In this chapter we will learn the SAS congruency rule with solved examples.

Let us first review the basics of congruency.

## What are congruent triangles?

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

This means that congruent triangle will perfectly overlap when placed against each other.

## What is SAS postulate?

The full form of SAS is **side – angle – side**.

It says that two triangles are congruent when they have two equal sides and angles between the sides are also equal.

**For Example;**

Consider the below triangle ABC and PQR

We can see that;

AB = PQ = 5 cm

∠A = ∠Q = 45 degree

AC = RQ = 4 cm

By SAS congruency, both the triangles are congruent.

Hence, ABC \mathtt{\cong } ▵PQR

I hope the concept is clear, let us look at some examples for better understanding.

## SAS Postulate – Solved Problems

**(01) In the below figure find value of ∠ACB**

**Solution**

Comparing triangle ACB and DBC

AC = DB = 4 cm

∠BAC = ∠CDB = 86 degree

AB = CD = 3 cm

By SAS congruency, triangles ACB and DBC are congruent.

Hence, ▵ACB \mathtt{\cong } ▵DBC

We know that congruent triangles have equal sides and angles.

We can say that;

∠ACB = ∠BDC = 40 degree.

(02) check if the triangles ABC and CDA are congruent or not.

**Solution**

Taking triangle ABC and CDA.

AB = CD = 4 cm

∠BAC = ∠ACD = 90 degree

AC = CA (common side)

By SAS congruency, both the triangles are congruent.

Hence, ▵ABC \mathtt{\cong } ▵CDA

(03) Check if triangle AOB and COD are congruent.

Solution

Taking triangle OAB and OCD

OA = OC = 3.1 cm

∠AOB = ∠COD = 87 degree

OB = OD = 3.4 cm

By SAS congruency, both the triangles are congruent.

Hence, ▵AOB \mathtt{\cong } ▵COD

(04) Check if the triangles ABC and PQR are congruent or not.

**Solution**Let us first find value of ∠PQR.

We know that sum of angles of triangle equals 180 degree.

∠P + ∠Q + ∠R = 180

51 + ∠Q + 63 = 180

∠Q + 114 = 180

∠Q = 180 – 114

∠Q = 66 degree.

Now comparing the triangles BAC and PQR

AB = PQ = 4 cm

∠A = ∠Q = 66 degree

AC = QR = 3.5 cm **By SAS congruency, both triangles are congruent.**

Hence, ▵BAC \mathtt{\cong } ▵PQR

**(05) Which of the following statement is true.**

(i) ▵ABD \mathtt{\cong } ▵CBD

(ii) ▵ABD \mathtt{\cong } ▵CDB

(iii) ▵ABD \mathtt{\cong } ▵DBC

Solution

Taking the triangles ABD and CDB

AB = CD = 3 cm

∠BAD = ∠BCD

AD = BC = 4 cm

Both the triangles are congruent by SAS congruency.

So, ▵ABD \mathtt{\cong } ▵CDB

Hence option (ii) is correct among the given three options.

Note:

While mentioning the congruency of triangles the sequence of alphabet is important.

Foe example, In ABD and CDB;

AB = CD { first two alphabet of both triangles}

∠B = ∠D { second alphabets }

AD = BC { first and last alphabet }

Note that the letters are synchronized as per congruency.

Now consider the third option;▵ABD \mathtt{\cong } ▵DBC

\mathtt{AB\ \neq \ DB} { first two letters }

Hence, the naming is not properly synchronized.

**Next chapter** : **Understand ASA congruency in detail**