Question

In the given figure, AOC is a straight line. If angle AOB = ${{50}^{\circ }}$, angle AOE = ${{90}^{\circ }}$ and angle COD = ${{25}^{\circ }}$. Find the measure of:

(i) angle BOC
(ii) angle EOD
(iii) obtuse angle BOD
(iv) reflex angle BOD
(v) reflex angle COE

Answer 1

Hint:We solve this question by first drawing the figure and representing the given angles. Then we use the fact that the angle made by a straight line on one side is ${{180}^{\circ }}$ and write it as a sum of angles as required and find the values of angles required. Then we consider the definition of reflex angle and we subtract the smaller angle we found from ${{360}^{\circ }}$ and find the required reflex angle.

Complete step by step answer:
We are given that in the given figure, angle AOB=${{50}^{\circ }}$, angle AOE=${{90}^{\circ }}$ and angle COD=${{25}^{\circ }}$.
Then let us now show those values in the figure. So, it looks like,

(i) angle BOC.
We are given that AC is a straight line.
Let us consider the fact that the angle made by a straight line on one side is ${{180}^{\circ }}$.
Using this we can say that angle made by line AC on the side with point B is ${{180}^{\circ }}$.
So, we get,
$\Rightarrow \angle AOB+\angle BOC={{180}^{\circ }}$
Now let us substitute the value of angle AOB in it. Then we get,
$\begin{align}
  & \Rightarrow \angle AOB+\angle BOC={{180}^{\circ }} \\
 & \Rightarrow {{50}^{\circ }}+\angle BOC={{180}^{\circ }} \\
 & \Rightarrow \angle BOC={{130}^{\circ }}............\left( 1 \right) \\
\end{align}$
Hence answer is ${{130}^{\circ }}$.

(ii) angle EOD.
The form above we can say the angle made by AC on the other side is also ${{180}^{\circ }}$.
So, we get that
$\Rightarrow \angle AOE+\angle EOC={{180}^{\circ }}$
Now let us substitute the value of angle AOE in it. Then we get,
$\begin{align}
  & \Rightarrow {{90}^{\circ }}+\angle EOC={{180}^{\circ }} \\
 & \Rightarrow \angle EOC={{90}^{\circ }} \\
\end{align}$
We can write angle EOC as,
$\Rightarrow \angle EOC=\angle EOD+\angle DOC$
Substituting the value of angle DOC, we get,
$\begin{align}
  & \Rightarrow \angle EOD+\angle DOC={{90}^{\circ }} \\
 & \Rightarrow \angle EOD+{{25}^{\circ }}={{90}^{\circ }} \\
 & \Rightarrow \angle EOD+{{25}^{\circ }}={{90}^{\circ }} \\
 & \Rightarrow \angle EOD={{65}^{\circ }} \\
\end{align}$
Hence answer is ${{65}^{\circ }}$.

(iii) obtuse angle BOD.
 From the figure above, we can say that,
$\Rightarrow \angle BOD=\angle BOC+\angle COD$
Now let us substitute the value of angle COD which is given and angle BOC from above equation (1).
Then we get,
$\begin{align}
  & \Rightarrow \angle BOD={{130}^{\circ }}+{{25}^{\circ }} \\
 & \Rightarrow \angle BOD={{155}^{\circ }} \\
\end{align}$
Hence answer is ${{155}^{\circ }}$.

(iv) reflex angle BOD
First, let us consider the concept of reflex angle.
A reflex angle is the larger angle made between the points.
Here from the above subpart (iii) we know that angle BOD is ${{155}^{\circ }}$.
As we need to find the other possible angle, let us subtract it from the total angle around the line, that is ${{360}^{\circ }}$. Then we get,
$\begin{align}
  & \Rightarrow {{360}^{\circ }}-{{155}^{\circ }} \\
 & \Rightarrow {{205}^{\circ }} \\
\end{align}$
Hence answer is ${{205}^{\circ }}$.

(v) reflex angle COE
First, let us consider the concept of reflex angle.
A reflex angle is the larger angle made between the points.
Here from the above subpart (ii) we know that angle COE is ${{90}^{\circ }}$.
As we need to find the other possible angle, let us subtract it from the total angle around the line, that is ${{360}^{\circ }}$. Then we get,
$\begin{align}
  & \Rightarrow {{360}^{\circ }}-{{90}^{\circ }} \\
 & \Rightarrow {{270}^{\circ }} \\
\end{align}$
Hence answer is ${{270}^{\circ }}$.
Note:
The common mistake one makes while solving this problem is one might subtract the smaller angle from ${{180}^{\circ }}$ instead of subtracting it from ${{360}^{\circ }}$. But here reflex angle means the larger angle made by the same lines so we need to subtract it from ${{360}^{\circ }}$.