We are given a diagram with several angles labeled. Specifically, $\angle STQ = m$, $\angle TUQ = 80^\circ$, $\angle UPQ = r$, $\angle PQU = n$, and $\angle RQT = 88^\circ$. We need to find the value of $m+n$.

1. Problem Description

We are given a diagram with several angles labeled. Specifically, $\angle STQ = m$, $\angle TUQ = 80^\circ$, $\angle UPQ = r$, $\angle PQU = n$, and $\angle RQT = 88^\circ$. We need to find the value of $m+n$.

2. Solution Steps

Since $\angle RQT = 88^\circ$ and $\angle PQR$ is a straight angle, we have:

$$\angle PQT + \angle RQT = 180^\circ$$

$$\angle PQT = 180^\circ - \angle RQT = 180^\circ - 88^\circ = 92^\circ$$

We are given that $\angle PQU = n$, so $n = \angle PQT = 92^\circ$.

In triangle $UQT$, we know $\angle TUQ = 80^\circ$ and $\angle UQT = 92^\circ$.

Since the sum of angles in a triangle is $180^\circ$, we have:

$$\angle TUQ + \angle UQT + \angle UTQ = 180^\circ$$

$$80^\circ + 92^\circ + \angle UTQ = 180^\circ$$

$$\angle UTQ = 180^\circ - 80^\circ - 92^\circ = 8^\circ$$

$\angle UTQ$ and $\angle STQ = m$ form a straight angle.

$$\angle UTQ + \angle STQ = 180^\circ$$

$$8^\circ + m = 180^\circ$$

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

We need to find the value of $m+n$. We have $m = 172^\circ$ and $n = 92^\circ$.

$$m + n = 172^\circ + 92^\circ = 264^\circ$$

3. Final Answer

The value of $(m+n)$ is $264$.