In this chapter we will learn AAS postulate of congruency with solved examples.

Let us first review the basics of congruency.

## What is congruency of triangles?

Two triangles are said to be congruent when** they have equal sides and angles.**

When the congruent triangles are placed against each other they will overlap fully.

## What is AAS theorem ?

AAS stands for **Angle – Angle – Side**.

When the **two angles of triangle and a non included sides are equal **then the given triangles are congruent.

Non included side is the one which do not lie between the equal angles.

**For Example;**

Consider the two triangles below;

In triangle ABC and PQR;

∠B = ∠Q = 60 degree

∠ C = ∠R = 55 degree

AB = PQ = 3.5 cm

Hence, by AAS congruency both the triangles are congruent.

▵ABC \mathtt{\cong } ▵PQR.

**Note:**

Here the side AB & PQ are non included sides.

This means that the sides does not lie between the equal angles.

I hope you understand the AAS concept of congruency. Let us now solve some problems for better understanding.

## AAS Postulate – Solved Problems

(01) Observe the below image. Prove that ▵ABD \mathtt{\cong } ▵ACD.

**Solution**

Taking triangle ABD and ACD;

∠ BAD = ∠CAD

∠ ABD = ∠ACD

BD = CD

By AAS congruency, both the triangle are congruent.

Hence, ▵ABD \mathtt{\cong } ▵ACD.

(02) Study the below image and prove ▵ABO \mathtt{\cong } ▵DCO

**Solution**

Taking triangle ABO and DCO.

∠ ABO = ∠DCO { given }

∠ AOB = ∠DOC { Vertically opposite angle }

AB = CD { given in image }

By AAS congruency, both the triangles are congruent.

▵ABO \mathtt{\cong } ▵DCO

**(03) Prove triangle ABD and CBD are congruent.**

**Solution**

Taking triangle ABD and CBD.

∠ BAD = ∠BCD { given }

∠ BDA = ∠BDC = 90 degree

BD = DB { common side }

By AAS congruency, both triangles are congruent.

Hence, ▵ADB \mathtt{\cong } ▵CDB

(04) In the below image line BO bisect ∠AOC. Prove that ▵ONW \mathtt{\cong } ▵OMW

**Solution**

As line BO bisect ∠AOC, we can write;

∠WON = ∠ WOM

Taking triangle ONW and OMW.

∠ONW = ∠OMW { both measure 90 degree }

∠WON = ∠ WOM

OW = WO { common side }

By AAS congruency, both the triangles are congruent.

▵ONW \mathtt{\cong } ▵OMW

**(05) In the below image line AB || CD. Prove that triangle ABD is congruent to triangle CDB.**

**Solution**

Taking triangle ABD and CDB.

∠ ADB = ∠ CBD { alternate interior angle }

∠ ABD = ∠ CDB { alternate interior angle }

AD = BC { given in image}

By AAS congruency theorem, both triangles are congruent.

Hence, ▵ABD \mathtt{\cong } ▵CDB

**N****ext chapter** :** Understand RHS congruency in detail**