anglesacute

Anglesacute is a term used in geometry to describe acute angles. An acute angle is a measure greater than 0 degrees and less than 90 degrees (in radians, 0 < θ < π/2). It is distinct from a right angle (90 degrees) and from obtuse angles (greater than 90 degrees but less than 180 degrees). The concept is central to many geometric definitions and theorems, and the idea of an acute angle is preserved under unit changes, so measurements can be given in degrees or radians.

In the context of intersecting lines, the acute angle formed by two lines is the smaller of the two angles that arise at their intersection. If the lines are not perpendicular, there are two acute angles and two obtuse angles, with measures θ, 180°−θ, θ, and 180°−θ, where 0 < θ < 90°. If the lines are perpendicular, all four angles are right angles (90°).

In triangles, a triangle is called acute-angled or an acute triangle when all three interior angles are acute, each measuring less than 90 degrees. Such triangles have specific properties: for example, the sum of interior angles is always 180 degrees, and the side lengths are constrained by the triangle inequality and by the Law of Cosines in relation to the angles.

Acute angles are fundamental in trigonometry, where many identities and functions are evaluated for angles within the range (0°, 90°). The term is often abbreviated or described in context by the symbol θ (or α, φ) to denote an angle that lies in this range.