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Corresponding parts of congruent triangles are congruent

When two triangles are congruent, we can know that all of their corresponding sides and angles are congruent too! Created by Sal Khan.

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  • leaf grey style avatar for user Shlomo Fingerer
    Who standardized all the notations involved in geometry?
    (204 votes)
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  • leaf orange style avatar for user Saman Israr
    How do we know what name should be given to the triangles?
    (12 votes)
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    • piceratops ultimate style avatar for user Lucian Xenos
      As far as I am aware, Pira's terminology is incorrect. The three types of triangles are Equilateral for all sides being equal length, Isosceles triangle for two sides being the same length and Scalene triangle for no sides being equal.

      Also, depending on the angles in a triangle, there are also obtuse, acute, and right triangle.
      (10 votes)
  • male robot hal style avatar for user Dylan Lam
    Who created Postulates, Theorems, Formulas, Proofs, etc. and why??
    (12 votes)
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  • starky ultimate style avatar for user Dev sapariya
    what is sss criterion?
    (2 votes)
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  • duskpin ultimate style avatar for user Mukta Patwari
    What does postulate mean?
    (4 votes)
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  • leaf orange style avatar for user August Berning
    I need some help understanding whether or not congruence markers are exclusive of other things with a different congruence marker. Is a line with a | marker automatically not congruent with a line with a || marker? Or is it just given that |s and |s are congruent and it doesn't rule out that |s may be congruent to ||s?

    Here is an example from a curriculum I am studying a geometry course on that I have programmed. The curriculum says the triangles are not congruent based on the congruency markers, but I don't understand why: https://www.khanacademy.org/computer-programming/a-graph-for-my-schoolwork-again/5793113371787264

    FYI, this is not advertising my program. This is the only way I can think of displaying this scenario. I also believe this scenario forces the triangles to be isosceles (the triangles are not to scale, so please take them for the given markers and not the looks or coordinates).

    As you can see, the SAS, SSS, and ASA postulates would appear to make them congruent, but the )) and ))) angles switch. Does that just mean ))s are congruent to )))s?

    I hope I haven't been to long and/or wordy, thank you to whoever takes the time to read this and/or respond! I will confirm understanding if someone does reply so they know if what they said sinks in for me :)
    (4 votes)
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    • aqualine ultimate style avatar for user Sabrina McNeil
      I think that when there is a single "|" it is meant to show that the line it's sitting on will only be congruent with another line that has a single "|" dash, when there are two "||" the line is congruent with another "||", etc.
      As for your math problem, the only reason I can think of that would explain why the triangles aren't congruent has to do with the lack of measurements. Since there are no measurements for the angles or sides of either triangle, there isn't enough information to solve the problem; you need measurements of at least one side and two angles to solve that problem. Since there are no measurements given in the problem, there is no way to tell whether or not the triangles are congruent, which leads me to believe that was meant to be a trick question in your curriculum.
      I hope that helped you at least somewhat :)
      (1 vote)
  • mr pants teal style avatar for user Gabrielle Phillip
    Are all congruent triangles similar?
    (3 votes)
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  • starky tree style avatar for user Greg Robbins
    Trick question about shapes... Would the Pythagorean theorem work on a cube? And if so- how would you do it? would it work on a pyramid... why or why not?
    (2 votes)
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    • leaf red style avatar for user Anthony Silva
      You can actually modify the the Pythagorean Theorem to get a formula that involves three dimensions, as long as it works with a rectangular prism. Let a, b and c represent the side lengths of that prism. You should have a^2+b^2+c^2=d^2. d would represent the length of the longest diagonal, involving two points that connected by an imaginary line that goes front to back, left to right, and bottom to top at the same time.
      (2 votes)
  • hopper jumping style avatar for user Ishan Khare
    What are corresponding angles?
    (0 votes)
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  • blobby green style avatar for user kikirabbit23
    Could we also call it the SSS congruence instead of calling it the SSS postulate?
    (2 votes)
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Video transcript

