# Difference between revisions of "Triangle"

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A '''triangle''' is a three-sided figure. | A '''triangle''' is a three-sided figure. | ||

− | In Euclidean geometry, each side of a triangle is perfectly straight, and the sum of the internal angles of a triangle is always 180º. | + | 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]]). | A [[right triangle]] has one 90º angle. Right triangles have special properties (see [[trigonometry]]). | ||

− | + | ==Congruence of triangles== | |

+ | Triangle can be proven [[congruence|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]] | ||

[[Category:Geometry]] | [[Category:Geometry]] |

## Revision as of 17:39, 20 May 2007

A **triangle** is a three-sided figure.

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).

## Congruence of triangles

Triangle 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.