A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.
In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space
Triangles can be classified according to the relative lengths of their sides:
* In an equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon.[1]
* In an isosceles triangle, two sides are of equal length (originally and conventionally limited to exactly two).[2] An isosceles triangle also has two equal angles: the angles opposite the two equal sides.
* In a scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.[3]
Equilateral Triangle Isosceles triangle Scalene triangle
Equilateral Isosceles Scalene
Triangles can also be classified according to their internal angles, described below using degrees of arc:
* A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle. Right triangles conform to the Pythagorean theorem, wherein the sum of the squares of the two legs is equal to the square of the hypotenuse, i.e., a2 + b2 = c2, where a and b are the legs and c is the hypotenuse. See also Special right triangles
* An oblique triangle has no internal angle equal to 90°.
* An obtuse triangle is an oblique triangle with one internal angle larger than 90° (an obtuse angle).
* An acute triangle is an oblique triangle with internal angles all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.
Right triangle Obtuse triangle Acute triangle
Right Obtuse Acute
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Oblique
[edit] Basic facts
Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). All triangles are two-dimensional.
A triangle with exterior angle.
The angles of a triangle in Euclidean space always add up to 180 degrees. An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle but not present within the triangle itself) is always equal to the two angles of a triangle that it is not adjacent/supplementary to; this is the exterior angle theorem. Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.
The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the triangle inequality. (In the special case of equality, two of the angles have collapsed to size zero, and the triangle has degenerated to a line segment.)
Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.
A few basic postulates and theorems about similar triangles:
* Two triangles are similar if at least two corresponding angles are equal.
* If two corresponding sides of two triangles are in proportion, and their included angles are congruent, then triangles are similar.
* If three sides of two triangles are in proportion, then the triangles are similar.
For two triangles to be congruent, each of their corresponding angles and sides must be equal (6 total). A few basic postulates and theorems about congruent triangles:
* SAS Postulate: If two sides and the included angle (angle between the two sides) of two triangles are congruent, then the triangles are congruent.
* ASA Postulate: If two angles and the included side (side between the two angles) of two triangles are congruent, then the triangles are congruent.
* SSS Postulate: If all three sides of two triangles are congruent, then the triangles are congruent.
* AAS Theorem: If two angles and a non-included side of two triangles are congruent, then the triangles are congruent.
* Hypotenuse-Leg (HL) Theorem: If a leg and hypotenuse of two right triangles are congruent, then the triangles are congruent.
* Hypotenuse-Angle Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to a hypotenuse and an acute angle of another right triangle, then the triangles are congruent
* Side-Side-Angle (or Angle-Side-Side) condition: if two sides and an angle that isn't included of two triangles are equal, then if the angle is obtuse, the opposite side is longer than the adjacent, or the opposite side is equal to the sine of the angle times the adjacent side, the triangles are congruent.
Even if three angles of two triangles are congruent (AAA), then you are still not guaranteed congruent triangles. The triangles will be similar (angles will be congruent, and the ratio of side measures will be equal on all three sides), but not necessarily congruent.
It should be noted that:
* The Side-Side-Angle condition does not guarantee the triangles to be congruent.
* For the Hypotenuse-Angle Theorem, the triangle must be a right-angled triangle. If it is not, then it would be classified as a Side-Side-Angle condition, which does not guarantee congruent triangles.
Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.
In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known.
The Pythagorean theorem
A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that
a^2 + b^2=c^2. \,
The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle.
Some other facts about right triangles:
* The acute angles of a right triangle are complementary.
a + b + 90^{\circ} = 180^{\circ} \implies a + b = 90^{\circ} \implies a = 90^{\circ} - b
* If the legs of a right triangle are equal, then the angles opposite the legs are equal, acute and complementary, and thus are both 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times the square root of two.
* In a 30-60 right triangle, in which the acute angles measure 30 and 60 degrees, the hypotenuse is twice the length of the shorter side.
* In all right triangles, the median on the hypotenuse is half of the hypotenuse.
For all triangles, angles and sides are related by the law of cosines and law of sines.
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