91Ó°ÊÓ

The provided diagram is,

The measurement of one angle in the diagram is $122°$. The numbered angles are 1, 2, 3, 4, 5, 6, 7 as given in the diagram.

The sum of the linear paired angles is $180°$, so, $122°+m∠1=180°$.

The measure of $∠1$ will be,

$\begin{array}{c}{122}^{∘}+m∠1={180}^{∘}\\ m∠1={180}^{∘}−122°\\ m∠1={58}^{∘}\end{array}$

Since vertically opposite angles are equal, $m∠1=m∠3$and $m∠2=122°$.

Therefore, $m∠2=122°$ and $m∠3=58°$.

From the given figure, $∠1$and $∠4$, and the angles $∠2$ and $∠6$ are two pairs of corresponding angles.

Since corresponding angles are equal, $m∠1=m∠4$and $m∠2=m∠6$.

Therefore, the measures of $∠4$ and $∠6$ will be,

$\begin{array}{l}m∠4=m∠1\\ m∠4=58°\\ \\ m∠6=m∠2\\ m∠6=122°\end{array}$

Since vertically opposite angles are equal, $∠5=∠6$and $∠4=∠7$.

Therefore, the measures of $∠5$ and $∠7$ will be,

$\begin{array}{l}m∠5=m∠6\\ m∠5=122°\\ \\ m∠7=m∠4\\ m∠7=58°\end{array}$

The measures of the numbered angles are $m∠1=58°$, $m∠2=122°$, $m∠3=58°$, $m∠4=58°$, $m∠5=122°$, $m∠6=122°$, and $m∠7=58°$.