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The SSS and SAS Congruence Postulates are two 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.
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 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.
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 and
C. the measures of sides and
D. the measures of sides and
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 is side . The side in between the two marked angles in is side You would need the measures of sides and to prove congruence. The correct answer is C.
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.
Example 2
What information would you need to prove that these two triangles were congruent using the AAS theorem?
A. the measures of sides and
B. the measures of sides and
C. the measures of and
D. the measures of angles and
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 and 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 is which is between the unmarked angle and the angle with two arcs. However in , 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.
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 and , we can conclude from the third angle theorem that . This is because the sum of the measures of the three angles in each triangle is and if we know the measures of two of the angles, then the measure of the third angle is already determined. Thus, marking , the diagram becomes this:
Now we can see that (A), (S), and (A), which shows that by ASA.
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: is the bisector of , and
Prove:
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 , 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.
Statement | Reason |
1. is the bisector of |
1. Given |
2. |
2. Definition of an angle bisector (a bisector divides an angle into two congruent angles) |
3. |
3. Given |
4. |
4. Reflexive Property |
5. |
5. AAS Congruence Theorem (if two pairs of angles and the corresponding non-included sides are congruent, then the triangles are congruent) |
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.
In this lesson, we explored triangle congruence. Specifically, we have learned to:
These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.
Now that you have been exposed to the SAS and AAS postulates, there is one more triangle congruence postulates to explore. The next lesson deals with the HL theorem.
The following questions are for your own benefit. The answers are below for you to check your work and understanding.
Use the following diagram for exercises 1-3.
Use the following diagram for exercises 4-6 .
Use the following diagram for exercises 7-9.
Given , and
Prove:
Note: You cannot assume that and are collinear or that and are collinear.
Statement | Reason |
1. | 1. Given |
2. | 2. ________ |
3. ________ | 3. Given |
4. _______ | 4. _______ triangle congruence postulate |
5. | 5. ________________________________ |
Statement | Reason |
1. |
1. Given |
2. |
2. Given |
3. |
3. Given |
4. |
4. AAS Triangle Congruence Postulate |
5. |
5. Definition of congruent triangles (if two triangles are then all corresponding parts are also ). |