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[[Image:Triangle.png|right]] | |||
A '''triangle''' is a three-sided [[geometric]] [[shape]]. | |||
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 triangle is defined by any three points that are not [[collinear]]. | |||
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''' | |||
== | ==Naming conventions== | ||
Usually, the vertices of a triangle are counter-clockwise denoted by big Latin letters, while small Latin letters are used for the sides: Each side will have the same letter as the opposite vertex. | |||
The angles are denoted by Greek letters, if possible, the letter of an angle will correspond to the letter of the adjacent vertex. | |||
Often, the sides will be referenced by the adjacent vertices: in the triangle on top of this page, we have <math>a = \overline{BC} = \overline{CB}, b = \overline{CA} = \overline{CA}, c = \overline{BA} = \overline{AB} </math>. | |||
Similarly, the angles are denoted as <math>\alpha = \angle CAB, \beta = \angle ABC, \gamma = \angle BCA</math>. | |||
== | ==Types of triangles== | ||
< | A [[right triangle]] has one 90º angle. Right triangles have special properties (see [[trigonometry]]). | ||
[[Category: | |||
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''' -- that is, a triangle with no [[congruent]] angles. | |||
An '''obtuse triangle''' has one angle that measures more than 90<sup>o</sup>. | |||
{| class="wikitable" | |||
|- | |||
|<!--col1-->[[Image:Right-triangle.png|150px]] | |||
|<!--col2-->[[Image:Isoscles-triangle.png|150px]] | |||
|<!--col3-->[[Image:Equilateral-triangle.png|150px]] | |||
|<!--col4-->[[Image:Obtuse-triangle.png|150px]] | |||
|- | |||
|<!--col1-->Right Triangle | |||
|<!--col2-->Isosceles Triangle | |||
|<!--col3-->Equilateral Triangle | |||
|<!--col4-->Obtuse Triangle | |||
|- | |||
|<!--col1--><math>\angle CBA =90^\circ = \pi/2</math> | |||
|<!--col2--><math>\angle BCA = \angle CAB </math> | |||
|<!--col3--><math>\angle ABC = \angle BCA = \angle CAB</math> | |||
|<!--col4--><math>\angle ACB > 90^\circ</math> | |||
|}<!--end wikitable--> | |||
==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.<br> | |||
'''Side-Side-Side (SSS)''': If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.<br> | |||
'''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.<br> | |||
'''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.<br> | |||
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. | |||
==See also== | |||
*[[polygon]] | |||
*[[Pascal's triangle]] | |||
*[[Sierpinski triangle]] | |||
*[[Bermuda Triangle]] | |||
[[Category:Geometry]] | |||