Question

If each pair of opposite sides of a quadrilateral is equal, then the quadrilateral is a parallelogram.
Can you reason out why?

Answer 1

Hint: We know that, if two opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is known as parallelogram.
For this proof, we will try to find the congruence of triangles.
By using the congruent property, we can make the alternate angles are equal and the opposite sides are parallel.

Complete step-by-step solution:
It is given that; each pair of opposite sides of a quadrilateral is equal.
We have to show that the quadrilateral is a parallelogram.
Let us draw the diagram.

Here, \[ABCD\] is a quadrilateral in which \[AB = CD\] and \[AD = BC\].
We have to prove that \[ABCD\] is a parallelogram. Moreover, we have to prove \[AB\parallel CD\] and \[AD\parallel BC\].
Let us join \[A\] and \[C\].
In \[\vartriangle ABC\] and \[\vartriangle CDA\]
It is given that,\[AB = CD\]
It is given that, \[AD = BC\]
\[AC\] is the common side.
So, by SSS condition both the triangles are congruent.
That is, \[\vartriangle ABC \cong \vartriangle CDA\]
Then, by common property of common triangles we get,
\[\angle ACD = \angle BAC\]
\[\angle DAC = \angle ACB\]
Since, the alternate angles are equal, thus the lines are parallel.
Therefore, \[AB\parallel CD\] and \[AD\parallel BC\].
Since both the pairs of opposite sides of a quadrilateral are parallel, \[ABCD\] is a parallelogram.
Hence, proved.

Note: We have mind that, there are three condition for congruence,
SSS - The triangles are said to be congruent if all the three sides of one triangle are equal to the three corresponding sides of another triangle.
SAS - The triangles are said to be congruent if the correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle.
AAS - The triangles are said to be congruent if two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle.