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# Triangles in Geometry (Definition, Shape, Types, Properties & Examples)

## What is a Triangle ?

A triangle is a geometric shape that is defined by three points in space, connected by three line segments. The sum of the interior angles of a triangle always adds up to 180 degrees.

Triangles are classified based on their sides and angles, including equilateral triangles (all sides equal and all angles equal), isosceles triangles (two sides equal), scalene triangles (all sides unequal), right triangles (one angle equal to 90 degrees), and acute triangles (all angles less than 90 degrees).

Triangles are commonly used in mathematics, engineering, and design and have many practical applications. ## Types of Triangles :-

(a) On the basis of length of the sides.

1. Scalene Triangle :–  A scalene triangle is a type of triangle in which all sides are of different lengths and all angles are of different measures. In other words, a scalene triangle has no two sides that are equal in length and no two angles that are equal in measure. Scalene triangles can have acute, right, or obtuse angles, and their sides can be of any length, as long as they meet the triangle inequality theorem.

This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Scalene triangles are important in geometry, trigonometry, and navigation.

2. Isosceles Triangle :-

An isosceles triangle is a type of triangle where two sides have the same length, and the third side has a different length. The two equal sides are called the “base” of the triangle and the third side is called the “height” or “altitude”. The two equal sides of the triangle are also referred to as the “legs”. The interior angles of an isosceles triangle are also special. The two angles that are formed by the two equal sides are congruent, meaning they are equal in measure. The third angle is also a right angle in an isosceles right triangle.

Isosceles triangles are important in mathematics and geometry for their symmetry and for the many theorems and formulas that are associated with them. They are used in a variety of applications, including construction, engineering, and computer graphics.

It’s important to note that while isosceles triangles are a specific type of triangle, they are still subject to the general properties of all triangles, such as the sum of the interior angles being equal to 180 degrees.

3. Equilateral Triangle :-

An equilateral triangle is a special type of triangle where all three sides have the same length and all three interior angles have the same measure. The three equal sides make the triangle highly symmetrical and visually appealing. An equilateral triangle has three congruent interior angles, each measuring 60 degrees. Because of this, equilateral triangles are often used to construct regular polyggon shapes, such as squares and hexagons.

Equilateral triangles also have several important properties that make them useful in mathematics and geometry. For example, they provide a simple way to demonstrate the Pythagorean theorem, as all three sides are equal in length. They are also used in trigonometry, as the interior angles of an equilateral triangle can be used to find missing side lengths or angle measures in a given triangle.

In summary, equilateral triangles are highly symmetrical and visually appealing triangles that have a number of important properties and applications in mathematics and geometry.

(b) On the basis of measurement of the angles.

1. Acute Angle Triangle :-

An acute triangle is a triangle where all three interior angles measure less than 90 degrees. In other words, all three angles are considered “acute”. Acute triangles are characterized by their sharp, pointy shape and are commonly used in a variety of mathematical and geometric applications. For example, they can be used in trigonometry to find missing side lengths or angle measures in a given triangle.

It’s important to note that while acute triangles are a specific type of triangle, they are still subject to the general properties of all triangles, such as the sum of the interior angles being equal to 180 degrees. Additionally, while all three interior angles in an acute triangle are less than 90 degrees, they can have any combination of measures, making them a flexible and useful type of triangle in many different contexts

2.Right Angle Triangle :-

A right triangle is a triangle in which one of the interior angles measures exactly 90 degrees. This angle is called the “right angle”. The other two angles in a right triangle are called the “acute angles”. The right angle in a triangle is represented by a small square symbol drawn in the corner of the angle. This symbol is used to distinguish right triangles from other types of triangles and to indicate the presence of the 90 degree angle.

Right triangles are an important type of triangle in mathematics and geometry. They are the basis for trigonometry and are used in a wide range of applications, including navigation, surveying, construction, and engineering.

One of the most famous theorems in mathematics, the Pythagorean theorem, applies only to right triangles. The theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In summary, right triangles are triangles with one interior angle measuring exactly 90 degrees and are an important type of triangle in mathematics and geometry, with many important properties and applications.

3. Obtuse Angle Triangle :-

An obtuse triangle is a triangle in which one of the interior angles measures greater than 90 degrees. In other words, one of the angles in the triangle is considered “obtuse”. Obtuse triangles are characterized by their wide, flattened shape and are less commonly used in mathematical and geometric applications than other types of triangles, such as acute triangles and right triangles. However, they still have important properties and can be useful in certain situations.

It’s important to note that while obtuse triangles are a specific type of triangle, they are still subject to the general properties of all triangles, such as the sum of the interior angles being equal to 180 degrees. Additionally, while one angle in an obtuse triangle is greater than 90 degrees, the other two angles can have any combination of measures.

In summary, obtuse triangles are triangles with one interior angle measuring greater than 90 degrees and are a less commonly used type of triangle, but still have important properties and applications in mathematics and geometry.