# Ex.7.1 Q2 Triangles Solution - NCERT Maths Class 9

## Question

\(ABCD\) is a quadrilateral in which \(AD = BC\) and \(\angle DAB = \angle CBA\) (See the given figure).

Prove that

(i) \(\Delta ABD \cong \Delta BAC\)

(ii) \(BD = AC\)

(iii) \(\Delta ABD=\Delta BAC\)

## Text Solution

**What is Known?**

\[{AD}={BC}\ \text{and}\ \angle {DAB}=\angle {CBA}\]

**To prove:**

(i) \(\Delta {{ABD}} \cong \Delta {{BAC}}\)

(ii) \({{BD}} = {{AC}}\)

(iii) \(\angle {{ABD}} = \angle {{BAC}}\)

**Reasoning:**

We can show two sides and included of are equals to corresponding sides and included angle of by using \(\rm{SAS}\) congruency criterion both triangles will be congruent. Then we can say corresponding parts of congruent triangle will be equal.

**Steps:**

In \(\Delta ABD\) and \(\Delta BAC\),

\[\begin{align} AD &= BC \;\text{(Given)}\\\Delta DAB &=\Delta CBA \;\text{(Given)}\\AB &= BA \;\text{(Common)}\\\therefore \Delta ABD &\cong \Delta BAC\\& \text{(By congruence}\,\text{rule})\\\\\end{align}\]

\[\begin{align}\therefore BD &= AC \;\text{(By CPCT )}\\& \text{and}\\\angle ABD &=\angle BAC \;\text{(By CPCT )}\end{align}\]