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The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line segments). [3] Only one of these angles contains the third side of the triangle in its interior, and this angle is called an interior angle of the triangle. [4]
When θ = π /2, ADB becomes a right triangle, r + s = c, and the original Pythagorean theorem is regained. One proof observes that triangle ABC has the same angles as triangle CAD, but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.)
The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof. An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.
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]
Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ ′ = 180° − γ. The figure on right shows the point C, the side b and the angle γ as the first solution, and the point C ′, side b ′ and the angle γ ′ as the ...
In Euclid's Elements, the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry.One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri–Legendre theorem, which states that the sum of the measures of the angles in ...
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]
Lie's theorem (Lie algebra) Lie's third theorem ; Lie–Palais theorem (differential geometry) Lindemann–Weierstrass theorem (transcendental number theory) Lie–Kolchin theorem (algebraic groups, representation theory) Liénard's theorem (dynamical systems) Lindelöf's theorem (complex analysis) Lindström's theorem (mathematical logic)