Question

In the given figure, if m || n and ∠1 and ∠2 are in the ratio 3 : 2, determine all the angles from 1 to 8.

Answer 1

Given $\angle$1 : $\angle$2 = 3 : 2.

Let $\angle$1 = 3x${}^{\circ }$ and $\angle$2 = 2x${}^{\circ }$

⇒ $\angle$1 + $\angle$2 =180${}^{\circ }$ [linear pair of angles]

⇒ 3x${}^{\circ }$ + 2x${}^{\circ }$ = 180${}^{\circ }$

⇒ 5x${}^{\circ }$ = 180${}^{\circ }$

⇒ x = 36${}^{\circ }$

⇒ $\angle$1 = 3x${}^{\circ }$ = (3 x 36)${}^{\circ }$ =108${}^{\circ }$

and, $\angle$2 = 2x${}^{\circ }$ = (2 x 36)${}^{\circ }$ = 72${}^{\circ }$
$\angle$1 = $\angle$3, $\angle$2 = $\angle$4 [Vertically Opposite Angles]

⇒ $\angle$3 = 108${}^{\circ }$, $\angle$4 = 72${}^{\circ }$

$\angle$6 = $\angle$2, $\angle$3 = $\angle$7 [Corresponding angles]

⇒ $\angle$6 = 72${}^{\circ }$, $\angle$7 = 108${}^{\circ }$

$\angle$5 = $\angle$7, $\angle$8 = $\angle$6 [Venically Opposite Angles]

⇒ $\angle$5 = 108${}^{\circ }$, $\angle$8 = 72${}^{\circ }$

Hence, $\angle$1 = 108${}^{\circ }$, $\angle$2 = 72${}^{\circ }$, $\angle$3 =108${}^{\circ }$, $\angle$4 = 72${}^{\circ }$, $\angle$5 = 108${}^{\circ }$, $\angle$6 = 72${}^{\circ }$, $\angle$7 = 108${}^{\circ }$ and $\angle$8 = 72${}^{\circ }$.l