Question

Let ABCD be a quadrilateral such that AD is equal to BC and AB is equal to DC . Prove that ABCD is a parallelogram

Answer 1

Let's start off by determining the general property of parallelograms.
A parallelogram consists of exactly two pairs of opposite sides that are BOTH parallel and congruent.

(In the question above) it has been stated that,
           In ABCD, 
                 AD  =  BC
                 AB  =  DC

Now, how to prove that quadrilateral ABCD is a parallelogram?

There are 5 ways to do so:
1) If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.
2) If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
3) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
4) If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.
5) If one angle is supplementary to both consecutive angles in a quadrilateral, the quadrilateral is a parallelogram.

*** Be sure to remember the second method, as it may save you time when working a proof.

In the end,
if ABCD fulfils all the requirements of a parallelogram,
then it simply is a parallelogram.

Now, take out your rulers and protractor! Complete the required measurements. Show your workings. Show why or why not ABCD is a parallelogram.

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