Question

If two straight lines intersect in such a way that one of the angles formed measures 90°, show that each of the remaining angles measures 90°.

Answer 1

We know that if two lines intersect, then the vertically-opposite angles are equal.

$\angle AOC=90°.\text{T}\mathrm{hen},\angle AOC=\angle BOD=90°.$
And let $\angle BOC=\angle AOD=x$
Also, we know that the sum of all angles around a point is $360°$
$$\begin{gathered} \therefore \angle AOC+\angle BOD+\angle AOD+\angle BOC=360° \\ \Rightarrow 90°+90°+x+x=360° \\ \Rightarrow 2x=180° \\ \Rightarrow x=90° \end{gathered}$$
Hence, $\angle BOC=\angle AOD=90°$
$\therefore \angle AOC=\angle BOD=\angle BOC=\angle AOD=90°$
Hence, the measure of each of the remaining angles is 90^o.