Question

Find the measures of $\angle 1$and $\angle 2$.

Short answer

The measure of $\angle 1\text{and}\angle 2$are $\text{35°and125°}$respectively.

Step-by-step solution

Step 1. Observe the figure and highlight angle 1 and angle 2.

Step 2. Description of step.

Consider $\overline{MN}$is parallel to $\overline{TR}$and $\overline{MT}$is a transversal. So, the interior angles on the same side of the transversal is supplementary.

$\begin{array}{c}m\angle TMN+m\angle MTR=180\\ 85+m\angle MTR=180\end{array}$

$\begin{array}{c}m\angle MTR=180-85\\ =95\end{array}$

Step 3. Apply the definition of exterior angles.

From the figure, it can be observed that $\angle 1\text{and}\angle S$are remote interior angles. Therefore,

$\begin{array}{c}m\angle 1+m\angle S=30+m\angle MTR\\ m\angle 1+90=30+95\\ m\angle 1=35\end{array}$

Step 4. Description of step.

Since $\overline{TS}$parallel to $\overline{NR}$and $\overline{RS}$, the value is $\angle R=90$.

If two lines are parallel and the line is perpendicular to one line then it is perpendicular to other line.

$$\begin{gathered} m\angle NRT+\angle 1=90 \\ m\angle NRT=90-35 \\ =55 \end{gathered}$$

Step 5. Description of step.

Since the sum of the same side interior angles is supplementary.

$$\begin{gathered} m\angle 2+55=180 \\ m\angle 2=180-55 \\ =125 \end{gathered}$$

Therefore, the measure of $\angle 1\text{and}\angle 2$are $35°\text{and125}°$respectively.