Question

In the given figure, ray OC stands on the line AB; ray OL and ray OM are angle bisectors of $\mathrm{\angle }AOC$ and $\mathrm{\angle }BOC$ respectively. Find the value of $\mathrm{\angle }LOM$.

A. ${90}^{\circ }$
B. ${100}^{\circ }$
C. ${180}^{\circ }$
D. ${120}^{\circ }$

Answer 1

Hint: Here, AOB is a straight line. So, $\mathrm{\angle }AOB={180}^{\circ }$. From figure, $\mathrm{\angle }AOL=\mathrm{\angle }COL$ and $\mathrm{\angle }BOM=\mathrm{\angle }COM$. Add all four angles in the figure and by substitution we have only two unknown angles. Equate the sum with ${180}^{\circ }$ to get the result.

Complete step by step solution:
In the given figure, $AB$ is a straight line and $OC$ is a ray stand on $AB$ at $O$.
Thus, $\mathrm{\angle }AOC$ and $\mathrm{\angle }BOC$ form a linear pair. Therefore the sum of these two angles is ${180}^{\circ }$.
[Sum of all angles which together forms a straight line is ${180}^{\circ }$ ]
i.e. $\mathrm{\angle }AOC$+$\mathrm{\angle }BOC$=${180}^{\circ }$
Since ray $OL$ and ray $OM$ are the angle bisector of $\mathrm{\angle }AOC$ and $\mathrm{\angle }BOC$ respectively.
So, OL divides $\mathrm{\angle }AOC$ into two equal parts and $OM$ divides $\mathrm{\angle }BOC$ into two equal parts.
$\Rightarrow \mathrm{\angle }AOL=\mathrm{\angle }COL={\displaystyle \frac{1}{2}}\mathrm{\angle }AOC$ and $\mathrm{\angle }BOM=\mathrm{\angle }COM={\displaystyle \frac{1}{2}}\mathrm{\angle }BOC$
Now we have,
$\mathrm{\angle }AOC$+$\mathrm{\angle }BOC={180}^{\circ }$


$\Rightarrow 2\mathrm{\angle }COL+2\mathrm{\angle }COM={180}^{\circ }$
As we have $\mathrm{\angle }COL+\mathrm{\angle }COM=\mathrm{\angle }LOM$
$\Rightarrow 2\mathrm{\angle }LOM={180}^{\circ }$
$\Rightarrow \mathrm{\angle }LOM={90}^{\circ }$

$\therefore$ The value of $\mathrm{\angle }LOM$$={90}^{\circ }$. Hence, option (A) is correct.

Note:
In these types of questions, we can geometrically analyse the answer of the questions asked. With the help of some properties and definitions we can prove it.
Some important points:
Angle bisector is the line that bisects the angle. There are three angle bisectors in a triangle. The angle bisectors of angles of a triangle meet at a point. Point at which angle bisectors of angles of a triangle meet is called in-centre of triangles. It always lies inside a triangle. In-centre is the point that is equidistant from all sides of the triangle. This distance is called the in-radius of the triangle. Thus, we can draw a circle taking in-centre and in-radius, this circle is called in-circle. If the angle bisectors of two angles of a triangle meet at a point thus angle formed by them is half of the third angle more than right angle.
Two angles form a linear pair if their sum is ${180}^{\circ }$.
The angle bisectors of angles of linear pair stand at the right angle.