91Ó°ÊÓ

The measure of the angle $∠6$ is known.

Transversal and parallel lines:

If two parallel lines are intersected by a transversal line then the corresponding angles are equal, alternate interior angles and alternate exterior angles are equal.

Vertically opposite angles are equal.

The sum of the linear paired angles is$180°$, $m∠5+m∠6=180°$.

The measure of $∠5$ will be,

$\begin{array}{c}m∠5+m∠6={180}^{∘}\\ m∠5={180}^{∘}−m∠6\end{array}$

Since vertically opposite angles are equal, $m∠6=m∠7$and $m∠5=m∠8$.

Therefore, the measures of $∠7$ and $∠8$ will be,

$\begin{array}{l}m∠7=m∠6\\ \\ \text{And,}\\ \\ m∠8=m∠5\\ m∠8=180°−m∠6\end{array}$

From the given figure, $∠1$and $∠5$, and the angles $∠3$ and $∠7$ are two pairs of corresponding angles.

Since corresponding anglesare equal, $m∠1=m∠5$and $m∠3=m∠7$.

Therefore, the measures of $∠1$ and $∠3$ will be,

$\begin{array}{l}m∠1=m∠5\\ m∠1=180°−m∠6\\ \\ \text{And,}\\ \\ m∠3=m∠7\\ m∠3=m∠6\end{array}$

From the given figure, $∠2$and $∠6$, and the angles $∠4$ and $∠8$ are two pairs of corresponding angles.

Since corresponding anglesare equal, $m∠2=m∠6$and $m∠4=m∠8$.

Therefore, the measures of $∠2$ and $∠4$ will be,

$\begin{array}{l}m∠2=m∠6\\ \\ \text{And,}\\ \\ m∠4=m∠8\\ m∠4=180°−m∠6\end{array}$

The measures of the numbered angles are $m∠1=180{}^{∘}-m∠6$, $m∠2=m∠6$, $m∠3=m∠6$, $m∠4=180{}^{∘}-m∠6$, $m∠5=180{}^{∘}-m∠6$, $m∠7=m∠6$, and , $m∠8=180{}^{∘}-m∠6$.