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.

## ASA Postulate (Angle-Side-Angle)

*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.

### Exercise 1

**Solution:**

Let’s start off this problem by examining the information we have been given. Since

segments ** PQ** and

**are parallel, this tells us that**

*RS*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

**is congruent**

*?PRQ*to

**. Let’s further develop our plan of attack.**

*?SQR* 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

**by the**

*?SQR***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

**and our parallel lines**

*RQ*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.

## AAS Postulate (Angle-Angle-Side)

*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.

### Exercise 2

**Solution:**

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

**, we can show that two**

*?ERV*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

**is equal**

*RN*to itself. Let’s look at our new figure.

Finally, by the **AAS Postulate**, we can say that ** ?ENR??VNR**. Note

that our side

**is not included. If it were included, we would use**

*RN*the

**ASA Postulate**to prove that the triangles are congruent. The two-column

proof for this exercise is shown below.