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### Course: Integrated math 1>Unit 17

Lesson 2: Triangle congruence from transformations

# Proving the ASA and AAS triangle congruence criteria using transformations

We can prove the angle-side-angle (ASA) and angle-angle-side (AAS) triangle congruence criteria using the rigid transformation definition of congruence. Created by Sal Khan.

## Want to join the conversation?

• hey y'all:
i just wanted to post something for anyone who wanted a quick conclusion/recap on the aas, sas, and sss theorems. def not as in-depth as sal tho!

sss: if all the sides have equal length, then the triangles are congruent. ex: in both △abc and △def, the side lengths are 3, 4, 6: they r congruent.

sas: if you have two sides that have the same length in both triangles and an angle joining them that is also the same length on the other triangle, the third side will have to be the same length on both triangles, and therefore the triangles are congruent. that was a lot of words so ex: △ghi has sides 3, 2, x, and an angle of 45 degrees joining the 3 and 2 side, and △jkl sides 3, 2, y, and an angle of 45 degrees joining the 3 and the 2 side, then the angles will line up on both triangles. if you draw a line from h to i and j to l, you will see that they match up. congruent!

aas: if two angles are the same on both triangles, then the third angle will be the same and they will be congruent. ex: both △mno and △pqr have angles 50, 70, and x degrees. since we know that all angles in a triangle add up to 180 degrees, 50 + 70 = 120 and 180 - 120 = 60, leaving us with 50, 70, and 60 degree angles in both triangles. from there, we just connect the points to form congruent line segments, and this turns into the sss theorem, which we already proved.

aaaaaaaah that was a lot of typing lol
hope this helps and don't give up!
• Why is he over-complicating such a simple series of rigid transformations to map triangle a to b, I was able to make that map in 5 seconds in my head without any rigorous proof...
• Math doesn't work like this, you can't just look at a picture and say anythinig without proving it. If triangles look the same, it doesn't mean that they are equiavalent. It's very important to understand every concept in math, even the most bacis one. And what he says in the videos is not obvious at all...
• would it be correct, to call AAS congruence, the same as SAA congruence, (sorry im a bit rusty in triangles)
• Yes, you could do that. You can think about reading the triangle from right to left, where you get either AAS or SAA, or from left to right, where you would get the other. As long as the middle letter is between the left and right letter on the triangle, it works.
• Since this video explains that both of them work, what is the difference? I mean, you can't just have 2 repeating congruence criteria, right?
• Once a triangle is proved congruent, all of the congruence theorems will work. Which ones you can use to determine congruence in the first place is purely based on what is given in the problem.
• I know this if off topic but, where can I learn to draw triangles of specific lengths, using only a ruler and a compass?
• Why isn't there a `SAA` theorem?
(1 vote)
• SAA is the exact same thing as AAS, so you do not need both.
• im still a little confused is SSA and AAS the same thing
• Nope.SSA is side-side-angle, and AAS is angle-angle-side.
• Why is it important to use a projactor? Why cant you just use a ruler?
• Using a protractor helps us determine the angle measurement so we can label it as acute, right or obtuse. Every protractor is a little bit different, but all will have a location on the bottom edge where we align the vertex of the angle we are measuring.
• I still don't get what AAS means, can you explain it. I kind of found Sal's explanation confusing.
(1 vote)
• AAS means that if two triangles have two pairs of congruent angles and a pair of congruent sides (and the sides are not the sides between the angles), then the triangles are congruent.

If two triangles have two angles in common, they must have the third angle in common as well, since the angles of a triangle sum to 180. So the triangles are definitely similar; that is, they look the same, possibly resized.

Then having a side in common guarantees that the triangles are in fact the same size as well. This is true regardless of where the side is in the triangles, which is why AAS and ASA are both theorems.