Question

If two arms of one angle are respectively parallel to the arms of another angle, then the two angles are either or

Answer 1

Let the required angles be $\angle ABC$ and $\angle PQR$

There are two possible cases :

$\left(1\right)$
Here, $\angle 1=\angle 2\left(\text{corresponding angles}\right)$
$\angle 2=\angle 3\left(\text{corresponding angles}\right)$

$\therefore \angle 1=\angle 3$
i.e., $\angle ABC=\angle PQR$



$\left(2\right)$

Here, $\angle 1+\angle 2={180}^{\circ }\left(\text{co-interior angles}\right)$
and $\angle 2=\angle 3\left(\text{corresponding angles}\right)$
$\therefore \angle 1+\angle 3={180}^{\circ }$
$\angle ABC+\angle PQR={180}^{\circ }$

Hence, if two arms of one angle are respectively parallel to the arms of another angle, then the two angles are either equal or supplementary.