Question

In figure m || n and $\angle$ 1 and $\angle$ 2 are in the ratio 3 : 2. Determine all the angles from 1 to 8.

Answer 1

It is given that $\angle 1:\angle 2=3:2.$ So, let
$\angle 1=3x^{\circ }$ and $\angle 2=2x^{\circ }$
But $\angle 1and\angle 2$ form a linear pair.
$\therefore \angle 1+\angle 2={180}^{\circ }$
$\Rightarrow 3x^{\circ }+2x^{\circ }={180}^{\circ }\Rightarrow 5x^{\circ }={180}^{\circ }$

$\Rightarrow x=\frac{{180}^{\circ }}{5}={36}^{\circ }$
$\therefore \angle 1=3x^{\circ }=(3\times 36{)}^{\circ }={108}^{\circ }$
and, $\angle 2=2x^{\circ }=(2\times 36{)}^{\circ }={72}^{\circ }$
Now, $\angle 1=\angle 3and\angle 2=\angle 4$ [Vertically opposite angles]
$\therefore \angle 4={72}^{\circ }and\angle 3={108}^{\circ }$
Now, $\angle 6=\angle 2and\angle 3=\angle 7$ [Corresponding angles]
$\Rightarrow \angle 6={72}^{\circ }and\angle 7={108}^{\circ }$ $[\because \angle 2={72}^{\circ }]$
[Vertically opposite angles]
Again, $\angle 5=\angle 7and\angle 8=\angle 6$
$\therefore \angle 5={108}^{\circ }and\angle 8={72}^{\circ }$
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 }$.