Prove all right angles are congruent?

Question

Prove all right angles are congruent.

I only have to prove one side to this argument, so I just need to the the other argument.

So basically, if two angles are right, then they must be congruent is what I am trying to prove.

All I have is my assumption that the two angles are right. And conclusion, therefore the angles are congruent.

Answer 1

We say that the angle $\measuredangle AOB$ is the supplement of the angle $\measuredangle Y$ if the latter is congruent to an adjacent angle $\measuredangle BOC$ to $\measuredangle AOB$ such that the points $A$, $O$ and $C$ are colineal.

Using this definition and the fact that an angle is Right iff it's congruent to one of its supplements (by definition), you can prove that all right angles are congruent as follows:

Let $\measuredangle AOB$ be a right angle, then it's congruent to one of its supplements (and therefore to all of them). Let $\measuredangle BOC$ be an adjacent supplement of $\measuredangle AOB$, then $\measuredangle AOB \cong \measuredangle BOC$ and $A$, $0$, $C$ are colineal.

Now let $Y$ be any other right angle and consider $D$ an exterior point of $\measuredangle AOB$ such that $\measuredangle AOD$ is a right angle congruent to $Y$. Here you have to prove that: $B$, $O$ and $D$ are colineal and once you have this prove that $\measuredangle AOB\cong \measuredangle AOD$ (using "vertex opposites" arguments), I'm leaving that to you.

Answer 2

A straight angle has two right angles. If two angles are such that a supplement of the one equals itself then each must be a right angle. Since only a single parameter is involved congruence is established.