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# Classifying Triangles by sides or Angles

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 Sub Topics We know that a triangle is a Polygon which has three sides, 3 angles & 3 vertices. Now the Triangles can be classified further on the basis of the length of their sides & also on the basis of their angle measure. We will first learn about classifying triangles by sides. On the basis of the measure of the sides of a triangle, we classify those follows: Scalene Triangle: If all the sides of a triangle are unequal, it is called scalene. Example: The triangle with sides 5m, 2m, 4m. Isosceles Triangle: If two sides of a triangle are equal, it is called isosceles. Example: A triangle with sides 3cm, 3cm, 6cm. Equilateral Triangle: If all sides of a triangle are equal, it is called equilateral. Example: A triangle with sides 4cm, 4cm, 4cm. After classifying the triangles by sides, we’ll learn classifying triangles by angles. For this, we must first recall the following kinds of angles: Acute: An angle which is more than 0 degree but less than 90 degrees is acute. Right: An angle of exactly 90 degrees. Obtuse: An angle which is more than 90 degree but less than 180 degrees is obtuse. For classifying triangles by angles, we check the measure of angles of the triangle. On this basis, the angles are classified in the following groups: Acute angled triangle: If all the angles of a triangle are acute, it is called an acute angled triangle. Right angled triangle: If one of the angles of a triangle measures 90 degrees, it is called right angled triangle. Obtuse angled triangle: If one of the angles of a triangle is obtuse, it is called obtuse angled triangle. Remember that in a right angled triangle as well as an obtuse angled triangle, the other two angles are always acute.

## Equilateral Triangle

We know that a triangle is a closed figure made up of 3 line segments. It is a Polygon with 3 sides, 3 vertices & 3 angles. On two different basis, the Triangles are grouped into two categories. One is on the basis of measure of length of its sides & the other, on the basis of the measure of its angles.

On the basis of measure of length of sides, the triangles may be scalene with all sides of different length, isosceles in which two sides are of the same length or equilateral in which all the three sides are equal.

On the basis of measure of angles , the triangles may be classified as Acute Triangle , where all angles are acute ,i.e., they measure less than 90 ̊, right triangle in which one of the angles is right, i.e., it measures exactly 90 ̊ & obtuse triangle in which one of the angle is obtuse ,i.e., it measures more than 90 ̊. The other two angles in the later two categories are acute angles.

Let us learn about equilateral triangles in detail here. An equilateral triangle is also called an Equiangular Triangle. It is a polygon with 3 sides, 3 angles & 3 vertices. What is special about an equilateral triangle that differentiates it from other triangles is that all the sides & also all the angles of such a triangle are equal. This means all of them are of the same measure. As all the angles are equal in an equilateral triangle, the measure of the angles is bound to be 60 degrees irrespective of the length of its sides, although the length of all sides is equal but it may vary from triangle to triangle whereas the angle measure is always 90 degrees each.

## Acute Triangle

We know that a triangle is a closed rectilinear figure made up of 3 line segments. It is a Polygon with 3 sides (or edges), 3 vertices (or corners) & 3 angles. The Triangles are grouped into two categories on two different criterions; one is on the basis of measure of length of its sides & the other on the basis of the measure of its angles.
On the basis of measure of length of sides, the triangles may be scalene in which all sides have different length, isosceles in which two of the sides have same length or equilateral in which all the three sides are of equal length.
On the basis of measure of angles , the triangles may be classified as acute triangle , where all angles are acute ,i.e., they measure less than 90 ̊ , right triangle in which one of the angles is right , i.e., it measures exactly 90 ̊ whereas the other two angles are acute & obtuse triangle in which one of the angle is obtuse ,i.e., it measures more than 90 ̊ while the other two angles in this category also are acute angles.
Let us know about acute triangles in detail here. An acute triangle is also called an acute angled triangle. It is a 3 sided polygon with 3 angles & 3 vertices. What makes an acute triangle special & different from other triangles is that all the angles of such a triangle are acute angles. This means all of them measure less than 90 ̊. As mentioned earlier, two of the angles in Right Triangle as well as in Obtuse Triangle are also acute but in an acute triangle all the three angles are acute. The 3 angles may or may not be equal but they should all be acute.

