Question

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

Short answer

Yes. The measure of all the angles can be determined.

Step-by-step solution

Step 1. Given Information.

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

Step 2. Concepts used.

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.

Step 3. Find the measure of angle 5.

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

The measure of $\angle 5$ will be,

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

Step 4. Find the measure of angle 7 and 8.

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,

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

Step 5. Find the measure of angle 4 and 5.

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,

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

Step 6. Find the measure of angle 2 and 4.

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,

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

Step 7. Conclusion.

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 , $m\angle 8=180{}^{\circ }-m\angle 6$.