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The sum of the internal angle and the external angle on the same vertex is π radians (180°). The sum of all the internal angles of a simple polygon is π (n −2) radians or 180 (n –2) degrees, where n is the number of sides. The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180 ...
The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.
v. t. e. In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles ...
Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple ...
The 'exterior' or 'external bisector' is the line that divides the supplementary angle (of 180° minus the original angle), formed by one side forming the original angle and the extension of the other side, into two equal angles. [1] To bisect an angle with straightedge and compass, one draws a circle whose center is the vertex. The circle ...
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. [ 1 ] Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called ...
This is a direct consequence of the inscribed angle theorem and the exterior angle theorem. There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression. [26] If a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex-bicentric.
The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of the length of side AB to the length of side AC: and conversely, if a point D on the side BC of ABC divides BC in the same ratio as the sides AB and AC, then AD is the angle bisector of angle ∠ A.