Question 1013669
recall: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

A parallelogram have:
- 2 sets of parallel sides
- 2 sets of congruent sides
- opposite angles congruent
- consecutive angles supplementary
- diagonals bisect each other
- diagonals form 2 congruent triangles

you can use: 
If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel and congruent, the quadrilateral is a parallelogram. 

first we sketch given points and figure they make:

({{{-2}}},{{{3}}}), ({{{-2}}},{{{-1}}}) ({{{1}}},{{{1}}}) and ({{{1}}},{{{5}}})

{{{drawing( 600,600, -10, 10, -10, 10,
circle(-2,3,.12),circle(-2,-1,.12),circle(1,1,.12),circle(1,5,.12),
green(line(-2,3,-2,-1)),green(line(-2,-1,1,1)),green(line(1,5,1,1)),
green(line(-2,3,1,5)),
 graph( 600,600, -10, 10, -10, 10, 0)) }}} 


opposite sides of a quadrilateral are sides made by points 
({{{-2}}},{{{3}}}), ({{{-2}}},{{{-1}}})
and
({{{1}}},{{{1}}}) and ({{{1}}},{{{5}}})

to prove that BOTH parallel and congruent, we need to find out if the lines passing through these points  have equal slopes: 

a. the line passing through the points 

 ({{{x[1]}}}, {{{y[1]}}}) = ({{{-2}}}, {{{3}}}) and ({{{x[2]}}}, {{{y[2]}}}) = ({{{-2}}}, {{{-1}}}) has a slope {{{m}}}:

{{{ m=(y[2]-y[1])/(x[1]-x[1])}}}

{{{ m=(-1-3)/(-2-(-2))}}}

{{{ m=-4/(-2+2)}}}

{{{ m=-4/0}}}.........which means a slope is undefined, and an undefined slope is the slope of a vertical line  


b. the line passing through the points 
({{{x[1]}}}, {{{y[1]}}}) =({{{1}}},{{{1}}}) and 
({{{x[2]}}}, {{{y[2]}}}) = ({{{1}}}, {{{5}}}) has a slope {{{m}}}

{{{ m=(y[2]-y[1])/(x[1]-x[1])}}}

{{{ m=(5-1)/(1-1)}}}

{{{ m=4/0}}}.........which means a slope is undefined, and an undefined slope is the slope of a vertical line 

so, since both lines have an undefined slope, both are vertical, we can conclude that both are {{{parallel}}}

now we need to prove that the distance between points is same:

({{{-2}}},{{{3}}}), ({{{-2}}},{{{-1}}})

*[invoke distance_formula -2, 3, -2, -1]

and
({{{1}}},{{{1}}}) and ({{{1}}},{{{5}}})

*[invoke distance_formula 1, 1, 1, 5]

distances are same


since both are that contain given points are {{{parallel}}} and distances are same, we can conclude that the quadrilaterals with vertices ({{{-2}}},{{{3}}}), ({{{-2}}},{{{-1}}}) ({{{1}}},{{{1}}}) and ({{{1}}},{{{5}}}) {{{IS}}} a parallelogram