A triangle can be examined using some basic information about it: the angles between the sides and the length of the sides. We can compare two or more triangles using intriguing measurements; for instance, we can measure each triangle's HL (hypotenuse-leg), ASA (two angles and one side), and so on. This helps us recognize the similarities and congruencies of different triangles with the least information given. Let us take a look at such theorems:
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.
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:
Two congruent triangles according to AAS Theorem, StudySmarter Originals
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:
Two congruent triangles due to AAS, StudySmarter Originals
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:
Equally oriented right triangles, StudySmarter Originals
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:
Differently position right triangles, StudySmarter Originals
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
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?
Three triangles sharing one side, StudySmarter Originals
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?
Two right triangles sharing one side, StudySmarter Originals
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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