in the given figure, two line segments AC and BD intersect each o… - Vidyadip

MCQ

in the given figure, two line segments $AC$ and $BD$ intersect each other at the point $P$ such that $PA = 6cm, PB = 3cm, PC = 2.5cm, PD = 5cm,$ $\angle\text{APB}=50^\circ$ and $\angle\text{CDP}=30^\circ$ then $\angle\text{PBA}=?$

A
$50^\circ$
B
$30^\circ$
C
$60^\circ$
✓
$100^\circ$

✓

Answer

Correct option: D.

$100^\circ$

In $\triangle\text{PBA}\sim\triangle\text{PCD}$
$\frac{\text{AP}}{\text{PD}}=\frac{\text{PB}}{\text{PC}}\dots\Big(\therefore\frac{\text{AP}}{\text{PD}}=\frac{6}{5 }\text{ and }\frac{\text{BP}}{\text{ BC}}=\frac{3}{2.5}=\frac{6}{5}\Big)$
So, $\angle\text{APB}=\angle\text{DPC}$ ....(Vertically opposite angles)
$\Rightarrow\triangle\text{CAD}\sim\triangle\text{PQR}$ ....(AA criterion for similarity)
$\angle\text{PBA}=\angle\text{DCP}$
In $\triangle\text{PCD},$
$\angle\text{PCD}=180^\circ-\angle\text{DPC}-\angle\text{PDC}$
$\Rightarrow\angle\text{PCD}=180^\circ-50^\circ-30^\circ$
$\Rightarrow\angle\text{PCD}=100^\circ$
So, $\angle\text{PBA}=\angle\text{DCP}=100^\circ.$

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