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# AB,CD and EF are three concurrent lines passing through the point O such that OF bisects  $\angle {\text{BOD}}$. ${\text{If }}\angle {\text{BOF = 3}}{{\text{5}}^0}{\text{, find }}\angle {\text{BOC and }}\angle {\text{AOD}}{\text{.}}$ ${\text{A}}{\text{.}}\angle {\text{BOC = 11}}{{\text{0}}^0} \\ {\text{B}}{\text{.}}\angle {\text{AOD = 11}}{{\text{0}}^0} \\ {\text{C}}{\text{.}}\angle {\text{AOD = 7}}{{\text{0}}^0} \\ {\text{D}}{\text{.None of these}}{\text{.}} \\$

Last updated date: 13th Jul 2024
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Hint: In this type of question which involves geometry, we have to first draw the diagram. For drawing the diagram information will be given in the question. Here it is given that three lines are concurrent which means they have the same starting point i.e. point O. After drawing the diagram for the question, use the property of straight angle and angle bisector to further solve the question.

This is a basic question based on finding the unknown angle.
In the question three lines AB, CD, and EF have same starting point O. i.e. they are concurrent and it is also given that OF line bisects$\angle {\text{BOD}}$ and $\angle {\text{BOF = 3}}{{\text{5}}^0}$
Based on this information the diagram is:

Line OF is the angle bisector of $\angle {\text{BOD}}$.
It is given that $\angle {\text{BOF = 3}}{{\text{5}}^0}.$
Since, OF is the angle bisector. So it will divide $\angle {\text{BOD}}$ into two equal parts.
$\therefore \angle {\text{BOF = 3}}{{\text{5}}^0} = \angle {\text{FOD}}.$
Now we know that angle made on a line or straight angle is equal to ${180^0}.$
Therefore, we can write:
$\Rightarrow \angle {\text{BOF + }}\angle {\text{FOD + }}\angle {\text{AOD = 18}}{{\text{0}}^0}$ …..(1)
Putting the values of $\angle {\text{BOF and }}\angle {\text{FOD in equation 1, we get:}}$
$\Rightarrow {35^0}{\text{ + 3}}{{\text{5}}^{0}}{\text{ + }}\angle {\text{AOD = 18}}{{\text{0}}^0} \\ \Rightarrow \angle {\text{AOD = 18}}{{\text{0}}^0} - {70^0} = {110^0}. \\$
Also, we know that when two lines intersect then the vertically opposite angles are equal.
So, we can write:
$\angle {\text{AOD = }}\angle {\text{BOC = }}{110^0}$.
So option A and B are correct.

Note: Before solving this type of problems which are simply based on angle calculation. You have to first draw the diagram from the information given in the question. You should remember the definition of vertically opposite angles and straight angles and their property. Straight angle is the angle which a straight line makes and is equal to ${180^0}$. Vertically opposite angles are always equal.