Question

In Example 3, suppose that $m\angle 1=135°$. Find $m\angle 2$, $m\angle 3$, and $m\angle 4$.

Short answer

The measures of the other three angles $m\angle 2,m\angle 3,$ and $m\angle 4$ are $45{}^{\circ },135{}^{\circ },$ and $45°$ respectively.

Step-by-step solution

Step 1. Given Information.

$m\angle 1=135°$. The figure is shown below.

Step 2. Concept used.

• When two lines intersect, the angles opposite each other are called ‘vertical angles’ and they have the same measure.
• If two adjacent angles form a straight angle, then the two angles are supplementary angles and the sum of their measures is$180°$.

Step 3. Calculation.

In the given figure, $\angle 1$and $\angle 3$ are vertical angles. So, they should have the same measure. Therefore,

$m\angle 3=m\angle 1=135°$

Also, in the given figure, $\angle 1$ and $\angle 2$ are supplementary angles as they are adjacent and form a straight angle. So, the sum of their measures should be $180°$.

Therefore,

$\begin{array}{c}m\angle 1+m\angle 2=180°\\ m\angle 2=180°-m\angle 1\\ =180°-135°\\ =45°\end{array}$

Similarly, $\angle 1$ and $\angle 4$ are supplementary angles.

Therefore,

$\begin{array}{c}m\angle 1+m\angle 4=180°\\ m\angle 4=180°-m\angle 1\\ =180°-135°\\ =45°\end{array}$