## Equiangular Triangle

A triangle is a Polygon formed by joining three line segments and thus forming a closed figure. There are different types of Triangles. We name here some of them as acute angled triangle, obtuse angle triangle, right angled triangle. If we look at the triangles, they are also classified based on the length of the sides of the triangle.
Here we are going to work on equiangular triangle. In Geometry we say that the equiangular triangles are the triangles which have all the three line segments of the same measure. Also we can say that when all the three sides of any triangle are equal, then the three angles of the triangles are also equal.
Thus as we know that the angle sum of any triangle is equal to 180 degrees thus if we find the measure of each angle of an Equilateral Triangle, we get each angle of an equilateral triangle = 180 / 3 = 60 degrees. If we take two equilateral triangles, then it does not Mean that they are congruent. Yes, the sides of the two equilateral triangles are in same proportion, it means that the two equilateral triangles are always similar, but they are not always congruent. The two equilateral triangles are called congruent to each other when we have the measures of one triangle exactly equal to another triangle. No matter what ever be the measure of each side of the equilateral triangle, but it is sure that the measure of each angle of all the equilateral triangles is always 60 degrees. So we also call this triangle as equiangular triangle. By word equiangular we mean that the angles of any given polynomial are equal, and if it is an equiangular triangle, then each angle is 60 degrees and so it’s all sides are also equal.

## Scalene Triangle

The figures, which have the same starting and the ending points are called the closed curves. Polygons are also the examples of the closed curves, which are formed by line segments. These are the figures formed by joining 3 or more line segments and so they form the closed figures.
We say that the polygons with three line segments are Triangles. And this way more different names are given to the polygons formed by joining different line segments. Here we will learn about scalene triangle in this unit. In Geometry we say that the triangles can be classified based on the length of their line segments. We observe that if line segments of the triangle are same, then we call it the Equilateral Triangle. On another hand we say that if the two sides of the triangles are equal, then the triangles are called Isosceles Triangle. Here we are learning about scalene triangles. The triangles which have the different length of the line segments, then the triangle so formed are called, scalene triangles. In scalene triangles, we say that the length measure of the triangles is different, so the length measure of the angles of the triangles are also of different measure. As the different sides of the triangle are not of same length in the scalene triangle are not same, but the sum of the angles sum of the triangle always remains same.
This sum of angles of the triangle is 180 degree. Thus we say that the property of sum of angles is equal is called angle sum property of the triangle. In scalene triangle, we say that if the two angles of the triangle are known, then we can find the third angle of the triangle by simply subtracting the sum of two angles of the triangle by 180 degrees.

## Isosceles Triangle

A Polygon with three sides is called a triangle. A triangle is called an isosceles triangle, when we find that the two sides of the triangle are equal, and then the Triangles are called isosceles triangles. In case the triangle is isosceles, we observe that as its two sides are equal, then the angles formed by its two sides are also equal. So if one of the angles of an isosceles triangle is known, we say that other two angles can be calculated by using the property of the angle sum property of the triangle. Let us try it with an example: if one of the angles of the isosceles is 50 degrees, then find the two equal angles of the given triangle.
Let the measure of each equal angle of the triangle be ‘x’ degree.
Then we say that by angle sum property of the triangle we can have:
X + x + 50 = 180 degrees,
Or 2x + 50 = 180,
Or 2x = 180 – 50,
Or 2x = 130,
Or x = 65 degrees.
We can check it as follows:
If we have the three angles as 65, 65 and 50 degrees, then we check if the triangle is formed or not?
So we add the three angles and get:
65 + 65 + 50 = 180 degree, so we conclude that the triangle is formed by these three angles. Also we observe that the two angles are of the same measure, so the corresponding sides of the two equal angles will also be equal. Thus we observe that the two sides of the given triangle are equal.
So the triangle so formed is called an isosceles triangle.
In the same way, if the two angles of same measure are known, we can find the third angle of the triangle.