# Ex.7.1 Q6 Triangles Solution - NCERT Maths Class 9

## Question

In the given figure, \(AC = AE\), \(AB = AD\)

and \(\angle BAD = \angle EAC\).

Show that \(BC = DE\).

## Text Solution

**What is Known?**

\(AC = AE, AB = AD\) and

\(\angle BAD = \angle EAC.\)

**To prove:**

\(BC = DE.\)

**Reasoning:**

We can show two triangles BAC and DAE congruent by using SAS congruency rule and then we can say corresponding parts of congruent triangles will be equal. To show both triangles congruent two pair of equal sides are given and add angle DAC on both sides in given pair of angles BAD and angle EAC to find the included angle BAC and DAE.

**Steps:**

It is given that

\(\angle BAD = \angle EAC\)

\(\angle BAD\!+\!\angle DAC\!=\!\angle EAC\!+\!\angle DAC\)

\(\angle BAC = \angle DAE\)

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

\[\begin{align} AB & =AD\,(\text{Given})\, \\ \angle BAC&=\angle DAE\, \\ & (\text{Proven}\,\,\text{above})\, \\ & \\ AC&=AE\,(\text{Given})\, \\ \therefore \Delta BAC&\cong \Delta DAE\, \\ (\text{By}\,\text{SAS cong}&\text{ruencerule}) \\ & \\\therefore BC & =DE\,(ByCPCT) \\ \end{align}\]