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Triangle Congruence Using
ASA and AAS
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C ONCEPT
1
1
Triangle Congruence Using
ASA and AAS
Learning Objectives
•
•
•
•
Understand and apply the ASA Congruence Postulate.
Understand and apply the AAS Congruence Theorem.
Understand and practice two-column proofs.
Understand and practice flow proofs.
Introduction
The SSS Congruence Postulate is one of the ways in which you can prove two triangles are congruent without
measuring six angles and six sides. The next two lessons explore other ways in which you can prove triangles
congruent using a combination of sides and angles. It is helpful to know all of the different ways you can prove
congruence between two triangles, or rule it out if necessary.
ASA Congruence
One of the other ways you can prove congruence between two triangles is the ASA Congruence Postulate. The
“S” represents “side,” as it did in the SSS Theorem. “A” stands for “angle” and the order of the letters in the name
of the postulate is crucial in this circumstance. To use the ASA postulate to show that two triangles are congruent,
you must identify two angles and the side in between them. If the corresponding sides and angles are congruent, the
entire triangles are congruent. In formal language, the ASA postulate is this:
Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side in one triangle are congruent
to two angles and the included side in another triangle, then the two triangles are congruent.
To test out this postulate, you can use a ruler and a protractor to make two congruent triangles. Start by drawing a
segment that will be one side of your first triangle and pick two angles whose sum is less than 180◦ . Draw one angle
on one side of the segment, and draw the second angle on the other side. Now, repeat the process on another piece
of paper, using the same side length and angle measures. What you’ll find is that there is only one possible triangle
you could create—the two triangles will be congruent.
Notice also that by picking two of the angles of the triangle, you have determined the measure of the third by the
Triangle Sum Theorem. So, in reality, you have defined the whole triangle; you have identified all of the angles in
the triangle, and by picking the length of one side, you defined the scale. So, no matter what, if you have two angles,
and the side in between them, you have described the whole triangle.
Concept 1. Triangle Congruence Using ASA and AAS
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Example 1
What information would you need to prove that these two triangles are congruent using the ASA postulate?
A. the measures of the missing angles
B. the measures of sides AB and BC
C. the measures of sides BC and EF
D. the measures of sides AC and DF
If you are to use the ASA postulate to prove congruence, you need to have two pairs of congruent angles and the
included side, the side in between the pairs of congruent angles. The side in between the two marked angles in
4ABC is side BC. The side in between the two marked angles in 4DEF is side EF. You would need the measures
of sides BC and EF to prove congruence. The correct answer is C.
AAS Congruence
Another way you can prove congruence between two triangles is using two angles and the non-included side.
Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side in one triangle are congruent to two corresponding angles and a non-included side in another triangle, then the triangles are congruent.
This is a theorem because it can be proven. First, we will do an example to see why this theorem is true, then we will
prove it formally. Like the ASA postulate, the AAS theorem uses two angles and a side to prove triangle congruence.
However, the order of the letters (and the angles and sides they stand for) is different.
The AAS theorem is equivalent to the ASA postulate because when you know the measure of two angles in a triangle,
you also know the measure of the third angle. The pair of congruent sides in the triangles will determine the size of
the two triangles.
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Example 2
What information would you need to prove that these two triangles were congruent using the AAS theorem?
A. the measures of sides TW and XZ
B. the measures of sides VW and Y Z
C. the measures of 6 V TW and 6 Y XZ
D. the measures of angles 6 TWV and 6 XZY
If you are to use the AAS theorem to prove congruence, you need to know that pairs of two angles are congruent
and the pair of sides adjacent to one of the given angles are congruent. You already have one side and its adjacent
angle, but you still need another angle. It needs to be the angle not touching the known side, rather than adjacent to
it. Therefore, you need to find the measures of 6 TWV and 6 XZY to prove congruence. The correct answer is D.
