High school geometry
Corresponding angles in congruent triangles
We write the letters of congruent triangles so their order tells us which parts are corresponding. In this video, Sal uses this notation to solve some angles. Created by Sal Khan.
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- At2:52how can we be sure that the three angles equal 180 degrees? It may appear to be a straight line, but without any notation ensuring that it is, how can we be sure?(13 votes)
- Hi J,
Right at the very beginning of the video, at0:01, Sal uses the words "in this larger triangle here" as he outlines triangle ABE. If this was a straight word question however, the question would start out with something like, "Given triangle ABE, prove that ..."
Hope that helps!(15 votes)
- what if it does not give that information? how do you know if they are all congruent or not?(4 votes)
- no sometimes the assignment is to figure out the congruent.(6 votes)
- I may have forgotten from a previous video, but how do you recognize, without a protractor or ruler, what the corresponding angles are? I know they appear when there are two or more congruent triangles, but do you eye them out? Or is their a mathematical way to know. Thanks(5 votes)
- Does the order of how the triangles are written matter?(2 votes)
- Very much so - corresponding angles should match on congruency statements.
For example if we say △ ABC ≅ △ EFG, that tells us a lot about the triangles, ∠ A ≅ ∠ E, ∠B ≅ ∠F, ∠C ≅ ∠G, AB ≅ EF, BC ≅ FG, CA ≅ GE.(6 votes)
I do not understand.
I need help with everything.
I have no clue what I'm doing, or what to do.
I struggle with math a lot and I'm not so bright when it comes to math.
Can you help me, please?(3 votes)
- If we know two figures are congruent, then we know matching parts are congruent. So lets use two of the three triangles triangle BCD is congruent to triangle ECD. Letters in the same spot (first, middle, and last) give congruent angles. So <B is congruent to <E, <C is congruent to <C (note they are different angles in the two triangles, but both at vertex C) and <D congrent to <D (once again, two different angles with common vertex). Similarly, matching sides are congruent, so BC congruent of EC (first two letters), CD is congruent to CD (any length is always congrent to itself), and DB is congruent to DE. So given a congruence statement, we can match angles and sides.(3 votes)
- although it's obvious enough to look at it, how do we know that the large shape ABE is definitely a triangle? is there a proof that a shape made of 3 congruent triangles must also be a triangle (and not some extreme quadrilateral where angle ACE could be not 180, but 179.99 degrees or something)?(4 votes)
- Maybe this is a silly question...but is Sal saying that the farthest left triangle is congruent to the two other? If so, could someone please explain why.
- it only explains the information given in the question.(4 votes)
- why does ~ on top of = mean congruent(3 votes)
- Someone created it and it stuck. The ~ alone means similar. Websites attribute it as noted below, the symbol did not transfer from the website correctly:
A similar looking symbol, º, meaning “congruent,” was credited to Johann Gauss in 1801. From http://mathserver.neu.edu/~bridger/U201/History_of_Math_Mathematical_Symbols.htm(2 votes)
- Don't we need to be given a few more pieces of information? Without being told on the diagram:
Can we actually assume that A, C, and E are colinear?
Same for B, D, and E.
Do we need to assume that C is the midpoint of AE?
Same for D on BE.
And do we need to assume that CD is the altitude of triangle BCE (therefore perpendicular to BE)?
*Edit: Took the word "assume" out of first sentence.(3 votes)
- youcant asuume anything in geometry(1 vote)
- I Still i don't how the triangles are congruent(3 votes)
So we have this larger triangle here and inside of that, we have these other triangles, and we're given this information right over here. That triangle BCD is congruent to triangle BCA, which is congruent to triangle ECD. And given just this information, what I want to do in this drawing, I want to figure out what every angle on this drawing is. What's the measure of every angle? So let's see what we can do here. So let's just start with the information that they've actually given us. So we know that triangle BCD is congruent to-- well, we know all of these three triangles are congruent to each other. So, for example, BCD is congruent to ECD, and so their corresponding sides and corresponding angles will also be congruent. So just looking at the order in which they're written B, vertex B corresponds, in this triangle, BCD, corresponds to vertex B in BCA, so this is the B vertex in BCA, which corresponds to the E vertex in ECD. So all-- everything that I've done in magenta, all of these angles are congruent, and then we also know that the C angle. So in BCA-- sorry, BCD, this angle right over here, is congruent to the C angle in BCA. BCA, the C angle is right over here, or C is the vertex for that angle in BCA. And that is also the C angle, I guess we could call it, in ECD. But in ECD, we're talking about this angle right over here. So these three angles are going to be congruent. And I think you could already guess a way to come up with the values of those three angles. But let's keep looking at everything else that they're telling us. Finally, we have vertex D over here. So angle-- so this is the last one in where we listed-- so in triangle BCD, this angle right over here corresponds to the A vertex angle in BCA. So BCA, that's going to correspond to this angle right over here. It's really the only one that we haven't labeled yet. And that corresponds to this angle, this vertex right over here, that angle right over there. And just to make it consistent, this C should also be circled in yellow. And so we have all these congruences, and now we can come up with some interesting things about them. First of all, here, angle BCA, angle BCD, and angle DCE, they're all congruent, and when you add them up together, you get to 180 degrees. If you put them all adjacent, as they all are right here, they end up with a straight angle, if you look at their outer sides. So you have, if these are each x, you have three of them added together have to be 180 degrees, which tells us that each of these have to be 60 degrees. That's the only way you have three of the same thing adding up to 180 degrees. Fair enough. What else can we do? Well, we have these two characters up here. They are both equal and they add up to 180 degrees. They are supplementary, the only way you can have two equal things that add up to 180 is if they're both 90 degrees. So these two characters are both 90 degrees. Or we could say this is a right angle, that's a right angle. And this is congruent to both of those, so that is also 90 degrees, and then we're left with these magenta parts of the angle. And here, we could just say, well, 90 plus 60 plus something is going to add up to 180. 90 plus 60 is 150. So this has to be 30 degrees to add up to 180. And if that's 30 degrees, then this is 30 degrees. And then this thing right over here is 30 degrees. And then the last thing-- we've actually done what we said we would do, we found out all of the angles. We could also think about these outer angles. So this-- or not the outer angles, or these combined angles. So angle say AC-- or say, angle ABE, so this whole angle we see is 60 degrees. This angle is 90 degrees, and this angle here is 30. So what's interesting is these three smaller triangles, they all have the exact same angles, 30, 60, 90, and the exact same side lengths. We know that because they're congruent. And what's interesting is when you put them together this way, they construct this larger triangle, triangle ABE, that's clearly not congruent. It's a larger triangle. It has different measures for its lengths, but it has the same angles 30, 60, and then 90. So it's actually similar to all of the triangles that it's made up of.