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In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180 ...
The congruence theorems side-angle-side (SAS) and side-side-side (SSS) also hold on a sphere; in addition, if two spherical triangles have an identical angle-angle-angle (AAA) sequence, they are congruent (unlike for plane triangles). [9] The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles. [10]
Suppose S is the common starting point of two rays, and two parallel lines are intersecting those two rays (see figure). Let A, B be the intersections of the first ray with the two parallels, such that B is further away from S than A, and similarly C, D are the intersections of the second ray with the two parallels such that D is further away ...
For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number", but holds if it is taken to mean ...
The intersection point falls within the first line segment if 0 ≤ t ≤ 1, and it falls within the second line segment if 0 ≤ u ≤ 1. These inequalities can be tested without the need for division, allowing rapid determination of the existence of any line segment intersection before calculating its exact point.
There are several theorems that guarantee triangle congruence in Euclidean geometry, namely Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), Side-Angle-Side (SAS), and Side-Side-Side (SSS). In taxicab geometry, however, only SASAS guarantees triangle congruence. [11] Take, for example, two right isosceles taxicab triangles whose angles measure ...
Axiom of line completeness: An extension (An extended line from a line that already exists, usually used in geometry) of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from ...
For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line.