Why are the opposite sides of a parallelogram equal?

Answer 1

It is difficult to give a proof without diagrams, but here goes:

Suppose you have the parallelogram with ABDC and ADBC

Then angle ABC = angle CDA.

Now draw the diagonal AC.

Then angle ACB = angle CAD (alternate angles).

So, in triangles ABC and ACD,

two pairs of angles are equal so that all three pairs are equal (in each triangle the three add to 180 deg).

and AC is a common side.

So the two triangles are congruent and so AB = DC and AD = BC