When you use AAS (or any triangle congruence postulate) to show that two triangles are congruent, you need to
make sure that the corresponding pairs of angles and sides actually align. For instance, look at the diagram below:
Even though two pairs of angles and one pair of sides are congruent in the triangles, these triangles are NOT
congruent. Why? Notice that the marked side in 4TVW is TV , which is between the unmarked angle and the angle
with two arcs. However in 4KML, the marked side is between the unmarked angle and the angle with one arc. As
the corresponding parts do not match up, you cannot use AAS to say these triangles are congruent.
Concept 1. Triangle Congruence Using ASA and AAS
4
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AAS and ASA
The AAS triangle congruence theorem is logically the exact same as the ASA triangle congruence postulate. Look
at the following diagrams to see why.
Since 6 C ∼
= 6 Z and 6 B ∼
= 6 Y , we can conclude from the third angle theorem that 6 A ∼
= 6 X. This is because the
◦
sum of the measures of the three angles in each triangle is 180 and if we know the measures of two of the angles,
then the measure of the third angle is already determined. Thus, marking 6 A ∼
= 6 X, the diagram becomes this:
Now we can see that 6 A ∼
= 6 X (A), AB ∼
= XY (S), and 6 B ∼
= 6 Y (A), which shows that 4ABC ∼
= 4XY Z by ASA.
Proving Triangles Congruent
In geometry we use proofs to show something is true. You have seen a few proofs already—they are a special form
of argument in which you have to justify every step of the argument with a reason. Valid reasons are definitions,
postulates, or results from other proofs.
One way to organize your thoughts when writing a proof is to use a two-column proof. This is probably the most
common kind of proof in geometry, and it has a specific format. In the left column you write statements that lead
to what you want to prove. In the right hand column, you write a reason for each step you take. Most proofs begin
with the “given” information, and the conclusion is the statement you are trying to prove. Here’s an example:
Example 3
Create a two-column proof for the statement below.
Given: NQ is the bisector of 6 MNP, and 6 NMQ ∼
= 6 NPQ
Prove: 4MNQ ∼
= 4PNQ
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Remember that each step in a proof must be clearly explained. You should formulate a strategy before you begin
the proof. Since you are trying to prove the two triangles congruent, you should look for congruence between the
sides and angles. You know that if you can prove SSS, ASA, or AAS, you can prove congruence. Since the given
information provides two pairs of congruent angles, you will most likely be able to show the triangles are congruent
using the ASA postulate or the AAS theorem. Notice that both triangles share one side. We know that side is
congruent to itself (NQ ∼
= NQ), and now you have pairs of two congruent angles and non-included sides. You can
use the AAS congruence theorem to prove the triangles are congruent.
TABLE 1.1:
Statement
1.6 NQ is the bisector of 6 MNP
2.6 MNQ ∼
= 6 PNQ
Reason
1. Given
2. Definition of an angle bisector (a bisector divides an
angle into two congruent angles)
3. Given
4. Reflexive Property
5. AAS Congruence Theorem (if two pairs of angles
and the corresponding non-included sides are congruent, then the triangles are congruent)
3.6 NMQ ∼
= 6 NPQ
4.NQ ∼
= NQ
5.4MNQ ∼
= 4PNQ
Notice how the markings in the triangles help in the proof. Whenever you do proofs, use arcs in the angles and tic
marks to show congruent angles and sides.
Flow Proofs
Though two-column proofs are the most traditional style (in geometry textbooks, at least!), there are many different
ways of solving problems in geometry. We already wrote a paragraph proof in an earlier lesson that simply described,
step by step, the rationale behind an assertion (when we showed why AAS is logically equivalent to ASA). The twocolumn style is easy to read and organizes ideas clearly. Some students, however, prefer flow proofs. Flow proofs
show the relationships between ideas more explicitly by using a chart that shows how one idea will lead to the next.
Like two-column proofs, it is helpful to always remember the end goal so you can identify what it is you need to
prove. Sometimes it is easier to work backwards!
The next example repeats the same proof as the one above, but displayed in a flow style, rather than two columns.
Example 4
Create a flow proof for the statement below.
