Find the measures of the numbered angles in the diagram.

The provided diagram is,

The provided angle is $m\angle 1=65°$ and the numbered angles are 1, 2, 3, 4, 5, 6, 7 as given in the diagram.

When two lines are crossed by another line, it is known as transversal.Corresponding angles are angles that are at the same relative position at each intersection. The measure of two corresponding angles is equal.

Thesum of the linear paired angles is $180°$, $m\angle 1+m\angle 3=180°$.

The measure of $\angle 3$ will be,

$\begin{array}{c}m\angle 1+m\angle 3=180°\\ m\angle 3=180°-m\angle 1\\ m\angle 3=180°-65°\\ m\angle 3=115°\end{array}$

Since vertically opposite angles are equal, $m\angle 1=m\angle 4$and $m\angle 2=m\angle 3$.

Therefore, the measures of $\angle 2$ and $\angle 4$ will be,

$\begin{array}{l}m\angle 2=m\angle 3\\ m\angle 2=115°\\ \\ \text{And,}\\ \\ m\angle 4=m\angle 1\\ m\angle 4=65°\end{array}$

From the given figure, $\angle 1$and $\angle 5$, and the angles $\angle 2$ and $\angle 6$ are two pairs of corresponding angles.

Since corresponding anglesare equal, $m\angle 1=m\angle 5$and $m\angle 2=m\angle 6$.

Therefore, the measures of $\angle 5$ and $\angle 6$ will be,

$\begin{array}{l}m\angle 5=m\angle 1\\ m\angle 5=65°\\ \\ \text{and}\\ \\ m\angle 6=m\angle 2\\ m\angle 6=115°\end{array}$

From the given figure, $\angle 3$and $\angle 7$, and the angles $\angle 4$ and $\angle 8$ are two pairs of corresponding angles.

Since corresponding anglesare equal, $m\angle 3=m\angle 7$and $m\angle 4=m\angle 8$.

Therefore, the measures of $\angle 7$ and $\angle 8$ will be,

$\begin{array}{l}m\angle 7=m\angle 3\\ m\angle 7=115°\\ \\ \text{and}\\ \\ m\angle 8=m\angle 4\\ m\angle 8=65°\end{array}$

The measures of the numbered angles are $m\angle 2=115°$, $m\angle 3=115°$, $m\angle 4=65°$, $m\angle 5=65°$, $m\angle 6=115°$, $m\angle 7=115°$, and $m\angle 8=65°$.