Choose the correct answer from the given four options: In two lin…

MCQ

Choose the correct answer from the given four options:
In two line segments AC and BD intersect each other at the point P such that PA = 6cm, PB = 3cm, PC = 2.5cm, PD = 5cm, $\angle\text{APB}=50^\circ\text{ and} \ \angle\text{CDP}=30^\circ.$ Then $\angle\text{PBA}$ is equal to:

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

✓

Answer

Correct option: D.

$100^\circ$

$\angle\text{APB}=\angle\text{CPD}=50^\circ$ [vertically opposite angles]
$\frac{\text{AP}}{\text{PD}}=\frac{6}{5}\ ......(\text{i})$
$\text{and }\frac{\text{BP}}{\text{CP}}=\frac{3}{2.5}=\frac{6}{5}\ ......(\text{ii})$
From Eq. (i) and (ii)
$\frac{\text{AP}}{\text{PD}}=\frac{\text{BP}}{\text{CP}}$
$\therefore\triangle\text{APB}\sim\triangle\text{DPC}$ [by SAS similarity criterion]
$\therefore\angle\text{A}=\angle\text{D} = 30^\circ $ [Corresponding angles of similar triangles]
$\text{In}\ \triangle\text{APB},\ \ \angle\text{A}+ \angle\text{B}+ \angle\text{PBA}=180^\circ$ [Sum of angles of a triangle = 180°]
$\Rightarrow30^\circ+\angle\text{B}+50^\circ=180^\circ$
$\therefore\angle\text{B}=180^\circ-(50^\circ+30^\circ)=100^\circ$
$\text{i,e.,}\ \ \angle\text{PBA}=100^\circ$