Question

Can a quadrilateral ABCD be a parallelogram if
(i) $\angle D + \angle B = {180^ \circ }$?
(ii) AB=DC=8cm, AD=4cm and BC=4.4cm?
(iii) $\angle A = {70^ \circ }$ and $\angle C = {65^ \circ }$?

Answer 1

Hint: First, draw a quadrilateral to see all angles and sides. In case one, try to prove if opposite angles are equal or the sum of adjacent angles must be \[{180^ \circ }\]. From the result, we can conclude that quadrilateral may or may not be a parallelogram. Then, use the properties of parallelograms such that, opposite sides are equal and opposite angles are equal to check the conditions given in the (ii) and (iii) part.

Complete step by step Answer:

Let us first draw a diagram of a quadrilateral ABCD.

As it is known that, opposite angles of a parallelogram are equal.
We also know that sum of all angles of a quadrilateral is equal to ${360^ \circ }$.
This implies that
 \[\angle A + \angle B + \angle C + \angle D = {360^ \circ }\]
But, we are given that $\angle D + \angle B = {180^ \circ }$
Therefore,
$
   \Rightarrow \angle A + \angle C + {180^ \circ } = {360^ \circ } \\
   \Rightarrow \angle A + \angle C = {180^ \circ } \\
$
\[ \Rightarrow \angle A + \angle B = {180^ \circ }\]
Here, we cannot draw any conclusion because in a parallelogram the sum of adjacent angles is \[{180^ \circ }\] and opposite angles are equal.
Hence, the quadrilateral with $\angle D + \angle B = {180^ \circ }$may or may not be a parallelogram
Next, we have to find out if there exists a parallelogram with AB=DC=8cm, AD=4cm, and BC=4.4cm
In a parallelogram, opposite sides are of equal length.
That is, AB=AC and AD should be equal to BC, which is false in the given case.
Therefore, such a parallelogram is not possible.
Next, find if $\angle A = {70^ \circ }$ and $\angle C = {65^ \circ }$ are possible measures for a quadrilateral to be a parallelogram.
In a parallelogram, opposite angles are equal.
Therefore, $\angle A$ should be equal to $\angle C$
But, $\angle A = {70^ \circ }$ and $\angle C = {65^ \circ }$, therefore such a parallelogram is not possible.

Note: In a parallelogram, opposite sides are equal and parallel, opposite angles are equal, the sum of adjacent angles must be \[{180^ \circ }\] and diagonals bisect each other. Students should know the properties of a parallelogram in order to do this question directly.