Question 972033
I have come to the conclusion that all of geometry boils down to triangles and their properties.
You just have to find the right triangles, drawing extra lines/segments as needed.
 
In this case, you would need to use congruent triangles (and their properties and theorems),
and the facts that
when two lines are cut by a transversal, the alternate interior angles are congruent,
and that
if two lines are cut by a transversal, and the alternate interior angles are congruent, then the lines are parallel.
 
If you draw the lines that contain the sides,
and the transversal line that contains diagonal AC,
you find that the transversal line that intersects parallel lines BC and AD, and splits the quadrilateral into triangles BCA and DAC.
 {{{drawing(300,200,-3,33,-2,22,
line(11,-0.7,11,0.7),line(16,19.3,16,20.7),
arrow(2,0,14,0),arrow(7,20,19,20),
line(-3,0,33,0),line(-3,20,33,20),
line(-1,-4,6,24),line(24,-4,31,24),
blue(arrow(-3,-2,39,26)),
locate(-1.2,1.7,A),locate(25.2,1.7,D),
locate(3.5,20,B),locate(30,20,C),
green(triangle(1,0.4,24.6,0.4,29.6,19.3)),
red(triangle(0.2,0.4,5.2,19.8,29.5,19.8))
)}}} You have to prove those triangles are congruent.
Then you use the theorem (usually abbreviated CPCTC) that says that
Corresponding Parts of Congruent Triangles are Congruent,
to prove that the alternate interior angles the transversal forms with the other two sides are congruent.
 
You can find that proof (in the boring Statement?reason table format in http://www.regentsprep.org/regents/math/geometry/gp9/lparallelogram.htm. 
 
Statements 1 and 2 (or both lumped together as stament 1) could be 
Line BC is parallel to line DA, and Segment BC is congruent to Segment DA.
The reason for that is written as "Given", because it is information given to you as part of the problem.
 
Next Statement:
The pair of alternate interior angles BCA and DAC area congruent.
Reason: Because AC is a transversal line intersecting parallel lines BC and AD.

(A silly extra statement may be required; the diagonal as a side of one triangle is congruent to the diagonal as a side of the other triangle, for the reason that it is the same segment, and any segment is congruent with itself).
 
From there, you prove that the two triangles formed as the diagonal splits the quadrilateral are congruent.
The reason is that they have a pair of congruent sides flanking a congruent angle (SAS congruency).
 
Next statement is that angles ACD and BAC are congruent.
Since the triangles are congruent, their corresponding parts are congruent (you could write as reason "by CPCTC").
 
Next statement is that the other two sides (AB and CD) are parallel.
The reason is that if the alternate interior angles formed by a transversal (like AC) with two lines {like lines AB and CD) are congruent,
then those two lines are parallel.
 
Next statement is that ABCD is a parallelogram.
The reason is it has two sets of parallel sides (AD||CD and BC||AD),
and a quadrilateral with two sets of parallel sides is by definition a parallelogram.