Question

Adjacent angles in a parallelogram are
(A) complementary
(B) Supplementary
(C) ${120^ \circ }$
(D) None of the above.

Answer 1

Hint: The adjacent angles in a parallelogram are two angles on the same arm (side) of the parallelogram.
If the sum of these two angles is ${90^ \circ }$ then these are called complementary angles but if their sum is ${180^ \circ }$ then these angles are called supplementary angles.

Complete step-by-step solution:
Here, we have two find the sum of two angles on the same arm (side) of the parallelogram.

In the given $\angle A$ and $\angle B$ are adjacent angles. Similarly, $\angle B$ is adjacent to $\angle C$ and $\angle C$ is adjacent to $\angle D$ and $\angle D$ is adjacent to $\angle A$.
We know that the pair of opposite sides of a parallelogram are parallel.
So, $AB$ is parallel to $CD$ and $AD$ is parallel to $BC$.
We have studied the property of parallel lines that if a pair of parallel lines is intersected by an intersecting line then the sum of angles on the same side of the intersecting line is ${180^ \circ }$.
Here, line segment $BC$ is parallel to $AD$ and $AB$ is an intersecting line then by applying the above stated property. we can say that $\angle A + \angle B = 18{0^ \circ }$ because $\angle {\rm A}$ and $\angle B$ are on the same side of intersecting lines.
So, we get the sum of adjacent angle $\angle A + \angle B = 18{0^ \circ }$.
Thus, the sum of adjacent angles of a parallelogram is a supplementary angle.

Option (B) is correct.

Note: In parallelogram the opposite angles are equal.
The above stated can be alternatively proved as:
The total sum of all angles of parallelogram is ${360^ \circ }$ and the opposite angles are equal. So, the sum of adjacent angles is half of ${360^ \circ }$.