Question

OA and OB are opposite rays as shown in the figure. If \[x={{75}^{\circ }}\] then the value of \[y\] is:

(a) \[{{105}^{\circ }}\]
(b) \[{{75}^{\circ }}\]
(c) \[{{45}^{\circ }}\]
(d) \[{{15}^{\circ }}\]

Answer 1

Hint: We solve this problem by using the standard condition of angles on the straight lines.
We have the condition that angle in a straight line is equal to \[{{180}^{\circ }}\]
That is from the figure we have
\[\angle AOB={{180}^{\circ }}\]

Complete step by step answer:
We are given that OA and OB are opposite rays
We are also given that the angles made by ray OC as
\[\Rightarrow \angle COB=x={{75}^{\circ }}\]
\[\Rightarrow \angle COA=y\]
We know that the condition that angle in a straight line is equal to \[{{180}^{\circ }}\]
By using the above condition to given figure we get
\[\Rightarrow \angle AOB={{180}^{\circ }}\]
Here, we can see that the angle \[\angle AOB\] is divided into two angles by ray OC such that the sum of those two angles gives \[\angle AOB\]
By using this condition to above equation we get
\[\Rightarrow \angle COA+\angle COB={{180}^{\circ }}\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow y+{{75}^{\circ }}={{180}^{\circ }} \\
 & \Rightarrow y={{105}^{\circ }} \\
\end{align}\]
Therefore, we can conclude that the value of \[y\] is \[{{105}^{\circ }}\]
So, option (a) is the correct answer.

Note:
We can solve this problem by using other conditions.
We are given that OA and OB are opposite rays
We are also given that the angles made by ray OC as
\[\Rightarrow \angle COB=x={{75}^{\circ }}\]
\[\Rightarrow \angle COA=y\]
We know that the condition that the sum of all angles cut in a straight line is equal to \[{{180}^{\circ }}\]
By using the above condition to given figure we get
\[\Rightarrow \angle COA+\angle COB={{180}^{\circ }}\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow y+{{75}^{\circ }}={{180}^{\circ }} \\
 & \Rightarrow y={{105}^{\circ }} \\
\end{align}\]
Therefore, we can conclude that the value of \[y\] is \[{{105}^{\circ }}\]
So, option (a) is the correct answer.