Triangles

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.

There exist different types of triangles. They can be classified: 1) according to the relative lengths of their sides:

· In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. It is possible to construct an equilateral triangle of a given side length using just compasses and a straightedge.

· In an isosceles triangle two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite the two sides of the same length; this fact is the content of the Isosceles triangle theorem.

· In a scalene triangle all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles.

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Equilateral	Isosceles	Scalene

2) according to their internal angles, measured here in degrees:

· A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite the right angle is the hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs or catheti of the triangle.

· Triangles that do not have an angle that measures 90° are called oblique triangles.

· A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.

· A triangle that has one angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle.

· A triangle with an interior angle of 180° (and collinear vertices) is degenerate.

Right	Obtuse	Acute
	Oblique

Triangles have some distinctive properties which are common to all types. The two most common properties are: 1) the interior angles of a triangle always add up to 180°; 2) the exterior angles of a triangle always add up to 360°.

The area of a triangle can be calculated by using the following geometric formula: Area of a triangle = ½ Base × Height;

The perimeter of a triangle is equal to the sum of its three sides.

For all triangles, angles and sides are related by the law of cosines and law of sines (also called the cosinerule and sine rule).

The law of cosines is a statement about a general triangle that relates the lengths of its sides to the cosine of one of its angles. This law states that where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c.

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle, then cos(γ) = 0, and thus the law of cosines reduces to

The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.

The law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law, where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known.