Choose the correct answer from the given four options : In two li… - Vidyadip

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 $PA = 6\ cm, PB = 3\ cm, PC = 2.5\ cm, PD = 5\ cm, \angle\text{APB}=50^\circ$ 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 $\text{SAS}$ 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 $= 180^\circ ]$
$\Rightarrow30^\circ+\angle\text{B}+50^\circ=180^\circ$
$\therefore\angle\text{B}=180^\circ-(50^\circ+30^\circ)=100^\circ$
$\text{i,e.,}\ \ \angle\text{PBA}=100^\circ$

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