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Are legs of a trapezoid always, sometimes, or never congruent?

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4 Answers

You asked "Are legs of a trapezoid always, sometimes, or never congruent?"   
What does it mean for two lines to be congruent?  It means the are the same length, period!  
What are you referring to by the term "legs"?  
Basically a trapezoid is a type of irregular quadrilateral.  Up to 3 sides of a trapezoid can be the same length, but there usually is one side, one of the two parallel sides, which is not the same length.
Now I'm going to confuse you by saying "Unless, of course, the trapezoid is a rectangle";  because a rectangle is in fact a special case of a trapezoid you know, and so is a square.
The best way to learn about trapezoids is to look at some examples on the internet.  
Think of this like Biology; a Trapezoid is just a classification of a 4-sided figure.  Classifications are can be made as follows:  
An irregular quadrilateral only has to have 4 sides.  To be a Trapezoid, an irregular quadrilateral  has to have two of the sides which are parallel.  To be a Parallelogram a Trapezoid has to have two pairs of sides Parallel.  If a Parallelogram has all sides equal it's a Rhombus; but if the sides are not equal but the sides make right angles with each other, then the Parallelogram is is a Rectangle.  If a Rectangle has all four sides the same length or a Rhombus has sides which make right angles with each other, then the figure is a Square.
A trapezoid is a quadrilateral (4 sides) with 2 sides parallel.
The other two sides are sometimes congruent (have same length).