Question

In figure, OA and OB are opposite rays: if \[y={{35}^{\circ }}\] what is the value of x?

Answer 1

Hint: We know that linear pairs of angles are formed when two lines intersect each other at a point. The sum of adjacent angles formed is always equal to $ {{180}^{\circ }} $ the adjacent angles are the angles having a common vertex. We have already been given \[y={{35}^{\circ }}\] so, by using this property i.e. linear pair of angles we will add the given pair of angles and equate them to \[{{180}^{\circ }}\] to find the required value.

Complete step-by-step answer:
We have been given a figure as shown below in which \[y={{35}^{\circ }}\] and we have to find the value of x.

Since, \[\angle AOC\text{ and }\angle \text{BOC}\] are the adjacent angles formed when the line AB intersects at O by the ray OC. So, it forms a linear pair.
We know that the sum of adjacent angle formed is equal to \[{{180}^{\circ }}\]
\[\begin{align}
  & \Rightarrow \angle AOC+\angle BOC={{180}^{\circ }} \\
 & \Rightarrow 2y+5+3x={{180}^{\circ }} \\
\end{align}\]
Since, it is given that \[y={{35}^{\circ }}\]
On substituting the value of y, we get
\[\begin{align}
  & \Rightarrow 2\times {{35}^{\circ }}+{{5}^{\circ }}+3{{x}^{\circ }}={{180}^{\circ }} \\
 & \Rightarrow {{70}^{\circ }}+{{5}^{\circ }}+3{{x}^{\circ }}={{180}^{\circ }} \\
 & \Rightarrow {{75}^{\circ }}+3{{x}^{\circ }}={{180}^{\circ }} \\
\end{align}\]
On taking \[{{75}^{\circ }}\] to left side of equation, we get
\[\begin{align}
  & \Rightarrow 3{{x}^{\circ }}={{180}^{\circ }}-{{75}^{\circ }} \\
 & \Rightarrow 3{{x}^{\circ }}={{105}^{\circ }} \\
\end{align}\]
On dividing the equation by 3, we get
\[\Rightarrow x={{\dfrac{105}{3}}^{\circ }}={{35}^{\circ }}\]
Therefore, the value of x is equal to \[{{35}^{\circ }}\]

Note: Remember the axiom that if a ray stands on a line then the adjacent angle form a linear pair of angles and the sum of adjacent angle is equal to \[{{180}^{\circ }}.\] Also, remember that a pair of angles whose sum is equal to \[{{180}^{\circ }}\] is known as supplementary angles. Some students compute the angle \[\left( 2y+5 \right)\] and then subtract it from \[{{180}^{\circ }}\] directly without going through all the steps, since it is a simple question. But then they will get a value of 3x and not x. This will lead to wrong answers.