This creates 2 triangles with 2 equal sides - the diagonal BD is common to both, and AD=BC, as given. It will be helpful to create these triangles in a way that makes use of known sides that are equal, so since we are told that AD=BC, we should probably make them sides in the triangles we create. To show that two lines are parallel, we can use the converse of the Corresponding Angles Theorem, (that is, show that 2 corresponding angles are congruent) or the converse Alternate Interior Angles Theorem (show that the interior alternating angles or exterior alternating angles are congruent), whichever is easier.īut to show any such congruent angles, we will need to create some triangles, in order to use triangle congruency. Here, we are already given the fact that one pair is parallel - AD||BC, so we will need to do so for the other pair - that is, show that AB||DC. To show that a quadrilateral is a parallelogram using the definition of a parallelogram, we need to show that both pairs of opposite sides are parallel.

Show that ABCD is a parallelogram Strategy We will start with a fifth converse theorem - that if a quadrilateral has one pair of sides that is both parallel and equal in length it is a parallelogram. We will now show that the converse is true - that if one of these properties holds, the quadrilateral is a parallelogram.

Two pairs of consecutive angles are supplementary.

The two pairs of opposite angles are congruent.

The two pairs of opposite sides are of equal length.

We have said (and proven) that parallelograms have four basic properties: And only if their lengthsĪre equal are they parallel.A parallelogram is defined as a quadrilateral with two opposite pairs of sides are parallel. You could say opposite sides ofĪ quadrilateral are parallel if and only if their Have the same length, then you have a parallelogram. Have a parallelogram, opposite sides have Is parallel to BD by alternate interior angles. Interior angles are congruent, we know that they are parallel. Here, which we're not sure whether they're parallel. Well, once again, these couldīe alternate interior angles. So we're just saying thisĪngle is equal to that angle. Angle ACB is congruentĬorresponding angles congruent of congruent triangles. Of a transversal intersecting parallel lines. Is parallel to CD by alternate interior angles And because we have theseĬongruent alternate interior angles, we know that AB That these things that could be alternate interiorĪngles are congruent. Well, this is interesting,īecause here you have a line. So ABC is going toīe congruent to DCB, so these two angles are And you could say, byĬorresponding angles congruent of congruent triangles. Of the corresponding angles are going to be congruent. Well, what does that do for us? Well, it tells us that all So we know that triangleĪ- and we're starting at A, and then I'm going Of these two triangles are equal to each other. Into two triangles, triangle ACB and triangle DBC. That this is a parallelogram? Well, it's kind of theĭiagonal here, since we know a lot about triangles. Type of a quadrilateral, and we know that the Reason- corresponding sides of congruent triangles. And that's because theyĪre corresponding sides of congruent triangles. On this bottom triangle corresponds to side BA Well, what does that do for us? Well, if two trianglesĪre congruent, then all of the correspondingįeatures of the two triangles are going to be congruent.

And this comes out ofĪngle-side-angle congruency. To pink to green- ADB is congruent to triangle. Pink angle, side in common,Īnd then the green angle. These triangles, they have this pink angle. Realize that we've just shown that both of And we also see thatīoth of these triangles, triangle ADB and triangle CDB,īoth share this side over here. Interior angles are congruent when you haveĪ transversal intersecting two parallel lines. Interior angles of a transversal intersecting Way, then you immediately see that angle DBCĬongruent to angle ADB for the exact same reason. View this diagonal, DB- you could view it asĪ transversal of these two parallel lines, of the other You have a transversal-Īngle ABD is going to be congruent to angle BDC. That's that angleĪlternate interior angles. Way, you can pick out that angle ABD is going toīe congruent- so angle ABD. This diagonal DB- we can view it as a transversalįor the parallel lines AB and DC. So prove that AB is equal toĭC and that AD is equal to BC. Parallelogram ABCD, let's prove that the opposite Straightforward parallelogram-related proofs. Prove in this video is a couple of fairly

Quadrilateral abcd is a parallelogram