^\circ}$

Step 4 — Finding the first part of angle NOP

Now look at line RS and line TU. We know they are parallel lines. Line NO cuts across them. This is a transversal line. The angle $\angle\text{SNO}$ and the angle $\angle\text{NOT}$ are alternate interior angles. Alternate interior angles are equal. We just found $\angle\text{SNO}$ is $x^\circ = 56^\circ$. Let us call $\angle\text{NOT}$ as $y^\circ$. So, $y^\circ$ is equal to $x^\circ$.

$y^\circ = x^\circ$

$y^\circ = 56^\circ$

$\boxed{y^\circ = 56^\circ}$

Step 5 — Finding the second part of angle NOP

Now look at line TU and line PQ. They are parallel lines. Line OP cuts across them. This is a transversal line. The angle $\angle\text{TOP}$ and the angle $\angle\text{OPQ}$ are alternate interior angles. Alternate interior angles are equal. $\angle\text{OPQ}$ is given as $\mathbf{52^\circ}$. Let us call $\angle\text{TOP}$ as $z^\circ$. So, $z^\circ$ is equal to $\angle\text{OPQ}$.

$z^\circ = 52^\circ$

$\boxed{z^\circ = 52^\circ}$

Step 6 — Finding the total angle NOP

The angle we need to find is $\angle\text{NOP}$, which is $a^\circ$. From the diagram, $\angle\text{NOP}$ is made of two parts. These parts are $\angle\text{NOT}$ and $\angle\text{TOP}$. We found $\angle\text{NOT}$ is $y^\circ = 56^\circ$. We found $\angle\text{TOP}$ is $z^\circ = 52^\circ$. So, $a^\circ$ is the sum of $y^\circ$ and $z^\circ$.

$a^\circ = y^\circ + z^\circ$

$a^\circ = 56^\circ + 52^\circ$

$a^\circ = 108^\circ$

$\boxed{a^\circ = 108^\circ}$

Answer

The measure of angle $\angle\text{NOP}$ is $\mathbf{108^\circ}$.