Question

In Fig., line $l\parallel m$ and a transversal $n$ cuts them at $P$ and $Q$ respectively. If $\angle 1={75}^{\circ }$, find all other angles.

Answer 1

In the given figure, $l\parallel m$ , $n$ is a transversal line and $\angle 1={75}^{\circ }$
Thus, we have:

$\angle 1+\angle 2={180}^{\circ }$ [Since, Linear pair]

$\Rightarrow \angle 2={180}^{\circ }-\angle 1$

$\Rightarrow \angle 2={180}^{\circ }-{75}^{\circ }$

$\Rightarrow \angle 2={105}^{\circ }$

Now, $\angle 1=\angle 3={75}^{\circ }$ [Vertically opposite angles]

$\angle 1=\angle 5={75}^{\circ }$ [Corresponding angles]

$\angle 5=\angle 7={75}^{\circ }$ [Vertically opposite angles]

$\angle 2=\angle 6={105}^{\circ }$ [Corresponding angles]

$\angle 8=\angle 6={105}^{\circ }$ [Vertically opposite angles]

$\angle 4=\angle 2={105}^{\circ }$ [Vertically opposite angles]

Therefore,

$\angle 2={105}^{\circ }$

$\angle 3={75}^{\circ }$

$\angle 4={105}^{\circ }$

$\angle 5={75}^{\circ }$

$\angle 6={105}^{\circ }$

$\angle 7={75}^{\circ }$

$\angle 8={105}^{\circ }$