Question

Given lines \[{{l}_{1}},{{l}_{2}},{{l}_{3}}\] in figure 2 are parallel. The value of x is:

\[\left( \text{a} \right)\text{ }{{40}^{\circ }}\]
\[\left( \text{b} \right)\text{ 1}{{40}^{\circ }}\]
\[\left( \text{c} \right)\text{ 5}{{0}^{\circ }}\]
\[\left( \text{d} \right)\text{ 6}{{0}^{\circ }}\]

Answer 1

Hint: First, we will mark all the other angles in the figure as a, b, c, d from top to bottom in a criss-cross manner such that a is vertically opposite to the given angle. To solve the given question, we will first find out the value of ‘a’ by the use of the fact that ‘a’ and \[{{40}^{\circ }}\] are vertically opposite angles. Then, we will find the value of ‘b’ by using the fact that a and b are alternate interior angles. Then, we will find the value ‘c’ by using the fact that ‘b’ and ‘c’ are vertically opposite angles. Then, we will find the value of ‘d’ by using the fact that ‘c’ and ‘d’ are alternate interior angles. Then, we will find the value of x by using the fact that the sum of x and d will be \[{{180}^{\circ }}.\] The angles a, b, c and d are shown in the figure drawn.

Complete step-by-step answer:
To start with, we will redraw the figure given in the question with the necessary angles shown.

Now, first, we will derive the relation between the angle ‘a’ and \[{{40}^{\circ }}.\] The angle ‘a’ and \[{{40}^{\circ }}\] are vertically opposite angles as they are formed by the crossing of lines AB and \[{{l}_{1}}.\] Thus, we have,
\[a={{40}^{\circ }}......\left( i \right)\]
Now, we know that ‘a’ and ‘b’ are alternate interior angles because \[{{l}_{1}}\] and \[{{l}_{2}}\] are parallel lines. So, ‘a’ and ‘b’ are equal. Thus, we have,
\[a=b\]
\[\Rightarrow b={{40}^{\circ }}......\left( ii \right)\left[ \text{From (i)} \right]\]
Now, ‘b’ and ‘c’ are vertically opposite angles as they are formed by the crossing of lines AB and \[{{l}_{2}}.\] Thus, we have,
\[b=c\]
\[\Rightarrow c={{40}^{\circ }}\]
Now, ‘c’ and ‘d’ are alternate interior angles because \[{{l}_{2}}\] and \[{{l}_{3}}\] are parallel lines. So, ‘c’ and ‘d’ are equal. Thus, we have,
\[d=c\]
\[\Rightarrow d={{40}^{\circ }}\]
Now, the sum of ‘d’ and ‘x’ will be \[{{180}^{\circ }}\] because AB is a straight line. Thus, we have,
\[d+x={{180}^{\circ }}\]
\[\Rightarrow {{40}^{\circ }}+x={{180}^{\circ }}\]
\[\Rightarrow x={{180}^{\circ }}-{{40}^{\circ }}\]
\[\Rightarrow x={{140}^{\circ }}\]
Hence, option (b) is the right answer.

Note: The alternate way of solving the question is shown below. The lines \[{{l}_{1}}\] and \[{{l}_{3}}\] are parallel lines and the line AB acts as a transversal. Thus, \[{{40}^{\circ }}\] and d are pair of corresponding angles and thus they are equal i.e, \[d={{40}^{\circ }}.\] Now, the sum of d and x is \[{{180}^{\circ }}.\] Thus,
\[d+x={{180}^{\circ }}\]
\[\Rightarrow {{40}^{\circ }}+x={{180}^{\circ }}\]
\[\Rightarrow x={{180}^{\circ }}-{{40}^{\circ }}\]
\[\Rightarrow x={{140}^{\circ }}\]