Editing Right angle (section)

== Euclid ==
Right angles are fundamental in [[Euclid's Elements]]. They are defined in Book 1, definition 10, which also defines perpendicular lines.  Definition 10 does not use numerical degree measurements but rather touches at the very heart of what a right angle is, namely two straight lines intersecting to form two equal and adjacent angles.<ref>Heath p. 181</ref>  The straight lines which form right angles are called perpendicular.<ref>Heath p. 181</ref>  Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle).<ref>Heath p. 181</ref> Two angles are called [[Complementary angles|complementary]] if their sum is a right angle.<ref>Wentworth p. 9</ref>

Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with. Euclid's commentator [[Proclus]] gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions. [[Giovanni Girolamo Saccheri|Saccheri]] gave a proof as well but using a more explicit assumption. In [[David Hilbert|Hilbert]]'s [[Hilbert's axioms|axiomatization of geometry]] this statement is given as a theorem, but only after much groundwork. One may argue that, even if postulate 4 can be proven from the preceding ones, in the order that Euclid presents his material it is necessary to include it since without it postulate 5, which uses the right angle as a unit of measure, makes no sense.<ref>Heath pp. 200–201 for the paragraph</ref>