Proving Opposite Sides of a Parallelogram Are Equal Using Congruency

in the previous video we discovered that when you take a quadrilateral where the opposite sides

so opposite meaning like that and like that when those are parallel then we get a shape that's called a

parallelogram now a parallelogram has some really incredible features that we are going to examine

over the next couple of videos in this video we are going to prove that the opposite sides of a

parallelogram are equal and i mean we could think of it like this if we took two parallel

lines like that notice they're not the same length but they're definitely

parallel and we had to take two other parallel lines so let's say we took one like

this over here let's say those connect and those are parallel and then we took another side

that's parallel well look what would happen you would form a shape that doesn't even complete

and that is because these two sides are not the same and so if you can get two lines that are parallel like this

and you can get two other lines that are parallel to each other and if you could somehow combine those into a

four-sided shape such as let's say we do this and that over there and then you cut off

all the extra pieces what you would find is that those the opposite sides

are going to be exactly equal in length but now we're going to prove that in this video

and so something that we need to do that makes this really easy is to simply include

a diagonal line now what we can do so we are for example trying to prove that this length

is the same as that length and this length is the same as this length ah but now

remember we've just gone through a whole series of videos on congruency so imagine you could prove that this

triangle is the same as this triangle well then what that would mean is that

this side is going to be the same as that side and that side is the same as that side

so congruency is the key so let's go for it so we've looked at congruency in previous videos

so we can say in triangle a b c and triangle a c d because just remember at the

moment nothing on this diagram tells us that the opposite sides are equal

all we need though is for these to be parallel and for these to be parallel and we are going to prove that that will

automatically make the sides equal that is really really cool so let's find three things so we know

that ac is going to be equal to ac right because that line

is going to be in both triangles and remember we don't say given for that but rather we say

common now we don't have any other sides to use because no other sides are the same

but we have got parallel lines and when you see parallel lines you should think of these three so did you see the z

over here aha so because of that we know that this corner angle is going to be the same as

this corner angle now what is this corner angle you can't just say a because a could be this part or this

part and so imagine you went from b to a to c which angle did you just form well

that's this one over here so to show your teacher which angle you're talking about you say this

b a c with the a the angle that you're talking about must always be in the middle

well that's going to be equal to this angle over here and that angle can be thought

of as a c d because which corner do we form it's

this one and so we could say a c d now what's the reason for that

well remember we said it was because of alternating angles and so the reason we can say is alt

angles and then you need to say which sides are parallel well that's going to be a b

is parallel to c d so we've now found one and two things but now what about the z over here did

you see that one well because of that z we know that this corner angle

is going to be the same as that corner angle over there and so that's going to be that's going

to be b c a is going to be the same as dac and that's also going to be because

of alternating angles and that's now because of

bc being parallel to a d and so now we need to say that these two triangles

are congruent so we can say therefore triangle a b c is congruent two so a b

c is the top triangle now you need to choose the other triangle but the order is very important so which angle goes

with a well a was this one over here and we said that that matches this one so

that's going to be c b well that's going to be the same as this one over s that's d

and then c in the top triangle that's this one here well we said that that goes with this

one over here which is a and the reason for that ah so what's the reason now

well we've used one side so when we were busy proving this we used one side over here and then we used two

angles and so we're going to say side angle angle so all in all we have now proved that these two triangles are

exactly the same that means that ba is going to have the same length as cd

so we can say therefore a b is going to have the same length as cd and it also means that this length over

here is the same as this length over here so we can say bc is the same as 80. so i

hope you guys can actually appreciate what we've just done here we started off with a shape that only had parallel

lines no one told us that those lengths were the same

but by using congruency we were able to prove that when you have a parallelogram the

opposite sides have to be the same