Question

Explain how the given figure is a trapezium. Which of its two sides are parallel?

Answer 1

Hint: Here, we need to find which of the two sides of the given figure are parallel. A trapezium is a quadrilateral in which a pair of sides parallel. We will use the given angle measures to prove that the co-interior angles between two sides and on the same side of a transversal are supplementary.

Complete step-by-step answer:
We know that a quadrilateral is a closed figure with 4 sides.
We can observe that the given figure has four sides.
Therefore, \[KLMN\] is a quadrilateral.
Now, we need to prove that the quadrilateral \[KLMN\] is a trapezium.
As trapezium has a pair of sides that are parallel. Therefore, we will prove that one of the pairs of sides \[KL\] and \[MN\], or \[LM\] and \[KN\] are parallel.
We know that two sides are parallel if the co-interior angles between them are supplementary.
It is given that the angle \[LMN\] measures \[100^\circ \], and the angle \[MLK\] measures \[80^\circ \].
Adding the two angles, we get
\[
   \Rightarrow \angle LMN + \angle MLK = 100^\circ + 80^\circ \\
   \Rightarrow \angle LMN + \angle MLK = 180^\circ \\
\]
Now, we can observe that the co-interior angles between the sides \[KL\] and \[MN\] are \[\angle LMN\] and \[\angle MLK\], where \[LM\] is the transversal.
Since \[\angle LMN + \angle MLK = 180^\circ \], the angles \[\angle LMN\] and \[\angle MLK\] are supplementary angles.
The co-interior angles between the sides \[KL\] and \[MN\] are supplementary.
Therefore, the sides \[KL\] and \[MN\] are parallel.
Now, the pair of sides \[KL\] and \[MN\] of the quadrilateral \[KLMN\] are parallel.
Therefore, the quadrilateral \[KLMN\] is a trapezium where the sides \[KL\] and \[MN\] are parallel.

Note: We have used the concept of supplementary and co-interior angles to prove that the quadrilateral \[KLMN\] is a trapezium. Two angles are said to be supplementary only if their sum is equal to \[180^\circ \]. The co-interior angles are the angles which lie between two lines and are on the same side of the transversal.