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,
$\Rightarrow \mathrm{\angle }AOB+\mathrm{\angle }BOC={180}^{\circ }$
Now let us substitute the value of angle AOB in it. Then we get,
$\begin{array}{rl}& \Rightarrow \mathrm{\angle }AOB+\mathrm{\angle }BOC={180}^{\circ }\\ & \Rightarrow {50}^{\circ }+\mathrm{\angle }BOC={180}^{\circ }\\ & \Rightarrow \mathrm{\angle }BOC={130}^{\circ }............\left(1\right)\end{array}$
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
$\Rightarrow \mathrm{\angle }AOE+\mathrm{\angle }EOC={180}^{\circ }$
Now let us substitute the value of angle AOE in it. Then we get,
$\begin{array}{rl}& \Rightarrow {90}^{\circ }+\mathrm{\angle }EOC={180}^{\circ }\\ & \Rightarrow \mathrm{\angle }EOC={90}^{\circ }\end{array}$
We can write angle EOC as,
$\Rightarrow \mathrm{\angle }EOC=\mathrm{\angle }EOD+\mathrm{\angle }DOC$
Substituting the value of angle DOC, we get,
$\begin{array}{rl}& \Rightarrow \mathrm{\angle }EOD+\mathrm{\angle }DOC={90}^{\circ }\\ & \Rightarrow \mathrm{\angle }EOD+{25}^{\circ }={90}^{\circ }\\ & \Rightarrow \mathrm{\angle }EOD+{25}^{\circ }={90}^{\circ }\\ & \Rightarrow \mathrm{\angle }EOD={65}^{\circ }\end{array}$
Hence answer is ${65}^{\circ }$.

(iii) obtuse angle BOD.
 From the figure above, we can say that,
$\Rightarrow \mathrm{\angle }BOD=\mathrm{\angle }BOC+\mathrm{\angle }COD$
Now let us substitute the value of angle COD which is given and angle BOC from above equation (1).
Then we get,
$\begin{array}{rl}& \Rightarrow \mathrm{\angle }BOD={130}^{\circ }+{25}^{\circ }\\ & \Rightarrow \mathrm{\angle }BOD={155}^{\circ }\end{array}$
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,
$\begin{array}{rl}& \Rightarrow {360}^{\circ }-{155}^{\circ }\\ & \Rightarrow {205}^{\circ }\end{array}$
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,
$\begin{array}{rl}& \Rightarrow {360}^{\circ }-{90}^{\circ }\\ & \Rightarrow {270}^{\circ }\end{array}$
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 ${360}^{\circ }$.