# Ex.7.1 Q1 Triangles Solution - NCERT Maths Class 9

## Question

In quadrilateral \(ACBD, AC = AD\) and \(AB\) bisects \(\angle A\) (See the given figure). Show that \(\begin{align} \Delta ABC \cong \Delta ABD \end{align}\). What can you say about \(BC\) and \(BD\)?

## Text Solution

**What is Known?**

\(AC = AD\) and \(AB\) bisects

**To prove:**

\(\Delta {\text{ABC}} \cong \Delta {\text{ABD}}\) and, what can be said about \(BC\) and \(BD.\)

**Reasoning:**

We can show two sides and included angle of are equals to corresponding sides and included angle of by using SAS congruency criterion both triangles will be congruent and by CPCT, BC and BD will be equal.

**Steps:**

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

\[\begin{align} AC &= AD \text {(Given)}\\\\\Delta CAB &=\Delta DAB\\&(AB \text { bisects } \angle A)\\\\AB &= AB\\ &\text {(Common)}\\ \\ \therefore \Delta ABC &\cong \Delta ABD\\ &\text {(By congruence rule)}\\ \\\therefore BC&=BD\\&\left(\begin{array}{I} \text{By corresponding parts}\, \\ \text{of congruent triangles} \\ \end{array}\!\right)\end{align}\]

Therefore, \(BC\) and \(BD\) are of equal lengths.