[G.CO.8] Congruence #8

Objective

Common Core Text:

  • [G.CO.8] Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Said Differently:

    • That's a pretty good way to say it. Also know how to use the criteria for triangle congruence to solve problems

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Explanation

In the last objective, we talked about using CPCTC to find information about parts of triangles if we know they are congruent. This was useful because if we knew two triangles we're congruent, we instantly knew 6 parts (3 sides, 3 angles) that would be congruent too.

  • Congruent Triangles → Congruent Parts

However, in reality, how would someone know that triangles are congruent if they didn't already know that some of the parts were congruent? In this objective, we're going to talk about using congruent parts to figure out if triangles are congruent.

  • Congruent Triangles ← Congruent Parts

So let's stop and think. Why do we care if two triangles are congruent? Because it tells us which parts are congruent. But if we need to know parts are congruent before we can know triangles are congruent, aren't we just going in circles?

Here's what makes it all worthwhile:

  • We don't need to know that ALL 6 parts are congruent to know that two triangles are congruent. Only three!!!!

  • That means that if I can show 3 parts are congruent, I get the other 3 for free!!

Its very rare that you get 3 pieces of information for free in math, so this is pretty cool.

Unfortunately, its not just any three parts. It needs to meet one of the three criteria for triangle congruence.

Let's look at the 6 possible combinations. I'll see if I can draw two triangles that aren't congruent.

Before you can understand these, you should make sure you understand the word included.

AAA (Angle - Angle - Angle): three angles

In this picture,

  • ∠A ≅ ∠X

  • ∠B ≅ ∠Y

  • ∠C ≅ ∠Z

I managed to make two triangles with 3 congruent angles that aren't congruent. The one on the right is way bigger. Therefore, AAA is not a criteria for congruence

  • AAA does not guarantee congruence

AAS (Angle - Angle - Side): two angles and a non-included side

This is a tricky one. On one hand, I could draw this

In this picture,

  • ∠A ≅ ∠X

  • ∠B ≅ ∠Y

  • ≅

Well, that's two angles and a side that isn't between them, yet the triangle on the right is clearly bigger. So I guess AAS doesn't guarantee congruent triangles right?

Something's fishy here. The problem is, I used the side on the right from my first triangle, and the side on the left from my second triangle. I didn't use corresponding sides. Can I do that?

Well, its kinda up to you. If you do, then AAS clearly doesn't guarantee congruence. On the other hand, if you only use corresponding sides, AAS will guarantee congruence. That's a lot to remember, and that's why nobody likes AAS. In fact, in Britian

Fortunately, we don't even need to worry about AAS. Why? Because if we know two angles are congruent, we know the third angle must be congruent as well (because of the Triangle Sum Theorem). So, hey, if you want to skip the confusion of AAS, just do the following.

    • When you see a triangle with two angles congruent and a non-included side congruent

      • just make the third angle congruent (Triangle Sum Theorem says you can)

      • now you can just check for congruence using ASA instead!

Fun Fact: That's why in Great Briton, they don't even write AAS or ASA. They just combine it into AAcorrS

TLDR; Leave AAS alone, make all angles correspond and use ASA instead.

  • AAS does not guarantee congruence

ASA (Angle - Side - Angle): two angles and an included side

In this picture,

  • ∠A ≅ ∠X

  • ∠B ≅ ∠Y

  • ≅

One cannot make a picture with ASA where the triangles aren't congruent. If there are two angles and an included side, the two triangles will always be congruent

  • ASA guarantees congruence.

SSA (Side - Side - Angle): two sides and a non-included angle

This one is even more complicated than AAS! Way more complicated! You can check out what Wikipedia says about it here, but I'd recommend forgetting this about this one until college

SAS (Side - Angle - Side)

In this picture,

  • ∠A ≅ ∠X

  • ≅

  • ≅

One cannot make a picture with SAS where the triangles aren't congruent. If there are two sides and an included angle, the two triangles will always be congruent

  • SAS guarantees congruence.

SSS (Side - Side - Side)

In this picture,

  • ≅

  • ≅

  • ≅

One cannot make a picture with SSS where the triangles aren't congruent. If there are three sides congruent, the two triangles will always be congruent

  • SSS guarantees congruence.

Conclusion

When we cut out the one's that don't work, we're left with three conditions that guarantee congruence.

These are:

    • ASA (Angle-Side-Angle) two angles and an included side

      • Here, our three parts are two angles, and a side between them

    • SAS (Side-Angle-Side) two sides and an included angle

      • Here, our three parts are two sides, and an angle between them

    • SSS (Side-Side-Side) three sides

      • Here, our three parts are all three sides of the triangle