Question

Which pair of lines in the figure are parallel? Give reasons.

Answer 1

Hint:To solve the question given, we will make use of the fact that if the sum of the interior angles between any two given lines is \[{{180}^{o}},\] then the lines will be parallel to each other. So, we will find the sum of the angles between the lines and then check if the sum is \[{{180}^{o}}.\]

Complete step-by-step answer:
In this question, we are given a quadrilateral with four angles between their sides. A quadrilateral is a polygon with four sides/edges and four vertices/corners. Now, in this question, we are going to check that with which sides the line AB and BC will be parallel. To check this, we will make use of the fact if the sum of the interior angles between any two given lines is \[{{180}^{o}}.\] If this sum will be equal to \[{{180}^{o}}\] then the lines will be parallel and if it is not equal to \[{{180}^{o}}\] then the given pair of lines will be non-parallel. So, first, let us consider the lines AB and CD. The line BC can act as a joining line. Now, we will calculate the sum of the interior angles. In this case, the interior angles will be \[\angle ABC\text{ and }\angle DCB.\] The sum of the interior angles is given by:
\[\begin{align}
  & \text{Sum }=\angle ABC+\angle DCB \\
 & \text{Sum }={{65}^{o}}+{{115}^{o}} \\
 & \text{Sum }={{180}^{o}} \\
\end{align}\]
As the sum between them is \[{{180}^{o}},\] the lines AB and CD will be parallel to each other.
Now, we will check the same for line AD and BC. In this case, line AB can act as a joining line. The interior angles, in this case, are \[\angle DAB\text{ and }\angle ABC.\] The sum of the interior angles will be given by:
\[\begin{align}
  & \text{Sum }=\angle DAB+\angle ABC \\
 & \text{Sum }={{115}^{o}}+{{65}^{o}} \\
 & \text{Sum }={{180}^{o}} \\
\end{align}\]
As the sum of the interior angles is \[{{180}^{o}},\] the lines AD and BC are parallel.
Thus, AB||CD and AD||BC.

Note: The above question can also be solved by using the concept that in a parallelogram, opposite angles are equal. Also, in a parallelogram, the opposite pair of sides are parallel. In our question, we can see that, the opposite angles are equal, thus the quadrilateral given will be a parallelogram and in a parallelogram, the opposite pair of the sides are parallel, so AB||CD and AD||BC