Euclid made the right angle his universal angle yardstick
A right angle measures exactly 90 degrees or π/2 radians—a quarter turn. If a ray meets a line so adjacent angles match, both are right. Latin angulus rectus means upright; Greek orthos gives orthogonality. Perpendicular lines meet in right angles; rectangles pack four of them.
English right echoes Latin rectus, erect or upright, pointing to a vertical line standing on a horizontal base, and Greek orthos, straight, survives in orthogonality, the vector version of meeting squarely. A right angle is what makes a triangle a right triangle, which is why it sits at the root of trigonometry. Rectangles have four of them, and squares add equal sides.
Euclid’s Book 1, definition 10, describes the right angle without numbers, as the pair of equal adjacent angles formed when one straight line stands on another, and calls such lines perpendicular. Postulate 4 declares all right angles equal, letting him use one as a unit for other angles. Proclus tried to derive that postulate from the earlier ones, probably relying on hidden assumptions; Saccheri offered a proof with a more explicit assumption, and in Hilbert’s axioms it becomes a theorem.
Diagrams flag a right angle with a tiny square in the corner, though German-speaking countries and Poland sometimes use an arc with a dot instead. Unicode assigns it U+221F, not to be confused with the look-alike bottom-left corner sign U+231E. Measured in other units, a right angle equals 100 gradians.
Builders have long checked corners with the 3-4-5 rule: mark three units along one side and four along the other, and if the diagonal between the marks is exactly five, the corner is square. Thales’ theorem guarantees that any angle drawn in a semicircle, with its rays running to the ends of the diameter, is a right angle. On a sphere, an octant forms a triangle with three right angles and subtends a solid angle of π/2 steradians.
Source: Right angle
