What additional information is required to prove that these two triangles are congruent using the AAS angle-angle-side rule?

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AAS or Angle Angle Side congruence rule states that if two pairs of corresponding angles along with the opposite or non-included sides are equal to each other then the two triangles are said to be congruent. The 5 congruence rules include SSS, SAS, ASA, AAS, and RHS. Let us learn more about the AAS congruence rule, the proof, and solve a few examples.

What is AAS Congruence Rule?

By definition, AAS congruence rule states that if any two angles and the non-included side of one triangle are equal to the corresponding angles and the non-included side of the other triangle. The angles are consecutive and corresponding in nature while the sides are not included between the angles but in either direction of the angles. Look at the image below, we can see the two consecutive or next to each other angles of one triangle are equal to corresponding angles of another triangle. The sides of both the triangles are not included between the angles but are consecutive to the angles, hence the sides are also equal.

AAS Congruence Rule

AAS congruence rule or theorem states that if two angles of a triangle with a non-included side are equal to the corresponding angles and non-included side of the other triangle, they are considered to be congruent. Let us see the proof of the theorem:

Given: AB = DE, ∠B=∠E, and ∠C =∠F. To prove: ∆ABC ≅ ∆DEF

If both the triangles are superimposed on each other, we see that ∠B =∠E and ∠C =∠F. And the non-included sides AB and DE are equal in length. Therefore, we can say that ∆ABC ≅ ∆DEF.

Proof of AAS Congruence Rule

To prove the AAS congruence rule or theorem, we need to first look at the ASA congruence theorem which states that when two angles and the included side (the side between the two angles) of one triangle are (correspondingly) equal to two angles and the included side of another triangle.

The AAS congruence rule states that if any two consecutive angles of a triangle along with a non-included side are equal to the corresponding consecutive angles and the non-included side of another triangle, the two triangles are said to be congruent. We should also remember that if two angles of a triangle are equal to two angles of another, then their third angles are automatically equal since the sum of angles in any triangle must be a constant 180° (by the angle sum property).

To prove the AAS congruence rule, let us consider the two triangles above ∆ABC and ∆DEF. We know that AB = DE, ∠B =∠E, and ∠C =∠F. We also saw if two angles of two triangles are equal then the third angle of both the triangle is equal since the sum of angles is a constant of 180°. Hence,

In ∆ABC, ∠A + ∠B + ∠C = 180 ------ (i)

In ∆DEF, ∠D + ∠E + ∠F = 180 -------(ii)

From (i) and (ii) we get,

∠A + ∠B + ∠C = ∠D + ∠E + ∠F

Since we already know that ∠B =∠E and ∠C =∠F, so

∠A + ∠E + ∠F = ∠D + ∠E + ∠F

∠A = ∠D

In both the triangles we know that,

AB = DE, ∠A = ∠D, and ∠C =∠F

Therefore, according to the ASA congruence rule, it is proved that ∆ABC ≅ ∆DEF.

Related Topics

Listed below are a few topics related to the AAS congruence rule, take a look.

• What is Congruence?
• Corresponding Parts of Congruent Triangles

1. Example 1: From the below image, which triangle follows the AAS congruence rule?

   Solution:

   From the above-given pairs, we can see that pair number 4 fits the AAS congruence rule where two consecutive angles with a non-included angle of one triangle are equal to the corresponding consecutive angles with a non-included side of another triangle, then the triangles are considered to be congruent. The pairs are of the other congruence rules such as,

   Pair 1 = SSS Congruency Rule

   Pair 2 = SAS Congruency Rule

   Pair 3 = ASA Congruency Rule

2. Example 2: From the below triangle, we know that ∠Q = ∠R along with right angles on both sides of the triangle. Can we prove that ∆PQS ≅ ∆PRS?

   Solution: Given,

   ∠Q = ∠R and ∠PSQ = ∠PSR = 90°

   Since both the triangles share the same perpendicular line making the length of the line the same for both triangles. Hence, the sides of both triangles are also equal. According to the AAS congruence rule, we can say that ∆PQS ≅ ∆PRS.

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FAQs on AAS Congruence Rule

The Angle Angle Side Postulate (AAS) states that if two consecutive angles along with a non-included side of one triangle are congruent to the corresponding two consecutive angles and the non-included side of another triangle, then the two triangles are congruent.

How Do You Tell if a Triangle is ASA or AAS?

Both the triangle congruence theorems deal with angles and sides but the difference between the two is ASA deals with two angles with a side included in between the angles of any two triangles. Whereas AAS deals with two angles with a side that is not included in between the two angles of any two given triangles.

What is SSS, SAS, ASA, and AAS?

The 4 different triangle congruence rules are:

• SSS: Where three sides of two triangles are equal to each other.
• SAS: Where two sides and an angle included in between the sides of two triangles are equal to each other.
• ASA: Where two angles along with a side included in between the angles of any two triangles are equal to each other.
• AAS: Where two angles of any two triangles along with a side that is not included in between the angles, are equal to each other.

What is AAS and ASA Congruence Rule?

AAS stands for angle angle side and ASA refers to angle side angle. These two are triangle congruence theorems that help in proving if two triangles are congruent or not. ASA congruence rule states that if two corresponding angles along with one corresponding side (included in between the angles) are equal to each other, the two triangles are congruent. Whereas the AAS congruence rule states that if two corresponding angles with a non-included side are equal to each other, the two triangles are equal to each other.

What Does AAS Mean in Geometry?

