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### Course: Geometry (FL B.E.S.T.)>Unit 4

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?

• 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?
• 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 isn't there a `SAA` theorem?
(1 vote)
• SAA is the exact same thing as AAS, so you do not need both.
• 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.
• 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.
• Sal mentioned something interesting at , when he said that we know the measure of the third angle because all angles of a triangle add up to 180°.

Couldn't we use the Pythagorean theorem to figure out the last side measurement in SAS triangles?
• The issue is that Pythagorean Theorem is used when you have a right triangle, and there is no evidence to assume this to be true in the diagram. Further, you are given generalities, and Pythagorean Theorem is used when you know 2 sides of a right triangle and looking for the third side.
• What's the difference between ASA and AAS
(1 vote)
• ASA = Angle, Side, Angle
AAS = Angle, Angle, Side

For ASA, you can have the side with the angle on each of the two outsides, and by definition of a triangle, the third side can be connected

For AAS, one of the angles isn't in between the sides, or it doesn't have to be. This again leads to the connection of the third side by definition of a triangle.

Hope this helps,