Before, we assumed a figure was a parallelogram, and we showed that the figure had some special properties. Many of these statements are the converse statements of the proofs we came up with already.

Both pairs of opposite angles are congruent.One pair of opposite sides is both congruent and parallel.Both pairs of opposite sides are congruent.(This is the definition of a parallelogram.) The two pairs of opposite sides are parallel.Here are some ways you can convince the jury that the quadrilateral is guilty of being a parallelogram: Let's say you've accused a quadrilateral of being a parallelogram. On the flip side, we'd be hoarding a bunch of useless facts when we could be using them as parallelogram primers. Well, we could continue collecting these nuggets and eventually give Fort Knox a run for its money. 'Why do we care?' 'What does this matter?' 'How is this at all relevant to my life?' 'Why are these little golden nuggets of wisdom important?' So it's only reasonable that we ask our favorite questions about mathematics. We've established five different theorems about parallelograms. And it wasn't just to make your math teacher happy.

Oh, you thought they were over and done with, did you? Well, we hate to burst your bubble, bud, but we learned about those triangles for a reason. The key to this proof (and probably most proofs about quadrilaterals) is a theorem about triangles. Opposite sides of a parallelogram are congruent (1) Given that ABCD is a parallelogram, prove that its diagonals, AC and BD, bisect each other. Or, if you'd rather not, you can get the theorem for free! Just handle it yourself. But wait, there's more! Order in the next ten minutes and we'll throw in another theorem, absolutely free! Just pay $4.95 for shipping and handling. With all of these theorems about parallelograms, it's like we struck mathematical gold. So ∠ A and ∠ B are supplementary, as are ∠ C and ∠ D. Two consecutive angles are supplementary.

Both pairs of opposite angles are congruent So for ABCD, ∠ B ≅ ∠ D and ∠ A ≅ ∠ C.

Both pairs of opposite sides of a parallelogram are congruent.

We don't want to say we told you so, so we'll opt for, ' We informed you thusly.'īut you know what's even better? Thanks to CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we can prove three more theorems right away: Given that ABCD is a parallelogram, prove that Δ ABC and Δ CDA are congruent.īam! Congruent triangles. It's because the diagonals of a parallelogram split it up into two congruent triangles.ĭon't believe us? We can prove it. Yes, we did just refer to those diagonals as 'honorable and generous,' and it's not because they donate to charities or volunteer at soup kitchens. Since a parallelogram is a specific type of quadrilateral, it comes with all the goodies that quadrilaterals have, like 360° of internal angles, opposite angles, sides, and vertices, and two honorable and generous diagonals. Since parallelograms are as common as a brain freeze at an ice cream shop, let's get down and dirty with them. We'll cover those in more detail later on. In fact, squares, rhombi, and rectangles are all special cases of parallelograms.ĭon't you worry your pretty little head about those specific parallelograms just yet. Well, a rectangle has two sets of parallel sides, doesn't it? So it's just as much of a parallelogram as the one above. For the good of your bank account, we sincerely hope it doesn't look like that. Unless you did some serious damage to your computer screen, it probably doesn't look like that. When we think of parallelograms, we usually think of something like this. Unless you have a particularly wonky-looking screen, that is.Ī parallelogram is any quadrilateral with two sets of parallel sides. Right now, as you read this, you are looking at a parallelogram. Like towel rods and toilets, we use them every day and never give them much thought. They hide in plain sight, staring us straight in the face.

Prove that abcd is a parallelogram