All right angles are equal to each other

Question

Why is it that

All right angles are equal to each other

-a postulate in Euclid's Elements (Wikipedia). Shouldn't it be a congruence rather than an equivalence? Isn't this just a special case of the definition of congruent angles?

Answer 1

After looking at an article recently I found the answer. Throughout The Elements, Euclid only uses the measures of multiples of right angles. Although he says angles can be bigger or smaller he does not actually measure them.

Source: http://blogs.scientificamerican.com/roots-of-unity/whate28099s-the-deal-with-euclide28099s-fourth-postulate/

Answer 2

They are equal angles because they all are 90 degrees. If the right triangle was included in a triangle, then the triangles with right angles would be similar.

Answer 3

Yes you are correct, all right angles are congruent. But let us refer to the definition of angle congruence: equality of angle measure. Therefore, congruent angles have equality of measure. Saying right angles are equal implies congruence, and saying right angles are congruent implies equality.