Geometry (all content)
Proof: Opposite sides of a parallelogram
Sal proves that a figure is a parallelogram if and only if opposite sides are congruent. Created by Sal Khan.
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- For the second problem, couldn't you just have used the alt. int. angles theorem converse?(74 votes)
- I think so. As long as it makes sense and you get the desired conclusion, your proof is correct(26 votes)
- I still dont quite grasp the side angle thing (SSS AAA SAS ASA ASS SSA) I watched the vids explainin them a lot and still dont get it and I c thatthey keep poppin up on the other geometry vids!!! HELP?!!!?!(21 votes)
- They are just ways to identify similar or congruent triangles. The S stands for corresponding sides of equal measure on each triangle, and the A stands for corresponding angles of equal measure on each triangle. So for SSS, all three sides of a triangle would have the same lengths of all of the sides of another triangle. For AAA (or just AA, because you only need two of the angles) it would be the same thing, all three angles of a triangle would be the same as the angles on another triangle. However, because no sides must be related in this case, you are only getting similar triangles, not congruent ones. For ASA and SAS, two angles (ASA) or two sides (SAS) and the angle (for SAS) or a side (for ASA) that is surrounded by the two sides/angles; if these measures are equal to measures in the same position of another triangle, then they are congruent (an example of ASA would be at2:30). ASS and SSA don't actually work, but AAS and SAA work. For those comparisons, if two angles and a side that is not between them have the same measure as another triangle's two angles and an outside side, then both of those are congruent. It is really difficult to explain it without having any visuals, but I would have thought that the KA videos would have explained it well enough, but I haven't seen them, so I don't know.(64 votes)
- Isn't the second theorem just a converse of the first theorem ?(9 votes)
- It is just a converse of the first theorem.(7 votes)
- When labeling that the triangles are congruent by the ASA theorem. How do you identify the corresponding parts and then label?(7 votes)
- Pretend a triangle is ABC and XYZ. If you right that it means angle A = angle X, B = Y, C = Z, line AB = line XY, BC = YZ, and AC = XZ.(4 votes)
- how can we say that angle abd=bdc in the first instance?(4 votes)
- We can say that abd=bdc because the line he drew through the parallelogram is technically a transversal. Because it is a transversal, the two angles it forms are congruent, since we already know that the lines are parallel to each other. Hope this helps :)(6 votes)
- Alternate interior angles at1:02:: What are they?(3 votes)
- 'When two lines are crossed by another line (which is called the Transversal), the pairs of angles
• on opposite sides of the transversal
• but inside the two lines
are called Alternate Interior Angles.' Reference: www.mathsisfun.com(5 votes)
- What does the equal sign with the ~ mark on top mean?(3 votes)
- It is the congruent sign. It means that two shapes are congruent.(5 votes)
- I don't understand what this guy is teaching me may someone please help me understand?(3 votes)
- My suggestion to you is to watch the video, do some practice questions, and then if you don't understand try searching it up on YouTube. So you can get other people's points of view.(2 votes)
- Could you give a easy-to-understand explanation of the 'if and only if' logic used in math? (like in the example at8:16)
Any help would really be appreciated!(2 votes)
- Lets talk about a square.
If it is a square, then it is a quadrilateral with all right angles and congruent sides.
If a quadrilateral has all right angles and congruent sides, then it is a square.
So both the original statement and its converse (switching the hypothesis and conclusion) are both true. Thus, we can combine it into an if and only if statement, It is a square if and only if it is a quadrilateral with all right angles and congruent sides.
It does not work if the converse is not true such as if you live in Houston, then you live in Texas. The converse is If you live in Texas, then you live in Houston which is false, there a number of counterexamples such as Dallas, Austin, etc.(4 votes)
- For the first problem couldn't you have just drawn the diagonal and found they were congruent by the SSS postulate?(2 votes)
- The SSS postulate says 'if two traingles have all three pairs of sides congruent, then the triangles are congruent.' From the given information in the first problem, we don't know that the triangles have three congruent side pairs, so we can't use that postulate.(4 votes)
What we're going to prove in this video is a couple of fairly straightforward parallelogram-related proofs. And this first one, we're going to 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. And this diagonal, depending on how you view it, is intersecting two sets of parallel lines. So you could also consider it to be a transversal. Actually, let me draw it a little bit neater than that. I can do a better job. Nope. That's not any better. That is about as good as I can do. So if we 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 could also view this diagonal, DB-- you could view it as a transversal of these two parallel lines, of the other pair of parallel lines, AD and BC. And if you look at it that way, then you immediately see that angle DBC right over here 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 this 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. Then 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 to green-- ADB is congruent to triangle-- non-labeled to pink to green-- CBD. And this comes out of 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 on this bottom triangle corresponds to side BA on that top triangle. So they need to be congruent. 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. Draw it. That's pretty good. All right. So we obviously know that CB is going to be equal to itself. So I'll draw it like that. Obviously, because it's the same line. And then we have something interesting. We've split this quadrilateral into two triangles, triangle ACB and triangle DBC. And notice, all three sides of these two triangles are equal to each other. So we know by side-side-side that they are congruent. So we know that triangle A-- and we're starting at A, and then I'm going to the one-hash 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 that all of the corresponding angles are going to be congruent. So for example, angle ABC is going to be-- so let me mark that. You can say ABC is going to be congruent to DCB. And you could say, by corresponding angles congruent of congruent triangles. I'm just using some shorthand 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 line. And it's intersecting AB and CD. And we clearly see that these things that could be 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 a transversal intersecting parallel lines. Now, we can use that exact same logic. We also know that angle-- let me get this right. Angle ACB is congruent to angle DBC. And we know that by corresponding angles congruent of congruent triangles. So we're just saying 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 whether 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-- it's interesting. We've shown if you have a parallelogram, 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 could say opposite sides of a quadrilateral are parallel if and only if their lengths are equal. And you say if and only if. So if they are parallel, then you could say their lengths are equal. And only if their lengths are equal are they parallel. We've proven it in both directions.