Triangle
A triangle is a three-sided geometric shape, made by any three points that are not collinear.
In Euclidean geometry, each side of a triangle is perfectly straight, and the sum of the internal angles of a triangle is always 180º.
A right triangle has one 90º angle. Right triangles have special properties (see trigonometry).
An isosceles triangle has two equal angles, and two equal sides.
An equilateral triangle has three equal sides, and three 60º angles.
If a triangle is not one of the above, it is a scalene triangle.
Congruence of triangles
Triangles can be proven congruent in the following ways:
Side-Angle-Side (SAS): If two sides are equal and the included angle is equal to another triangle, then the triangles are congruent.
Side-Side-Side (SSS): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
Angle-Side-Angle (ASA): If two angles and the included side of one triangle are equal the ones of another triangle, then the triangles are congruent.
Angle-Angle-Side (AAS): If two angles and a side that is not included are equal to the ones of another triangle, then the triangles are congruent.
The SSA (Side-Side-Angle) cannot prove triangles congruent unless it is a right angle, where it is known as the HL (Hypotenuse-Leg) Theorem. AAA (Angle-Angle-Angle) cannot prove triangles congruent either. In hyperbolic geometry, however, it does prove congruence.
