In example 2, if you know only $m\angle 6$,can you find the measures of the other numbered angles? Explain.

The measure of the angle $\angle 6$ 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$180°$, $m\angle 5+m\angle 6=180°$.

The measure of $\angle 5$ will be,

$\begin{array}{c}m\angle 5+m\angle 6={180}^{\circ }\\ m\angle 5={180}^{\circ }-m\angle 6\end{array}$

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

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

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

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

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

Therefore, the measures of $\angle 1$ and $\angle 3$ will be,

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

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

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

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

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

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