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Exterior angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem. One can also consider the sum of all three exterior angles, that equals to 360° [9] in the Euclidean case (as for any convex polygon), is less than 360° in the spherical case, and is greater than 360° in the hyperbolic case.
In several high school treatments of geometry, the term "exterior angle theorem" has been applied to a different result, [1] namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel ...
The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. [34] The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter ...
In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.
Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle. In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2 n -gon, then the two sums of alternate interior angles are each equal to ( n -1) π {\displaystyle \pi } . [ 4 ]
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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 ...
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