Question

Find the measures of the numbered angles in the diagram.

Short answer

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 $m\angle 8=65°$.

Step-by-step solution

Step 1. Given Information.

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.

Step 2. Concepts used.

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.

Step 3. Find the measure of angle 3.

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

The measure of $\angle 3$ will be,

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

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

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,

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

Step 5. Find the measure of angle 5 and 6.

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,

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

Step 6. Find the measure of angle 7 and 8.

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,

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

Step 8. Conclusion.

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 $m\angle 8=65°$.