Question

In fig 3.31 , if $$ PQ | | ST , \angle PQR = 110^{\circ} and \angle RST = 130^{\circ} $$ , find $$\angle QRS $$
(Hint : Draw a line parallel to ST through point R )

Answer 1

Data : $$PQ | | ST $$ and $$ \angle PQR = 110^{\circ} and \angle RST = 130^{\circ}$$
To Prove : $$ \angle QRS = ? $$
Construction : Draw $$ ST | | UV $$ Through 'R'
Proof $$ PQ | | St $$ (Data )
$$ST | | UV $$ (Construction )
$$\therefore PQ | | ST | | UV $$
$$ PQ | | UV $$
$$\therefore \angle PQR + \angle URQ = 180^{\circ} $$ (Sum of interior angles )
$$ 110 + \angle URQ = 180^{\circ}$$
$$\therefore URO = 180 - 110$$
$$\therefore \angle URO = 70^{\circ} (i) $$
Similarly , $$ST | | UV $$
$$ \therefore \angle RST + \angle SRV = 180$$
$$\therefore \angle SRV = 180 - 130 $$
$$\angle SRV = 50 ^{\circ} (ii) $$
But , URV is a straight line .
$$\angle QRS+50+70 = 180 $$

$$\angle QRS = 180 - 120 $$
$$\therefore \angle QRS = 60^{\circ}$$