5 2 Proving a Quadrilateral is Parallelogram

what we're gonna talk about today is the five ways to prove that quadrilateral is parallelogram and many of these theorems that we're going to talk about should look familiar because they're converses of the properties of the parallelogram that we did yesterday so the first one and this first one the definition of parallelogram is the way that we can prove the other four ways so you can show that if both pairs of opposite sides are parallel by the definition of parallelogram hence our quadrilateral is parallelogram the second way both pairs both pairs of opposite sides are congruent that's the converse of the property of parallelogram that we did yesterday 3 this is different this isn't converse of what we did yesterday this is showing that one pair of opposite sides are both congruent and parallel so looking at quadrilateral if can show this side is congruent to that side and that is parallel to that we can now say that that is parallelogram and we're gonna prove that theorem on the next slot fourth again is converse of what we talked about yesterday showing that both pairs of opposite angles both pairs of opposite angles are congruent showing the diagonals bisect each other again converse of what we talked about yesterday so let's prove that third way showing one pair of opposite sides are both congruent and parallel so again one of the things that want you guys to be able to do before we leave this class is want you to be able to look at an if-then statement and want you to be able to come up with given approve picture and two column proof so if both pairs of opposite sides are both congruent and parallel remember that if part is what we are given so we have our quadrilateral opposite sides are congruent so let's say WX is congruent to zy is congruent to we could also prove wz congruent to but I'm gonna focus on WX congruent to zy and we also are given that those segments are parallel that WX is parallel to and we're trying to prove is parallelogram and that symbol we can use for parallelogram so first as always we start out with our giving our next step we're going to use the idea of congruent triangles to be able to prove that this figure is parallelogram we need to draw in an auxilary line we need to draw in the diagonal of so our reason is gonna be draw apologize our statement is gonna be draw our reason is gonna be two points make one again remember every statement has to be justified with reason so now because have parallel lines know that angle 1 is congruent to angle 2 that is one of our special angle parts and go 1 is congruent to angle 2 because have parallel lines those alternate interior angles congruent you know have outside have an angle have side have an angle well ZX is common in both triangles so ZX it's congruent to ZX diagonal is congruent to itself by reflexive so now have triangle congruent to triangle and that is by side-angle-side next can say is congruent to by cpctc lastly that figure is parallelogram because opposite sides congruent give me parallelogram again remember that each one of these on the last slide each one of these are reasons so didn't go through proving each one of these but can use this idea to prove another theorem so let's do an example with one of these values must X&Y have to make that figure parallelogram well let's think about this they give us our two diagonals do you know the diagonals of parallelogram bisect each other so if we set the two pieces of the diagonal equal to 3x minus 4 equal to if subtract from both sides and at 40 to both sides get 2x equal to 40 divide by 2 we get equal to 20 why becomes little bit more challenging we have that way square it in there so know the two pieces are going to be equal so set them equal with quadratic have to get one side to be zero so subtract everything over to the squared side now factor two numbers that multiply to be negative 30 bird add to be negative one negative 6 and positive five are gonna add to be negative one we're gonna multiply to be negative 30 so can be 6 negative 5 do check plug 6 in for both the values of in the original picture 6 does it make either segment negative get 5 what square negative 5 get 25 negative 5 plus 30 equals 25 so both of those are possibilities equals 20 is our other one Leslie paragraph proof how to prove different quadrilateral is parallelogram remember paragraph proofs you don't need to state some obvious things first of all we are given that okay LJ is parallelogram and okay it's equal to so mark up that diagram now notice I'm trying to prove is parallelogram it's common diagonal and SH is the diagonal of what we're trying to prove parallelogram the figure we're trying to prove is parallelogram so what we want to do is we want to show that those two diagonals bisect each other so first as always we start out with given we are given parallelogram & is equal to since diagonals of it parallelogram bisect each other is the mid point of both Jay and out now Jay is common diagonal in both of these figures there so keep that in mind so because we know the midpoints sets two that's two of these segments equal so because of the definition of midpoint is equal to now look at that have two pieces equal have ko equal to + FK equal to my segment addition postulate can combine those two pieces and say that is congruent to by segment addition postulate in two column proof that's going to be about three or four steps remember with paragraph proof we can kind of look at the picture and don't have to state everything that's completely obvious so since is congruent to by segment addition postulate therefore is the midpoint of because is the midpoint of and those segments bisect each other notice again that and our diagonals of what we're trying to prove is parallelogram so therefore thus alright thus thus is parallelogram because the diagonals bisect each other other hey again when did this as two column proof in my notes did this as two column proof this worked out to be about 10-step proof because when was proving the segment addition postulate had couple steps in there sometimes paragraph proof is going to be little bit shorter there are your lesson questions completely Google form with those less than questions and please make sure that is submitted by 8:00 a.m. on the due date