Question

In the given figure, find the value of y.

Answer 1

Hint: We can use the property of vertically opposite angles are equal to find the unknown angles. Then we can add all the angles around the point and equate it to $360^\circ $. Then we can solve for y to get the required solution.

Complete step by step answer:

We are given that $\angle AOB = 6y$. Consider the line AD and EB. Then, $\angle AOB$ and $\angle EOD$are vertically opposite angles. We know that vertically opposite angles are equal.
$ \Rightarrow \angle AOB = \angle EOD = 6y$… (1)
Consider the lines FC and EB. Then, $\angle FOE$and $\angle BOC$ are vertically opposite angles and are equal.
$ \Rightarrow \angle FOE = \angle BOC$
We know that $\angle FOE = 4y$,
$ \Rightarrow \angle BOC = \angle FOE = 4y$… (2)
Now consider the lines AD and CF. Then, $\angle FOA$and $\angle DOC$ are vertically opposite angles and are equal.
$ \Rightarrow \angle FOA = \angle DOC$
We know that $\angle DOC = 8y$,
$ \Rightarrow \angle FOA = \angle DOC = 8y$… (3)
Now we have the measures of all the angles around the point O. The sum of these angles will be $360^\circ $ as it completes a full turn around point O.
$ \Rightarrow \angle AOB + \angle BOC + \angle COD + \angle DOE + \angle EOF + \angle FOA = 360^\circ $
Using equations (1), (2), and (3), we get,
$6y + 4y + 8y + 6y + 4y + 8y = 360^\circ $
$ \Rightarrow 36y = 360^\circ $
Dividing throughout with 36, we get,
 \[
  y = \dfrac{{360^\circ }}{{36}} \\
   \Rightarrow y = 10^\circ \\
  \]
Therefore, the required value of y is \[10^\circ \].

Note: The properties that vertically opposite angles are equal and angle around a point is$360^\circ $ are used to solve this problem.
Another method to solve this problem is,
We are given that $\angle AOB = 6y$. Consider the line AD and EB. Then, $\angle AOB$and $\angle EOD$are vertically opposite angles. We know that vertically opposite angles are equal.
$ \Rightarrow \angle AOB = \angle EOD = 6y$ … (1)
Now from the figure, we can write,
$\angle COD + \angle DOE + \angle EOF = \angle COF$
We are given $\angle COD = 8y$,$\angle DOE = 6y$and $\angle EOF = 4y$.
We also know that $\angle COF = 180^\circ $as COF is a straight line.
$ \Rightarrow 8y + 6y + 4y = 180^\circ $
$ \Rightarrow 18y = 180^\circ $
Dividing throughout with 18, we get,
 \[
  y = \dfrac{{180^\circ }}{{18}} \\
   \Rightarrow y = 10^\circ \\
  \]
So, the value of y is\[10^\circ \]