Quadrilaterals
-
Quadrilateral Overview
-
Quadrilateral Properties
-
Quadrilateral types
-
Proof - Opposite Sides of Parallelogram Congruent
-
Proof - Diagonals of a Parallelogram Bisect Each Other
-
Proof - Opposite Angles of Parallelogram Congruent
-
Quadrilateral angles
-
Proof - Rhombus Diagonals are Perpendicular Bisectors
-
Proof - Rhombus Area Half Product of Diagonal Length
-
Area of a Parallelogram
-
Whether a Special Quadrilateral Can Exist
-
Quadrilaterals challenge
Proof - Opposite Sides of Parallelogram Congruent Proving that a figure is a parallelogram if and only if opposite sides are congruent
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- What we're going to prove in this video is a couple of fairly
- straight forward parallelogram related proofs
- And this first one we're gonna say,
- "Hey, if we have this parallelogram ABCD,
- let's prove that the opposite sides have the same length "
- So, prove that AB is equal to DC and that AD is equal to BC
- So, let me draw a diagonal here
- So, I'm gonna draw a diagonal
- And this diagonal, depending on how you view it is intersecting
- two sets of parallel lines so you can also consider it
- to be a transversal
- Actually let me draw a little bit neater than that
- I can do a better job
- So, nope that's not any better
- That is about as good as I can do
- So, if we look, view DB, this diagonal DB, we can view it
- as a transversal for the parallel lines AB and DC
- And if you view it that way, you can pick out that angle ABD
- is going to be congruent
- So, angle ABD, that's that angle right there is going to be
- congruent to angle BDC because they are alternate interior angles
- You have a transversal, parallel lines
- So, we know that angle ABD is going to be congruent
- to angle BDC
- Now you can also view this diagonal DB, you can view
- this as a transversal of these two parallel lines,
- of the other two pair of parallel lines, AD and BC
- And if you look at it that way you'll immediately see that angle
- DBC, right over here, angle DBC is going to be congruent to angle
- ADB for the exact same reason, they are alternate interior angles
- of a transversal intersecting these two parallel lines
- So, I could write this
- This is alternate interior angles are congruent when you have
- a transversal intersecting two parallel lines
- And we also see that both of these triangles,
- triangle ADB and triangle CDB both share the side over here
- It's obviously equal to itself
- Now, why is this useful
- Well, you might realize that we've just shown that both of these
- triangles, they have this pink angle and they have this side
- in common and then they have the green angle
- Pink angle, side in common and then the green angle
- So, we've just shown by angle-side-angle that these
- two triangles are congruent
- So, let me write this down
- We have shown that triangle -- I'll go from non-labeled to pink
- non-labeled to pink to green -- CBD and this comes out of
- angle-side-angle congruency
- So, this is from angle-side-angle congruency
- Well, what does that do for us
- Well, if two triangles are congruent then all of the corresponding
- features of the two triangles are going to be congruent
- In particular, side DC corresponds to side BA --
- side DC on this bottom triangle corresponds to side BA
- on the top triangle
- So, they need to be congruent
- So, DC
- So, we get DC is going to be equal to BA and that's
- because they are corresponding sides of congruent triangles
- So, this is going to be equal to that and by that exact same logic,
- AD corresponds to CB
- AD is equal to CB and for the exact same reason:
- corresponding sides of congruent triangles
- And then we're done!
- We've proven that opposite sides are congruent
- Now, let's go the other way
- Let's say that we have some type of a quadrilateral
- and we know that the opposite sides are congruent,
- can we prove to ourselves that this is a parallelogram
- Well, it's kind of the same proof in reverse
- So, let's draw a diagonal here,
- since we know a lot about triangles
- So, let me draw
- There we go
- That's the hardest part, let's see
- Draw at a -- that's pretty good
- Alright
- So, we obviously know that CB is going to be equal to itself
- So, I'll draw it like that
- We obviously 'cause that's the same line
- And then we have something interesting
- We've split this quadrilateral to two triangles: triangle ACB
- and triangle DBC
- And notice, they have all three sides of these two triangles
- are equal to each other
- So, we know that by side-side-side that they are congruent
- So, we know that triangle, I'm gonna start at A and I'm going
- to the one half side, so, ACB is congruent to triangle DBC
- and this is by side-side-side congruency
- Well, what does that do for us
- Well, it tells us all the corresponding angles
- are going to be congruent
- So, for example, ABC, angle ABC is going to be
- you can see ABC -- is going to be congruent to DCB,
- angle DCB and you can say by you can say corresponding angles
- congruent of congruent triangles
- I'm just using some short hand here to save some time
- So, ABC is going to be congruent to DCB
- So, these two angles are going to be congruent
- Well, this is interesting because here you have a long line
- and it's intersecting AB and CD and we clearly see that these
- things that could be alternate angles, alternate interior angles,
- are congruent
- And because we have these congruent alternate interior angles
- we know that AB must be parallel to CD
- So, this must be parallel to that
- So, we know that AB is parallel to CD by alternate interior angles
- of transversal intersecting parallel lines
- Now, we can use that exact same logic
- angle ACB is congruent to angle DBC
- and we know that by corresponding
- angles congruent of congruent triangles
- So, we're just saying that this angle is equal to that angle
- Well, once again these could be alternate interior angles,
- they look like they could be, this is a transversal
- and here's two lines here which we're not sure they're parallel
- but because the alternate interior angles are congruent
- we know that they are parallel
- So, this is parallel to that
- So, we know that AC is parallel to BD by alternate interior angles
- And we're done!
- So, what we've done is interesting
- We've shown if you have a parallelogram, opposite sides are,
- opposite sides have the same length
- And if opposite sides have the same length then you have
- a parallelogram
- And so, we've actually proven it in both directions
- And so we can actually make what you call
- an if-and-only-if statement
- You can say, "If opposite sides are parallel of a quadrilateral "
- Or you could say, "Opposite sides of quadrilateral are parallel
- if and only if their lengths are equal "
- And you can say "if and only if"
- So, if they are parallel then you can say their lengths are equal
- and only if their lengths are equal are they're parallel
- We've prove it in both directions
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.