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Triangle

A polygon is a two dimensional shape enclosing an area. A shape must require at least three sides to enclose something and the shape which has three sides is called a triangle.
Each point of intersection of a set of two sides is called as vertex.Thus, a triangle is the first member in the family of polygon.
The name is derived from Greek in which the prefix ‘tri’ refers to the number 3.
A triangle has a number of special properties and let us look into some of those. Before that we shall discuss about different types of triangles.

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Types of Triangles

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Triangles are classified into different types on the bases of different considerations.

Based on the number of sides and interior angles, triangles are classified as equilateral, isosceles and scalene.
  • In an equilateral triangle, all sides are congruent and as per geometrical theory all the internal angles are also congruent. In an equilateral triangle, the measure of each interior angle is $60^{\circ}$ irrespective of the size of the triangle.
  • An isosceles triangle has two congruent sides and accordingly the corresponding angles are also congruent.
  • In a scalene triangle, neither any set of sides nor any set of the interior angles are congruent.
Types of Triangles

Triangles are also classified on the types of interior angles.
  • If all the interior angles of a triangle are acute (measuring < 90o), then it is termed as an acute triangle.
  • A triangle is called obtuse triangle if any of the (it can be only one) internal angles is obtuse (measuring > 90o).
  • If one of the interior angles of a triangle is a right angle (measuring exactly 90o), the triangle is named as right angle triangle or simple as right triangle.

Properties of Triangles

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There are many properties associated with a triangle irrespective of its type.
  • The first and foremost property is, the sum of the measures of any two sides in any triangle is always greater than the measure of the remaining side.
  • In any triangle, the sum of the measures of all interior angles is $180^{\circ}$. This is the reason that each internal angle in an equilateral triangle measures exactly $60^{\circ}$.
  • The area of any triangle is given by the unique formula as  A = $\frac{1}{2}$ $\times$ b $\times$ h, where ‘b’ is the measure of the base (any of the three sides can be considered as base) and ‘h’ is the measure of the altitude drawn on the same base from the opposite vertex. There are several other formulas which are used depending upon the available information.
  • In any triangle, the interior angle bisectors, perpendicular bisectors of the sides, altitudes and medians are all concurrent and the points of concurrencies are respectively called as incenter, circumcenter, orthocenter and centroid. The highly interesting fact is that all such centers coincide at a single point in case of an equilateral triangle.
  • In a right triangle, a revolutionary fact was discovered by the great Greek mathematician and philosopher Pythagoras that the square of the measure of the hypotenuse is equal to the sum of the squares of the measures of the remaining sides. This statement is considered as one of most important theorems in mathematics and named as Pythagorean Theorem.
It is also the foundation for the vast topic known as "trigonometry".

More topics in  Triangle
Area of a Triangle Perimeter of a Triangle
Median of a Triangle Centroid of a Triangle
Scalene Triangle Isosceles Triangle
Equilateral Triangle Right Triangle
Acute Triangle Obtuse Triangle
How do you find the Perpendicular Bisector of a Triangle? Altitude of a Triangle
Similarity and Congruence
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