## ASA, AAS, and HL Congruence Theorems in Geometry

### What is HL?

This is the only theorem meant explicitly for determining the congruence of right triangles. But what is the meaning of HL? This acronym stands for "Hypotenuse-Leg." This means that if the hypotenuse and the respective leg are equal between two or more triangles, triangles, then the given triangles are congruent. You can think of HL the same as SSS for right triangles. Why do you think this is the case?

*I’ll give you a hint - there is a famous theorem that explains this.*

As you probably already know - this is because of the Pythagoras theorem.

$Hypotenus{e}^{2}=Le{g}^{2}+Le{g}^{2}\phantom{\rule{0ex}{0ex}}or\phantom{\rule{0ex}{0ex}}A{B}^{2}=A{C}^{2}+B{C}^{2}$

If you know two sides of a right triangle, you can precisely calculate the third side using this formula. This means that if two sides in a right triangle are equal to the respective sides of another right triangle, both triangles are congruent.

__If you can calculate the third side given any two sides of a right triangle, why is this condition for congruence called HL and not SS (as in Side-Side)?__

This is an excellent question to test your understanding of right triangles. Maybe you can think of the answer to this one?

First, SS could be mistaken for a universal condition of congruence, not only for right triangles, because the letters are very similar to other conditions - SSS, SAS, etc. HL stands out. But this is pretty trivial.

The main reason for the necessity of hypothenuse and leg for this condition is because, given a right triangle with two known legs, the condition would be SAS, that is, Side-Angle-Side. If this seems confusing, remember that we are talking about right triangles, which means we already know the angle between the two legs. This is a right angle, exactly 90º. So, if you know the length of both legs is equal to that of another right triangle, both triangles are congruent given the condition SAS.

HL is unique to right triangles - SAS is universal to all triangles. It’s important to note that when using the condition HL, the angle between the leg and hypothenuse may not be known and isn’t needed to prove congruence to another right triangle.

It’s always important to distinguish the legs from the hypotenuse to correctly use the condition HL for evaluating congruence. Let’s jump further!

### What is ASA?

The meaning for ASA is Angle-Side-Angle. This theorem for congruence tells that if two or more triangles have one equal side with equal angles on both ends of this side, then the given triangles are congruent. Check out the picture below to understand this better:

For ASA to work, the respective angles must be equal and located on both sides of the respective side.

## AAS Congruence Theorem

Angle-Angle-Side or AAS for short. As the name suggests, one angle must be at one end of a side, and the other angle is the "free" angle. This means that the angle not attached on one end of the respective side will be opposite it. Here's a picture to clarify:

The "free" angle is directly above the respective side - it's not attached to the other end of the side. So AAS can be formulated like this: if one side, an angle on one end of it, and an angle opposite to it are equal between two or more triangles, then these triangles are congruent.

## AAS Theorem

The AAS Theorem is the same as the AAS Congruence Theorem.

## ASA, AAS, and HL Examples

Let's see how we can use these theorems through some examples!

Three right triangles are given. All of them have equal hypotenuses. Does this mean that all of them are congruent? Here’s a picture to help you understand this a little better:

As you can see, equal hypotenuses don’t mean congruence right away. You also need to know the length of at least one leg from every triangle to prove congruence or non-congruence.

Two right triangles are positioned opposite one another like this:

The hypotenuses are equal, and the bottom leg of the triangle on the right is equal to the top leg of the triangle on the left. Are these triangles congruent?

As you can see – HL can be used to prove congruence. For both triangles, the hypotenuse and the respective leg are equal. This can be seen better if you rotate one triangle 180º. So we can now say

$\u2206ABC\cong \u2206DEF\cong \u2206GHI$

Three right triangles are given in the picture below. Assume no units of measurement, only numbers.

Info on the given triangles below.

ABC: AB = 10, BC = 5

DEF: DE = 10, EF = 5

GHI: IG = 5, HI = 10

Can congruence be proven in this example?

In this example, congruence can only be proven between the triangles ABC and DEF because IG and HI are both legs of the triangle GHI – the hypotenuse GH is approximately 11.18 in length.

ABC ≅ DEF

Three triangles share one side. It is known that the two of them have equal respective angles. Are all of the given triangles congruent?

All the triangles share the side AB, so we don't need to know the length of any side to continue proving congruency. We also don't know any angles of the given triangles, but we don't need to know the precise values of those too. Knowing that two triangles have two equal respective angles is sufficient. These angles are DAB, DBA, CAB, and CBA. As you can see - these equal angles are positioned on both ends of the shared side AB. Using ASA, we can prove congruency between two of the given triangles, which are ACB and ADB:

ACB ≅ ADB

To fully answer the question asked in this example - no, all of the given triangles aren't congruent. Only two of them are.

Two right triangles share the same hypotenuse and have one equal acute angle. Are these triangles congruent?

Both right triangles share the hypotenuse. Both acute angles in a right triangle are always at each end of the hypotenuse, and the right angle is always opposite the hypotenuse. So, two angles: one at the end of the shared side, the other - directly opposite it. All the respective angles are equal, and the side is shared. This gives us all the necessary info for the AAS theorem to prove congruency between both triangles:

ACB ≅ ADB

## HL, ASA, and AAS - Key takeaways

- HL is the only theorem meant explicitly for determining the congruence of
**right triangles.** - HL is derived from Hypotenuse-Leg and means if the hypotenuse and the respective leg are equal between two or more triangles, then the given triangles are congruent;
- ASA means Angle-Side-Angle and tells us that if two or more triangles have one equal side with equal angles on both ends, then the given triangles are congruent;
- AAS comes from Angle-Angle-Side and means if one side, an angle on one end of it, and an angle opposite to it are equal between two or more triangles, then these triangles are congruent.

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##### Frequently Asked Questions about HL ASA and AAS

What is ASA, AAS and HL?

ASA, AAS and HL are ** theorems** for proving triangle

__. All of the above are abbreviations that mean__

**congruence***Angle-Side-Angle*,

*Angle-Angle-Side*and

*Hypotenuse-Leg*, respectively. HL proves congruence exclusively between right triangles.

Is AAS same as HL?

AAS is ** not** the same as HL. AAS means

*Angle-Angle-Side*and is universal for proving congruence between

__of triangle. HL means__

**any type***Hypotenuse-Leg*and is

**.**

__unique to right triangles__Is AAS and ASA same?

AAS and ASA are ** not** the same. AAS means

*Angle-Angle-Side*, but ASA means

*Angle-Side-Angle*. Both are different theorems for proving triangle congruence.

How to find HL, ASA and AAS?

You can find which theorem to use for proving congruence between two or more triangles depending on the information given about the triangles in question.

What is HL, ASA and AAS theorem?

HL, ASA and AAS theorems are theorems for __ proving congruence__ between two or more triangles.

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