Question

In $△ABC$ and $△PQR$, $\angle ABC\cong \angle PQR$, seg $BD$ and seg $QS$ are angle bisector.
If $\frac{l\left(AD\right)}{l\left(PS\right)}=\frac{l\left(DC\right)}{l\left(SR\right)}$ Prove that:$△ABC\sim △PQR$

Answer 1

Given:$\frac{l\left(AD\right)}{l\left(PS\right)}=\frac{l\left(DC\right)}{l\left(SR\right)}$

$\frac{l\left(AD\right)}{l\left(DC\right)}=\frac{l\left(PS\right)}{l\left(SR\right)}$

According to angle bisector theorem,

$\frac{l\left(AD\right)}{l\left(DC\right)}=\frac{l\left(AB\right)}{l\left(BC\right)}$

$\frac{l\left(PS\right)}{l\left(SR\right)}=\frac{l\left(PQ\right)}{l\left(QR\right)}$

$\therefore ,\frac{l\left(AB\right)}{l\left(BC\right)}=\frac{l\left(PQ\right)}{l\left(QR\right)}$

and $\angle ABC\cong \angle PQR$ (given)

$\therefore △ABC\cong △PQR$ by SAS test