91Ó°ÊÓ

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

• 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 $180°$.

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

$m∠2=m∠1$

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

$m∠2=35°$

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

$m∠3+m∠4=180°$

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

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

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

$m∠2+m∠4+m∠5=180°$

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

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