In the given diagram, we have the following angle measures: $\angle STQ = m$, $\angle TUQ = 80^\circ$, $\angle UPQ = r$, $\angle PQU = n$, and $\angle RQT = 88^\circ$. The problem asks us to find the value of $m+n$.

1. Problem Description

In the given diagram, we have the following angle measures: $\angle STQ = m$, $\angle TUQ = 80^\circ$, $\angle UPQ = r$, $\angle PQU = n$, and $\angle RQT = 88^\circ$. The problem asks us to find the value of $m+n$.

2. Solution Steps

First, note that $\angle PQU$ and $\angle RQT$ are supplementary angles, which means that their sum is $180^\circ$.

Since $\angle RQT = 88^\circ$, we have

$n + 88^\circ = 180^\circ$.

$n = 180^\circ - 88^\circ = 92^\circ$.

Next, consider triangle $TUQ$. The sum of the angles in any triangle is $180^\circ$.

Thus, $\angle STQ + \angle TUQ + \angle TQU = 180^\circ$.

$\angle STQ = m$, $\angle TUQ = 80^\circ$, and $\angle TQU = n$.

Then we have $m + 80^\circ + n = 180^\circ$.

$m + n = 180^\circ - 80^\circ$.

$m + n = 100^\circ$.

Another way to solve is:

Since $n = 92^\circ$,

$m + 80^\circ + 92^\circ = 180^\circ$.

$m + 172^\circ = 180^\circ$.

$m = 180^\circ - 172^\circ = 8^\circ$.

$m + n = 8^\circ + 92^\circ = 100^\circ$.

Therefore, $m + n = 100$.

3. Final Answer

The value of $m+n$ is $100$.