- [Instructor] Let's talk a little bit about congruence, congruence. And one way to think about congruence, it's really kind of equivalence for shapes. So when, in algebra, when something is equal to another thing, it means that their quantities are the same. But, if we're now all of a sudden talking about shapes, and we say that those shapes are the same, the shapes are the same size and shape, then we say that they're congruent. And just to see a simple example here, I have this triangle right over there, and let's say I have this triangle right over here. And, if you are able to shift, if you are able to shift this triangle and rotate this triangle and flip this triangle, you can make it look exactly like this triangle, as long as you're not changing the lengths of any of the sides or the angles here. But you can flip it, you can shift it and rotate it. So you can shift, let me write this, you can shift it, you can flip it, you can flip it and you can rotate. If you can do those three procedures to make the exact same triangle and make them look exactly the same, then they are congruent. And, if you say that a triangle is congruent, and let me label these. So let's call this triangle A, B and C. And let's call this D, oh let me call it X, Y and Z, X, Y and Z. So, if we were to say, if we make the claim that both of these triangles are congruent, so, if we say triangle ABC is congruent, and the way you specify it, it looks almost like an equal sign, but it's an equal sign with this little curly thing on top. Let me write it a little bit neater. So we would write it like this. If we know that triangle ABC is congruent to triangle XY, XYZ, that means that their corresponding sides have the same length, and their corresponding angles, and their corresponding angles have the same measure. So, if we make this assumption, or if someone tells us that this is true, then we know, then we know, for example, that AB is going to be equal to XY, the length of segment AB is going to be equal to the length of segment XY. And we could denote it like this. And I'm assuming that these are the corresponding sides. And you can see it actually by the way we've defined these triangles. A corresponds to X, B corresponds to Y, and then C corresponds to Z right over there. So AB, side AB, is going to have the same length as side XY, and you can sometimes, if you don't have the colors, you would denote it just like that. These, these two lengths, or these two line segments, have the same length. And you can actually say this, and you don't always see it written this way, you could also make the statement that line segment AB is congruent, is congruent to line segment XY. But congruence of line segments really just means that their lengths are equivalent. So these two things mean the same thing. If one line segment is congruent to another line segment, that just means the measure of one line segment is equal to the measure of the other line segment. And so, we can go through all the corresponding sides. If these two characters are congruent, we also know, we also know that BC, we also know the length of BC is going to be the length of YZ, assuming that those are the corresponding sides. And we could put these double hash marks right over here to show that this one, that these two lengths are the same. And then, if we go to the third side, we also know that these are going to have the same length, or the line segments themselves are going to be congruent. So we also know that the length of AC, the length of AC is going to be equal to the length of XZ, is going to be equal to the length of XZ. Not only do we know that all of the corresponding sides are going to have the same length, if someone tells us that a triangle is congruent, we also know that all the corresponding angles are going to have the same measure. So, for example, we also know, we also know that this angle's measure is going to be the same as the corresponding angle's measure, and the corresponding angle is right over here. It's between this orange side and this blue side, or this orange side and this purple side, I should say, in between the orange side and this purple side. And so, it also tells us that the measure, the measure of angle, what's this, BAC, measure of angle BAC, is equal to the measure of angle, of angle YXZ, the measure of angle, let me write that angle symbol a little less like a, measure of angle YXZ, YXZ. We can also write that as angle BAC is congruent to angle YXZ. And, once again, like line segments, if one line segment is congruent to another line segment, it just means that their lengths are equal. And, if one angle is congruent to another angle, it just means that their measures are equal. So we know that those two corresponding angles have the same measure, they're congruent. We also know that these two corresponding angles have the same measure. I'll use a double arc to specify that this has the same measure as that. So we also know that the measure, the measure of angle ABC, ABC, is equal to the measure of angle XYZ, XYZ. And then, finally, we know, we finally, we know that this angle, if we know that these two characters are congruent, that this angle's going to have the same measure as this angle, as its corresponding angle. So we know that the measure of angle ACB, ACB, is going to be equal to the measure of angle XZY, XZY. Now, what we're gonna concern ourselves a lot with is how do we prove congruence 'cause it's cool. 'Cause if you can prove congruence of two triangles, then all of a sudden you can make all of these assumptions.