Is it possible to find angles inside a shape if I am only given the slopes and lengths of the lines? I was given a math problem listing 4 ordered pairs of points to plot. So I plotted them, found their slopes and distances. The question asks me to determine the most precise name of the quadrilateral. I know their opposite sides are parallel and all sides are congruent so I concluded it was a rhombus. But then I was thinking it could also be a square but to be a square all angles must be right angles. In short my question is with this information that I know is there any way to find out the angles so I can call the quadrilateral a square? Or is the most specific name for it a rhombus? Thank you. 
 A: Here's an approach to categorization based mainly on line lengths. Suppose the points are $a_1=(x_1,y_1),\dots, a_4=(x_4,y_4)$. If you find the point $O=\dfrac{a_1+a_2+a_3+a_4}4$ and define $a_i'=a_i-O$, this will move the center of the quadrilateral to the origin and make several patterns easier to spot.


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*The quadrilateral is a parallelogram iff $a_1'=-a_3'$ and $a_2'=-a_4'$. This is saying that both diagonals go through the center, and from this you can see that $|a_1-a_2|=|a_3-a_4|$ and $|a_1-a_3|=|a_2-a_4|$, so opposite sides are congruent.


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*A parallelogram is a rectangle iff $|a_1'|=|a_2'|$. This says that the diagonals have the same length, so adjacent sides are perpendicular.

*A parallelogram is a rhombus iff $a_1'$ is perpendicular to $a_2'$. If $a_1'=(x_1',y_1')$ and $a_2'=(x_2',y_2')$ this is equivalent to $x_1'x_2'+y_1'y_2'=0$. Alternatively you can check that $|a_1'-a_2'|=|a_3'-a_2'|$, which says directly that the edges have the same lengths.

*Finally, a parallelogram is a square iff it is a rectangle and a rhombus.
