Question

Find the measures of the numbered angles in the diagram.

Short answer

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

Step-by-step solution

Step 1. Given Information.

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.

Step 2. Find the measure of angle 1.

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

The measure of $\angle 1$ will be,

$\begin{array}{c}{122}^{\circ }+m\angle 1={180}^{\circ }\\ m\angle 1={180}^{\circ }-122°\\ m\angle 1={58}^{\circ }\end{array}$

Step 3. Find the measure of angle 2 and 3.

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

Therefore, $m\angle 2=122°$ and $m\angle 3=58°$.

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

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

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

Therefore, the measures of $\angle 4$ and $\angle 6$ will be,

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

Step 5. Find the measure of angle 5 and 7.

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

Therefore, the measures of $\angle 5$ and $\angle 7$ will be,

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

Step 6. Conclusion.

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