Question

In the figure, lines AB and CD intersect at O. If \[\angle {\text{AOC}} + \angle {\text{BOE}} = 70\] and \[\angle {\text{BOD}} = 40\], find \[\angle {\text{BOE}}\] and reflex \[\angle {\text{COE}}\].

Answer 1

Hint: First we will use AB and CD intersects, the vertically opposite angles are equal and then substitute the value of \[\angle {\text{AOC}}\] in the equation \[\angle {\text{AOC}} + \angle {\text{BOE}} = 70\]. Then we know that the reflex of \[\angle {\text{COE}}\], so \[{\text{Reflex}}\angle {\text{COE = 360}}^\circ - \angle {\text{COE}}\], where we will find the value of \[\angle {\text{COE}}\] by taking \[\angle {\text{AOC}}\], \[\angle {\text{COE}}\] and \[\angle {\text{BOE}}\] are a linear pair, their sum is 180 degrees.

Complete step-by-step answer:
We are given that \[\angle {\text{BOD}} = 40^\circ \] and \[\angle {\text{AOC}} + \angle {\text{BOE}} = 70\].
Since we know that AB and CD intersect, the vertically opposite angles are equal, we have
\[
   \Rightarrow \angle {\text{AOC}} = \angle {\text{BOD}} \\
   \Rightarrow \angle {\text{AOC}} = 40^\circ \\
 \]
Substituting the value of \[\angle {\text{AOC}}\] in the equation \[\angle {\text{AOC}} + \angle {\text{BOE}} = 70\], we get
\[ \Rightarrow 40^\circ + \angle {\text{BOC}} = 70^\circ \]
Subtracting the above equation by \[40^\circ \] on both sides, we get
\[
   \Rightarrow 40^\circ + \angle {\text{BOE}} - 40^\circ = 70^\circ - 40^\circ \\
   \Rightarrow \angle {\text{BOE}} = 30^\circ \\
 \]

Since we know that the reflex of \[\angle {\text{COE}}\], so \[{\text{Reflex}}\angle {\text{COE = 360}}^\circ - \angle {\text{COE}}\].
First, we will find the angle \[\angle {\text{COE}}\].
Since we know that the \[\angle {\text{AOC}}\], \[\angle {\text{COE}}\] and \[\angle {\text{BOE}}\] are a linear pair, their sum is 180 degrees.
Using the value of \[\angle {\text{AOC}}\] and \[\angle {\text{BOE}}\] in the sum, we get
\[
   \Rightarrow \angle {\text{COE}} + 40^\circ + 30^\circ = 180^\circ \\
   \Rightarrow \angle {\text{COE}} + 70^\circ = 180^\circ \\
 \]
Subtracting the above equation by \[70^\circ \] on both sides, we get
\[
   \Rightarrow \angle {\text{COE}} + 70^\circ - 70^\circ = 180^\circ - 70^\circ \\
   \Rightarrow \angle {\text{COE}} = 110^\circ \\
 \]
Finding the reflex of \[\angle {\text{COE}}\], we get
 \[
   \Rightarrow \angle {\text{COE}} = 360^\circ - 110^\circ \\
   \Rightarrow \angle {\text{COE}} = 250^\circ \\
 \]

Note: In solving these types of questions, students need to know the basic properties of angles on a straight line. The possibility of error in this question can be that you assume the sum is equal to 90 degrees, which is wrong. Also, avoid calculation mistakes. Students take the reflex of an angle to subtract the angle from 180 degrees, which is also wrong so you have to be careful.