nonrightangled

Nonrightangled is an adjective used in geometry to describe figures, most often triangles, that do not contain a right angle. By definition, no interior angle measures 90 degrees; such triangles are either acute, with all angles under 90, or obtuse, with one angle over 90.

Nonrightangled triangles may have any side-length combinations that satisfy the triangle inequality; they can be equilateral (all sides equal), isosceles (two sides equal), or scalene (all sides different). In the acute case, all three angles are less than 90 degrees; in the obtuse case, one angle exceeds 90 degrees.

Solving nonrightangled triangles relies on trigonometric relations such as the Law of Sines and the Law of Cosines. In particular, the Law of Cosines, c^2 = a^2 + b^2 − 2ab cos(C), is commonly used, since the Pythagorean theorem applies only to right triangles. The Law of Sines also provides connections between corresponding sides and angles, useful when certain measurements are known.

Example: a triangle with sides 2, 3, 4 is nonrightangled; the angle opposite the side of length 4 is arccos((2^2 + 3^2 − 4^2) / (2·2·3)) ≈ 104.5 degrees.

Nonrightangled geometry is a standard topic in trigonometry and geometry problem solving and has applications in fields such as computer graphics, engineering, and surveying, where many triangles do not form right angles.