Question

If one of the four angles formed by two intersecting lines is a right angle, then each of the four angles is a right angle.
This statement is (True/False).

Answer 1

Let us try to visualize the given statement using a diagram.
$\underset{–}{\text{Given :}}$ One angle is a right angle.
$\angle AOB$ and $\angle BOC$ form a linear pair.
Hence, sum of their measures must equal ${180}^o.$
$\because$ $m\angle AOB={90}^o$
$\therefore$ $m\angle BOC={180}^o-{90}^o={90}^o$

Here, we have two pairs of vertical angles:
$\bullet$ $\angle AOB$ and $\angle COD$
$\bullet$ $\angle BOC$ and $\angle DOA$

$\text{Vertical Angles Theorem}$ states that vertical angles are always congruent (equal in measure).
$\therefore$ $\angle AOB$ = $\angle COD$ and $\angle BOC$ = $\angle DOA$

But we already have,
$m\angle AOB=m\angle BOC={90}^o$
$\Rightarrow$ $m\angle COD=m\angle DOA={90}^o$

$\therefore$ Each of the four angles is a right angle.

Hence, The given statement is $\text{True}$.