Hi there. This is Ryan Malloy here at the Worldwide Center of Mathematics. In this video we are going to discuss the theorems for parallelograms. Suppose we're given some quadrilateral, what we want to know here is what information do we need to know about this quadrilateral in order to determine that it is a parallelogram. Well there's the trivial answer that if we already know that opposite sides are parallel, then by definition it's a parallelogram. So we'll skip that one for now. But a more interesting case is if we know that opposite sides are of the same length then it is also true that it is a parallelogram. Similarly if we know that we have one pair of opposite sides that are of the same length and they're parallel, this is also enough information. Go ahead and keep track of these over here. So we have opposite pairs of sides are congruent, that's enough information. We have one opposite pair of sides is congruent and parallel. That's also enough information. Lets talk for a moment about the diagonals. We have a diagonal from the bottom left to the upper right. And a diagonal from the top left to the bottom right. Now if it is true that the two diagonals bisect each other, meaning this component of a diagonal is congruent to this component, and vice versa, that's also enough information to determine that a shape is a parallelogram. So we keep track of that. Diagonals bisect each other. Okay, lets talk about the interior angles of this quadrilateral. If we know that opposite pairs of angles are congruent, meaning the bottom left and the upper right are congruent and the top left and bottom right are congruent, that's enough information. Since from there you can quickly determine that the opposite sides are parallel. So we have opposite angles are congruent. Okay. Finally, what color haven't I used yet. Lets use yellow. So let's take a look at this angle here. If we know that this angle plus this angle add up to 180 degrees, in another words that they're supplementary, and we also know that this angle and this angle are 180 degrees, then that's enough information to determine that a quadrilateral is a parallelogram. So in simplest terms if you can find any angle such that it's adjacent angles are supplementary to it, then it must be a parallelogram. So we have angle is supplementary to both adjacent angles. And there we have it. So we have a variety of ways of showing that a quadrilateral can be a parallelogram. If the opposite pairs of sides are congruent, if one opposite pair of sides are congruent and parallel, that's enough information. If the diagonals bisect each other, that's enough information. If opposite angles are congruent, that's enough. And if one angle is supplementary to both of it's adjacent angles, that's also enough information. And there we have it. My name is Ryan Malloy and we've just discussed theorems for parallelograms.