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### Course: High school geometry>Unit 3

Lesson 6: Theorems concerning quadrilateral properties

# Proof: Rhombus diagonals are perpendicular bisectors

Sal proves that the diagonals of a rhombus are perpendicular, and that they intersect at the midpoints of both. Created by Sal Khan.

## Want to join the conversation?

• What does SSS mean?
(14 votes)
• SSS stands for side/side/side and means that all 3 sides of two triangles are congruent and therefore the triangles are congruent.
SAS stands for side/angle/side and means that 2 sides and the angle they form in two triangles are congruent and therefore the triangles are congruent.
There are lots of videos about congruent triangles. here is one about SSS:
https://www.khanacademy.org/math/geometry/congruent-triangles/cong_triangle/v/congruent-triangles-and-sss
(31 votes)
• What is the the proof or theorem for all sides of a rhombus being congruent
(5 votes)
• There is no proof. It is simply a definition.
That's like asking for a proof of a triangle has 3 sides.
(11 votes)
• I get confused because sometimes Sal says to look at the figure, or that something is obvious from the figure. However, there always seems to be the caveat to not trust the drawing.. I'm confused as to when you can use the drawing and when you can't.
(7 votes)
• You are correct not to trust the diagram, but we can trust the markings of a diagram such as tick marks or equivalent angles or parallel sides. Sal ends up marking the diagram based on what we know about the rhombus (all sides equal) and what we know about parallelograms (diagonals bisect each other).
(6 votes)
• One other point of confusion I have in this proof is when he says that ∠AEB is supplementary to ∠CEB. Why is he able to simply state this? How is it proved that those angles are supplementary?
(6 votes)
• Two angles that form a line are by definition supplementary because a line is a straight angle (180 degrees) and the definition of supplementary is two angles that add to be 180 degrees.
(7 votes)
• I just wanted to say thank you to Sal for bettering my math education and helping me through educational rough spots
(8 votes)
• What helps for finding missing angles for a quadrilateral (this only works if you're good at finding missing angles in triangles) is if, say, one angle is 50, one angle is X, one is 90, and one is 70 (yes, simple numbers). divide the quad in half to get two triangles. Half X, 50, half 90 and half X, 70, half 90. Add 50 and 45 (half 90) and subtract from 180. You get half X (85). Multiply by 2 and there's X. (Just saying, there is probably a simpler way, I sometimes make things a challenge, BUT it works so you can't say there is something wrong with it)
(1 vote)
• Actually, your method does not work. It would only work if the shape was a square (or a rhombus), and in your case, it isn't.
The answer your method yields is 170, which is incorrect. The correct answer is 150. Simply add the three known angles together and subtract that from 360, because all the angles in a quadrilateral add up to 360.
The reason your method won't work is because you cannot divide an irregular polygon equally. Take a rectangle for example. If you draw a diagonal from one corner to another, the 90 degree angle is not split in half. One side will equal 30 and the other equals 60.
(11 votes)
• you know if you put the playback speed to 1.5 he actually sounds normal.
(6 votes)
• I'm usually a quick learner, but I don't understand the questions where they give you the statements, I need someone to clarify it a bit. I know how to make my own proofs but I don't understand the ones where they give you a multiple choice. Can someone clarify them a bit for me?
(5 votes)
• All sides of a rhombus are equal in length.
(2 votes)
• what's the difference between a parallelogram and a rhombus, I noticed that sal said that a rhombus has all equal sides but a parallelogram has opposite sides congruent.
(3 votes)
• A parallelogram has 2 pairs of parallel sides. A rhombus has 4 sides of equal length. Rhombuses are a sub-category of parallelograms, since all rhombuses have 2 pairs of parallel sides, so all rhombuses are parallelograms, but not all parallelograms are rhombuses. I hope this helps you.
(6 votes)
• Isn't this just a different kind of kite?
(4 votes)
• Yes. The only difference being, that the adjacent sides have the same length. Thus, we can say that - all rhombus are kites but not vice versa.
(4 votes)

## Video transcript

We're told that quadrilateral ABCD is a rhombus. And what they want us to prove is that their diagonals are perpendicular, that AC is perpendicular to BD. Now let's think about everything we know about a rhombus. First of all, a rhombus is a special case of a parallelogram. In a parallelogram, the opposite sides are parallel. So that side is parallel to that side. These two sides are parallel. And in a rhombus, not only are the opposite sides parallel-- it's a parallelogram-- but also, all of the sides have equal length. So this side is equal to this side, which is equal to that side, which is equal to that side right over there. Now, there's other interesting things we know about the diagonals of a parallelogram, which we know all rhombi are parallelograms. The other way around is not necessarily true. We know that for any parallelogram-- and a rhombus is a parallelogram-- that the diagonals bisect each other. So for example, let me label this point in the center. Let me label it point E. We know that AE is going to be equal to EC. I'll put two slashes right over there. And we also know that EB is going to be equal to ED. So this is all of what we know when someone just says that ABCD is a rhombus, based on other things that we've proven to ourselves. Now we need to prove that AC is perpendicular to BD. So an interesting way to prove it-- and you can look at it just by eyeballing it-- is if we can show that this triangle is congruent to this triangle and that these two angles right over here correspond to each other, then they have to be the same. And they'll be supplementary, and then they'll be 90 degrees. So let's just prove it to ourselves. So the first thing we see is we have a side, a side, and a side; a side, a side, and a side. So we can see that triangle-- let me write it here. Let me [? do this ?] in a new color. We see that triangle ABE is congruent to triangle CBE. And we know that by side-side-side congruency. We have a side, a side, and a side; a side, a side, and a side. And then once we know that, we know that all the corresponding angles are congruent. And in particular, we know that angle AEB is going to be congruent to angle CEB. Because they are corresponding angles of congruent triangles. So this angle right over here is going to be equal to that angle over there. And we also know that they are supplementary. Let me write it this way. They're congruent, and they are supplementary. These two are going to have the same measure, and they need to add up to 180 degrees. So if I have two things that are the same thing and they add up to 180 degrees, what does that tell me? So that tells me that the measure of angle AEB is equal to the measure of angle CEB, which must be equal to 90 degrees. They're the same measure, and they are supplementary. So this is a right angle, and then this is a right angle. And obviously, if this is a right angle, this angle down here is a vertical angle. That's going to be a right angle. If this is a right angle, this over here is going to be a vertical angle. And you see the diagonals intersect at a 90-degree angle. So we've just proved-- so this is interesting. A parallelogram, the diagonals bisect each other. For a rhombus, where all the sides are equal, we've shown that not only do they bisect each other but they're perpendicular bisectors of each other.