Given: 6 NQ is the bisector of 6 MNP and 6 NMQ ∼
= 6 NPQ
Prove: 4MNQ ∼
= 4PNQ
Concept 1. Triangle Congruence Using ASA and AAS
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As you can see from these two proofs of the theorem, there are many different ways of expressing the same
information. It is important that you become familiar with proving things using all of these styles because you
may find that different types of proofs are better suited for different theorems.
Lesson Summary
In this lesson, we explored triangle congruence. Specifically, we have learned to:
•
•
•
•
Understand and apply the ASA Congruence Postulate.
Understand and apply the AAS Congruence Postulate.
Understand and practice Two-Column Proofs.
Understand and practice Flow Proofs.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.
Points to Consider
Now that you have been exposed to the SAS and AAS postulates, there are even more triangle congruence postulates
to explore. The next lesson deals with SAS and HL proofs.
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Review Questions
Use the following diagram for exercises 1-3.
1. Complete the following congruence statement, if possible 4PQR ∼
= ________.
2. What postulate allows you to make the congruence statement in 1, or, if it is not possible to make a congruence
statement explain why.
3. Given the marked congruent parts, what other congruence statements do you now know based on your answers
to 1 and 2?
Use the following diagram for exercises 4-6 .
4. Complete the following congruence statement, if possible 4ABC ∼
= _______.
5. What postulate allows you to make the congruence statement in 4, or, if it is not possible to make a congruence
statement explain why.
6. Given the marked congruent parts in the triangles above, what other congruence statements do you now know
based on your answers to 4 and 5?
Use the following diagram for exercises 7-9.
Concept 1. Triangle Congruence Using ASA and AAS
8
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7. Complete the following congruence statement, if possible 4POC ∼
= ________.
8. What postulate allows you to make the congruence statement in 7, or, if it is not possible to make a congruence
statement explain why.
9. Given the marked congruent parts in the triangles above, what other congruence statements do you now know
based on your answers to 7 and 8?
10. Complete the steps of this two-column proof:
Given 6 L ∼
= 6 N, 6 P ∼
= 6 O , and LM ∼
= MN Prove: 6 PML ∼
= 6 OMN
Note: You cannot assume that P, M,and Nare collinear or that L, M,and Oare collinear.
TABLE 1.2:
Statement
1. 6 L ∼
=6 N
6
2. P ∼
=6 O
3. ________
4. 4LMP ∼
= _______
5. 6 PML ∼
= 6 OMN
Reason
1. Given
2. ________
3. Given
4. _______ triangle congruence postulate
5. ________________________________
11. Bonus question: Why do we have to use three letters to name 6 PML and 6 OMN, while we can use only one
letter to name 6 L or 6 N?
Review Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
4PQR ∼
= 4BCA
AAS triangle congruence postulate
PQ ∼
= BC, QR ∼
= CA, and 6 R ∼
=6 A
No congruence statement is possible
We can’t use either AAS or ASA because the corresponding parts do not match up
6 E ∼
= 6 B. This is still true by the third angle theorem, even if the triangles are not congruent.
4POC ∼
= 4RAM4PQR ∼
= 4BCA
ASA triangle congruence postulate
6 P∼
= 6 A, PO ∼
= RA, and PC ∼
= RM
TABLE 1.3:
Statement
1. 6 L ∼
=6 N
6
2. P ∼
=6 O
3. LM ∼
= MN
Reason
1. Given
2. Given
3. Given
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TABLE 1.3: (continued)
Statement
4. 4LMP ∼
= 4NMO
∼
6
5. PML = 6 OMN
Reason
4. AAS Triangle Congruence Postulate
5. Definition of congruent triangles (if two triangles are
∼
= then all corresponding parts are also ∼
=).
11. We can use one letter to name an angle when there is no ambiguity. So at point L in the diagram for 10 there
is only one possible angle. At point M there are four angles, so we use the “full name” of the angles to be
specific!
Concept 1. Triangle Congruence Using ASA and AAS
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