Triangle

A polygon with three sides and three interior angles. Its side lengths and angles determine its shape, while its angles always sum to 180 degrees in Euclidean geometry.

Euclidean geometry: The geometry of points, lines, angles, and shapes in flat space, governed by Euclid’s postulates. The familiar 180-degree angle sum follows from the parallel postulate of this geometry.
Triangle angle sum theorem: The theorem that a triangle’s interior angles sum to 180 degrees in Euclidean geometry. It constrains every possible combination of the triangle’s three angles.
Trigonometry: The branch of mathematics that studies relationships between angles and side lengths, especially in triangles. Triangle side-angle rules form the basis of trigonometric calculations.
Equilateral triangle: A triangle whose three sides are equal and whose three angles each measure 60 degrees. Equal sides force the most symmetric possible triangle.
Angle: A figure formed by two rays sharing an endpoint, or the measure of their rotation. The three interior angles distinguish a triangle’s shape and always total 180 degrees in Euclidean geometry.
Law of sines: A theorem relating each side of a triangle to the sine of its opposite angle. It converts known side-angle pairs into unknown sides or angles.
Triangulation: A method of locating a point by measuring directions or distances from known points. Surveyors use triangle geometry to infer inaccessible distances and positions.
Isosceles triangle: A triangle with at least two equal sides and equal angles opposite those sides. Its paired sides and base angles illustrate symmetry within the triangle family.
Side (geometry): A line segment forming one boundary of a polygon. A triangle is defined by having exactly three boundary segments.
Law of cosines: A formula relating a triangle’s side lengths to the cosine of one of its angles. It generalizes the Pythagorean theorem to triangles that need not be right-angled.

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