Question

In the diagram shown, $m\angle 1=35°$and $m\angle 3=110°$. What is $m\angle 5$?

1. Tell how $\angle 1$and $\angle 2$ arerelated. Then find $m\angle 2$.
2. Tell how $\angle 3$and $\angle 4$ arerelated. Then find $m\angle 4$.
3. Use what you know about the sum of the measures of the angles of a triangle to find $m\angle 5$.

Short answer

1. The measure of angle $m\angle 2$ is $35°$.

2. The measure of angle $m\angle 4$ is $70°$.

3. The measure of angle $m\angle 5$is$75°$.

Step-by-step solution

Step 1. Given Information.

The measures $m\angle 1=35°$ and $m\angle 3=110°$ in the following figure.

Step 2. Concept used.

• The measures of a pair of vertical angles are equal.
• The sum of measures of a pair of adjacent angles forming a straight angle (supplementary angles) is $180°$.
• The sum of the measures of the three angles of a triangle is $180°$.

Step 3. Calculation.

In the given figure, $\angle 1$ and $\angle 2$are vertical angles. So,

$m\angle 2=m\angle 1$

Put the given value of $m\angle 1=35°$ in the above equation to write

$m\angle 2=35°$

Now, in the figure, $\angle 3$and $\angle 4$are supplementary angles as they together form a straight angle. So,

$m\angle 3+m\angle 4=180°$

Put the given value of $m\angle 3=110°$ in the above equation to write,

$\begin{array}{c}m\angle 4+110°=180°\\ m\angle 4=70°\end{array}$

Finally, since in the figurecform a triangle, the sum of their measures should be equal to $180°$. So,

$m\angle 2+m\angle 4+m\angle 5=180°$

Put the earlier computed values of $m\angle 2$ and $m\angle 4$ in the above equation to write,

$\begin{array}{c}m\angle 5+70°+35°=180°\\ m\angle 5=180°-\left(35°+70°\right)\\ =180°-105°\\ =75°\end{array}$