Question

In the figure, $AD$ and $CE$ are the angle bisectors of $\angle A$ and $\angle C$ respectively. If $\angle ABC = 90^\circ $ then find $\angle AOC$.

Answer 1

Hint:
Here, we will use the angle sum property of triangles to find different angles. Then using the values of obtained angles and the given information we will find the required angle. Angle sum property of triangles states that the sum of interior angles of a triangle is $180^\circ $.

Complete Step by Step Solution:
According to the question, it is given that, $AD$ and $CE$ are the angle bisectors of $\angle A$ and $\angle C$ respectively and $\angle ABC = 90^\circ $ .
Now In $\vartriangle ABC$, by angle sum property of triangles, we get
$\angle A + \angle B + \angle C = 180^\circ $
But it is given that $\angle ABC = 90^\circ $, hence,
$ \Rightarrow \angle A + 90^\circ + \angle C = 180^\circ $
$ \Rightarrow \angle A + \angle C = 90^\circ $……………………………..$\left( 1 \right)$
Now, in $\vartriangle AOC$,
Again by angle sum property of triangles,
$\angle OAC + \angle AOC + \angle OCA = 180^\circ $…………………………….$\left( 2 \right)$
But, it is given that $AD$ and $CE$ are the angle bisectors of $\angle A$ and $\angle C$ respectively.
This means that they divide the angles $\angle A$ and $\angle C$ in two equal angles.
$\therefore \angle OAC = \dfrac{1}{2}\angle A$
Also, $\angle OCA = \dfrac{1}{2}\angle C$
Hence, from equation $\left( 2 \right)$, we get
$\dfrac{1}{2}\angle A + \angle AOC + \dfrac{1}{2}\angle C = 180^\circ $
$ \Rightarrow \dfrac{1}{2}\left( {\angle A + \angle C} \right) + \angle AOC = 180^\circ $
Now, substituting $\angle A + \angle C = 90^\circ $ in the above equation, we get
$ \Rightarrow \dfrac{1}{2}\left( {90^\circ } \right) + \angle AOC = 180^\circ $
$ \Rightarrow 45^\circ + \angle AOC = 180^\circ $
Solving further,
$ \Rightarrow \angle AOC = 180^\circ - 45^\circ = 135^\circ $

Therefore, the required value of $\angle AOC = 135^\circ $

Note:
A triangle is a 3-sides polygon having 3 sides, 3 angles and 3 vertices. Also, the sum of any two sides of a triangle is always greater than the third side. Whenever a bisector bisects the angle of a triangle, the angle bisector divides the triangle into two parts such that they are congruent to each other. The bisector divides its opposite side into two segments which are proportional to each other, hence, helping to prove the congruence of these triangles.