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A direct proof using classical geometry was developed by James Mercer in 1923. [2] This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles.
The nines' complement math breakthrough eliminated the significant mechanical complexity created when "borrowing" during subtraction. This drum was the key to miniaturizing the Curta. His work on the pocket calculator stopped in 1938 when the Nazis forced him and his company to concentrate on manufacturing precision instruments for the German army.
If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. Two congruent shapes are similar, with a scale factor of 1. However, some school ...
Classically the defect arises in two contexts: in the Euclidean plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180°. However, on a convex polyhedron , the angles of the faces meeting at a vertex add up to less than 360° (a defect), while the angles at some vertices of a nonconvex polyhedron may add ...
An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. 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]
Alternate angles are the four pairs of angles that: have distinct vertex points, lie on opposite sides of the transversal and; both angles are interior or both angles are exterior. If the two angles of one pair are congruent (equal in measure), then the angles of each of the other pairs are also congruent.
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. [ 1 ] They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in three dimensional linear algebra .
In Euclidean geometry, an angle or plane 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] Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.