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### Course: Class 8 (Foundation) > Unit 12

Lesson 3: Congruence of triangles# 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.

## Want to join the conversation?

- Who standardized all the notations involved in geometry?(219 votes)
- Elucid did. There is a video at the beginning of geometry about Elucid as the father of Geometry called "Elucid as the father of Geometry."

Hope that helps!(36 votes)

- How do we know what name should be given to the triangles?(13 votes)
- 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.(12 votes)

- Who created Postulates, Theorems, Formulas, Proofs, etc. and why??(14 votes)
- multiple people, to explain what was the best way to use mathematics.(5 votes)

- What does postulate mean?(6 votes)
- A postulate is a statement that is assumed true without proof. A theorem is a true statement that can be proven.

-Source Internet-(6 votes)

- what is sss criterion?(3 votes)
- It stands for "side-side-side". If two triangle both have all of their sides equal (that is, if one triangle has side lengths a, b, c, then so does the other triangle), then they must be congruent.(12 votes)

- 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 :)(6 votes)- 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 :)(3 votes)

- Are all congruent triangles similar?(4 votes)
- Yes, all congruent triangles are similar. Triangles can be called similar if all 3 angles are the same. This is true in all congruent triangles.(4 votes)

- Can someone help me summarize this? I'm really confused right now.(2 votes)
- Definition: Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.

One way to think about triangle congruence is to imagine they are made of cardboard. They are congruent if you can slide them around, rotate them, and flip them over in various ways so they make a pile where they exactly fit over each other.

Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of them to show that two triangles are congruent. Various groups of three will do. Triangles are congruent if:

SSS (side side side)

All three corresponding sides are equal in length.

See Triangle Congruence (side side side).

SAS (side angle side)

A pair of corresponding sides and the included angle are equal.

See Triangle Congruence (side angle side).

ASA (angle side angle)

A pair of corresponding angles and the included side are equal.

See Triangle Congruence (angle side angle).

AAS (angle angle side)

A pair of corresponding angles and a non-included side are equal.

See Triangle Congruence (angle angle side).

HL (hypotenuse leg of a right triangle)

Two right triangles are congruent if the hypotenuse and one leg are equal.

See Triangle Congruence (hypotenuse leg).

Hope this helps!

Calc-Ya-Later!(7 votes)

- when did descartes standardize all of the notations in geometry?(4 votes)
- I am pretty sure it was in 1637(2 votes)

- 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?(3 votes)
- 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.(3 votes)

## 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.