In geometry, AAS means angle angle side and is one of the congruence theorems among the 5 different theorems that prove the congruency of two triangles.

How Do You Find the Angle Angle Side?

In angle-angle side(AAS) if two angles and the one non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent.

Mathematics is a pure science, so you are almost never stopped on the street and challenged to test two triangles for congruence. If you were, though, you could test triangles for congruence in five ways. Knowing as many methods as possible helps you, giving you flexibility to deal with any situation, whether you are stopped on the street or stumped in the classroom. This method is the Angle Angle Side, or AAS Theorem.

Proving Congruent Triangles

Five methods exist for testing congruence in triangles, though one is restricted for use with right triangles. Here are all five:

1. Side Side Side -- SSS
2. Side Angle Side -- SAS
3. Angle Side Angle -- ASA
4. Hypotenuse Leg -- HL Reserved for right triangles
5. Angle Angle Side -- AAS Hey! That's the one we're on!

In other lessons we have illustrated the other methods, and no, we did not just randomly rearrange "Angle" and "Side" in as many ways as we could think of. Notice, for instance, you cannot find Angle Angle Angle as a congruence proof (that is reserved for similarity), nor can you cook up a Side Side Angle postulate.

Whichever term you see sandwiched between the others, that part is included. An included angle or side is physically between the others in the triangle. So Side Angle Side (SAS) means one side, the angle next to that side, and then the side next to that angle. That side is out there, all alone, not between the angles.

For every testing method, you are checking the three parts identified between the two triangles. If corresponding parts are congruent for those three parts, the two triangles are congruent. These testing methods or proofs allow you to establish congruence by checking only half the parts (from three possible sides and three possible angles).

AAS Theorem

Your textbook probably calls this a theorem, or it may be labeled a postulate; don't worry about it! Keep the concept, not the fussy words, in mind as you attempt to prove triangles congruent.

AAS Theorem Definition

The AAS Theorem says: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Notice how it says "non-included side," meaning you take two consecutive angles and then move on to the next side (in either direction). You do not take the side between those two angles! (If you did, you would be using the ASA Postulate).

To demonstrate with actual triangles, below we proudly present △GUM and △RED.

Are they congruent? Notice the little hatch marks that indicate all the congruencies, which in mathematical shorthand uses the symbol ≅.

The congruent parts are:

• ∠G ≅ ∠R
• ∠M ≅ ∠D
• Side GU ≅ Side RE

We know from these triangles that two interior angles are congruent (and consecutive, or next to each other), but we know nothing about the side between them. Instead, seemingly unhelpfully, we learn that another side is congruent.

Going through our toolbox full of triangle congruence testing methods, we can try each:

1. Side Side Side (SSS) -- That won't work, because we do not know about all three sides
2. Side Angle Side (SAS) -- That won't work, either, because we know two angles, not two sides
3. Angle Side Angle (ASA) -- This at first looks promising, but the side we know about is not an included side; it is sticking out there, past one of the two known angles
4. Hypotenuse Leg (HL) -- Forget about it! This is reserved for right triangles, which we don't have
5. Angle Angle Side (AAS) -- That's the ticket! This is the one (the only one) that we can use!

Why does the AAS Theorem work?

Quick, what do the interior angles of all triangles add up to?

We hope you said 180°, because that is the correct answer. If you know two angles of a triangle, then you know three angles of a triangle. That is not magic; it's mathematics:

Solving for ∠U now gives you two angles with an included side. Did you see that? We did an end run around that side that was just sticking out there, all alone, and put it between two identified angles, ∠G and ∠U. So, where once we had AAS, we scooted around the triangle and turned it into ASA, which is already a postulate.

If two angles and their included side of one triangle are all congruent to two corresponding angles and their included side of another triangle, the two triangles are congruent.

AAS Theorem Example

Here we offer two new triangles, △LEG and △ARM. Notice all the little hatch marks indicating congruent angles and sides:

∠L ≅ ∠A

∠E ≅ ∠R

Side LG ≅ Side AM

Knowing the interior angles are congruent as listed, what else do you know?

We hope you said that ∠G ≅ ∠M, because:

• 180° - ∠L - ∠E = ∠G
• 180° - ∠A - ∠R = ∠M
• ∠G ≅ ∠M

What does that allow you to do now? Deploy ASA and declare the two triangles congruent, since:

∠L ≅ ∠A

Side LG ≅ Side AM

∠G ≅ ∠M

You have no need of proving the third angle's congruence and then deploying ASA, since we have, ready and waiting, the AAS Theorem. So real mathematicians and geometricians just leap right to AAS and declare the two triangles congruent.

If you have to explain this theorem to another student, friend, or random stranger on the street, you cannot make the leap from two angles to the mysterious third angle without some explanation. Then you may need to explain how we are essentially giving up on one of our original angles in favor of the third angle.

It is that mental shift, from a given angle to the newly identified third angle, that allows you to tap the awesome power of ASA and gather our previously outlying side into the proof.

Finally, after walking your pal through those steps, hit 'em with the efficiency and even more awesome power of AAS, where any two angles and a non-included side can be used to identify congruence between triangles. Pretty impressive, isn't it?

Lesson Summary

Now that you have tinkered with triangles and studied these notes, you are able to recall and apply the Angle Angle Side (AAS) Theorem, know the right times to to apply AAS, make the connection between AAS and ASA, and (perhaps most helpful of all) explain to someone else how AAS helps to determine congruence in triangles.

Next Lesson:

HA Theorem