How does anyone know how to prove parallel lines, especially when there are two pairs of them?

Step-by-step explanation:

If two lines are cut by a transversal so that any pair of alternate interior angles are congruent, then the lines are parallel. If two lines are cut by a transversal so that any pair of corresponding angles are congruent, then the lines are parallel. You can use these converses to decide whether two lines are parallel.

A).  To go from ABCD to A'B'C'D' takes two transformations.

Step-1 :  Reflect the original figure around the x-axis.

Step-2 :  Slide the new figure 7 units to the right.

See the attached drawing.  It shows the whole process
including the intermediate status and coordinates of
the four points after Step-1.  You'll love it.

B).  I looked up a definition of 'congruent'.   It said that two figures
are congruent if they have have the same shape and size after
being moved, rotated, or reflected.  (I was worried about the
'reflected' part.)

With that definition, I'll let you decide whether these figures
are congruent.

In conclusion, let me express my gratitude for the generous
5-points bounty with which I've been showered by answering
this question.  The green crust and the cup of tepid turgid water
have been delicious, and by working quickly and efficiently, I was
able to construct the attached file in less than an hour.