Question

In the quadrilateral ABCD, AB = DC and AD = BC, then the sides AB and DC are parallel to each other.
If the above statement is true, mention the answer as 1, else mention 0 if false.

Answer 1

Hint: Here we have to draw a quadrilateral and then prove that the triangle ABD and the triangle CBD are congruent to each other. After that using the CPCT rule (Corresponding parts of congruent triangles) we can prove that alternate interior angles are equal. And this will prove sides as parallel to each other.

Complete step-by-step answer:

As we can know that sides AB and D are equal and sides AD and BC are also equal.
Let us take triangle ABD and CBD and check whether they are congruent to each other or not.
So, in the triangle \[\Delta ABD\] and \[\Delta CBD\].
AB = DC (given)
AD = BC (given)
BD = BD (common side)
Now as we know by SSS congruence rule of the two triangles if all three sides of the two triangles are equal then those two triangles will be congruent by SSS congruence Rule.
So, \[\Delta ABD \cong \Delta CBD\] (By SSS congruence rule)
Now according to the CPCT theorem where CPCT stands for Corresponding parts of Congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.
So, in triangle \[\Delta ABD\] and \[\Delta CBD\]
We can say that \[\angle ABD = \angle BDC\](CPCT Rule).
Now as we can see that \[\angle ABD\] and \[\angle BDC\] are the alternate interior angles. And we know that if two alternate interior angles of two sides are equal then the side must be parallel to each other.
So, AB is parallel to DC
So, AB || DC
Hence, the given statement is true. So, the answer will be 1.

Note: Whenever we come up with this type of problem then first, we join two opposite sides of the given quadrilateral and after that we had to from that the triangles form from the lines are congruent to each other and after that using CPCT Rule for congruent triangles. We easily prove that the alternate angles are equal. And we know that if the alternate interior angles of two sides are equal then the sides must be parallel to each other. This will be the easiest and efficient way to prove the result.