In the figure given below, AB, CD, and EF are three concurrent lines passing through the point O such that OF bisects $\angle \mathrm{BOD}.$If $\angle \mathrm{BOF}={35}^{°},$ then what are the measures of $\angle \mathrm{BOC}$ and $\angle \mathrm{AOD}$?
 

 

It is given that AB, CD, and EF are three concurrent lines passing through the point O such that OF bisects $\angle \mathrm{BOD}$ such that $\mathbf{\angle }\mathbf{BOF}\mathbf{=}{\mathbf{35}}^{\mathbf{°}}\mathbf{.}$

As OF bisects $\angle \mathrm{BOD},$ then:
$\begin{array}{l}\angle \mathrm{DOF}=\angle \mathrm{BOF}\\ \Rightarrow \mathbf{\angle }\mathbf{DOF}\mathbf{=}{\mathbf{35}}^{\mathbf{°}}\end{array}$

Also, $\mathbf{AB}$ is a straight line and the sum of angles formed by a straight line is ${\mathbf{180}}^{\mathbf{°}}\mathbf{.}$

$\therefore \angle \mathrm{AOD}+\angle \mathrm{DOF}+\angle \mathrm{BOF}={180}^{°}$

$\Rightarrow \angle \mathrm{AOD}+{35}^{°}+{35}^{°}={180}^{°}$

$\Rightarrow \angle \mathrm{AOD}={180}^{°}-{70}^{°}$

$\Rightarrow \mathbf{\angle }\mathbf{AOD}\mathbf{=}{\mathbf{110}}^{\mathbf{°}}$

Since $\angle \mathrm{AOD}$ and $\angle \mathrm{BOC}$ are vertically opposite angles. 

$\begin{array}{l}\therefore \angle \mathrm{AOD}=\angle \mathrm{BOC}\\ \Rightarrow \mathbf{\angle }\mathbf{BOC}\mathbf{=}{\mathbf{110}}^{\mathbf{°}}\end{array}$

Therefore,  $\mathbf{\angle }\mathbf{AOD}\mathbf{=}{\mathbf{110}}^{\mathbf{°}}\mathbf{,}\mathbf{ }\mathbf{\angle }\mathbf{BOC}\mathbf{=}{\mathbf{110}}^{\mathbf{°}}$