We've just studied two postulates that will help us prove congruence between triangles. However, these postulates were quite reliant on the use of congruent sides. In this section, we will get introduced to two postulates that involve the angles of triangles much more than the SSS Postulate and the SAS Postulate did. Understanding these four postulates and being able to apply them in the correct situations will help us tremendously as we continue our study of geometry. Let's take a look at our next postulate.
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
In a sense, this is basically the opposite of the SAS Postulate. The SAS Postulate required congruence of two sides and the included angle, whereas the ASA Postulate requires two angles and the included side to be congruent. An illustration of this postulate is shown below.
We conclude that ?ABC??DEF by the ASA Postulate because the triangles' two angles and included side are congruent.
Let's practice using the ASA Postulate to prove congruence between two triangles.
Let's start off this problem by examining the information we have been given. Since segments PQ and RS are parallel, this tells us that we may need to use some of the angle postulates we've studied in the past. Now, let's look at the other piece of information we've been given. We know that ?PRQ is congruent to ?SQR. Let's further develop our plan of attack.
We have been given just one pair of congruent angles, so let's look for another pair that we can prove to be congruent. We can say ?PQR is congruent to ?SQR by the Alternate Interior Angles Postulate. Recall, we can only use this postulate when a transversal crosses a set of parallel lines. In this case, our transversal is segment RQ and our parallel lines have been given to us.
Now that we've established congruence between two pairs of angles, let's try to do something with the included side. The included side is segment RQ. By using the Reflexive Property to show that the segment is equal to itself, we now have two pairs of congruent angles, and common shared line between the angles. Our new illustration is shown below.
We conclude our proof by using the ASA Postulate to show that ?PQR??SRQ. Let's look at our two-column geometric proof that shows the arguments we've made.
Aside from the ASA Postulate, there is also another congruence postulate that involves two pairs of congruent angles and one pair of congruent sides. Let's take a look at this postulate now.
If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
In order to use this postulate, it is essential that the congruent sides not be included between the two pairs of congruent angles. If the side is included between the angles, we would actually need to use the ASA Postulate. The correct use of the AAS Postulate is shown below.
We conclude that ?ABC??DEF by the AAS Postulate since we have two pairs of congruent angles and one pair of congruent sides not included between the angles.
Let's use the AAS Postulate to prove the claim in our next exercise.
Before we begin our proof, let's see how the given information can help us. We have been given that ?NER??NVR, so that is one pair of angles that we do not need to show as congruent.
Now, we must decide on which other angles to show congruence for. We may be able to derive a key component of this proof from the second piece of information given. Since segment RN bisects ?ERV, we can show that two congruent angles are formed. By the definition of an angle bisector, we have that ?ERN??VRN.
The only component of the proof we have left to show is that the triangles have congruent sides. Luckily for us, the triangles are attached by segment RN. So, we use the Reflexive Property to show that RN is equal to itself. Let's look at our new figure.
Finally, by the AAS Postulate, we can say that ?ENR??VNR. Note that our side RN is not included. If it were included, we would use the ASA Postulate to prove that the triangles are congruent. The two-column proof for this exercise is shown below.