Find the measures of the numbered angles in the diagram.

The provided diagram is,

The measurement of one angle in the diagram is$80°$. The numbered angles are 1, 2, 3, 4, 5, 6, 7 as given in the diagram.

The sum of the linear paired angles is $180°$, $80°+m\angle 1=180°$.

The measure of $\angle 1$ will be,

$\begin{array}{c}{80}^{\circ }+m\angle 1={180}^{\circ }\\ m\angle 1={180}^{\circ }-{80}^{\circ }\\ m\angle 1={100}^{\circ }\end{array}$

Since vertically opposite angles are equal, $m\angle 1=m\angle 2$and $m\angle 3=80°$.

Therefore, $m\angle 2=100°$ and $m\angle 3=80°$.

From the given figure, $\angle 2$and $\angle 5$, and the angles $\angle 3$ and $\angle 4$ are two pairs of alternate interior angles.

Since alternate interior angles are equal, $m\angle 5=m\angle 2$and $m\angle 3=m\angle 4$.

Therefore, the measures of $\angle 4$ and $\angle 5$ will be,

$\begin{array}{l}m\angle 5=m\angle 2\\ m\angle 5=100°\\ \\ m\angle 4=m\angle 3\\ m\angle 4=80°\end{array}$

Since vertically opposite angles are equal, $m\angle 5=m\angle 6$and $m\angle 4=m\angle 7$.

Therefore, the measures of $\angle 4$ and $\angle 5$ will be,

$\begin{array}{l}m\angle 6=m\angle 5\\ m\angle 6=100°\\ \\ m\angle 7=m\angle 4\\ m\angle 7=80°\end{array}$

The measures of the numbered angles are $m\angle 1=100°$, $m\angle 2=100°$, $m\angle 3=80°$, $m\angle 4=80°$, $m\angle 5=100°$, $m\angle 6=100°$, and $m\angle 7=80°$.