**Properties Of Triangles:** Triangle is an important geometrical shape that is taught in school from primary classes till Class 12. A lot of different concepts related to Triangles, from simple to more complex, are covered under Geometry, Mensuration, and Trigonometry. It is important not only for school-level and board exams but also for various other competitive exams, like CAT, MAT, and exams for government job recruitment. So, in this article, we will discuss the definition, types, and properties of triangles, along with diagrams and important formulas.

## Properties Of Triangles

Let us first understand what is a triangle.

*A triangle is the simplest form of a polygon. The name ‘triangle’ derives from the fact that it has three angles formed by joining three line segments end to end.* So, basically, a triangle is a closed geometric shape that has three angles, three sides, and three vertices. The sum of the three angles of a triangle is 180°.

Let us now look into the different types of triangles.

### Classification Of Triangles

Triangles can be classified:

a. Based on the length of sides, and

b. Based on the internal angles

### Types Of Triangles Based On Length Of Sides

There are three types of triangles based on the length of the sides:

a. Scalene Triangle

b. Isosceles Triangle

c. Equilateral Triangle

Let us understand each of these types of triangles:

**a. Scalene Triangle:** If the three sides of a triangle have different lengths, then it is called a scalene triangle. This also means that the three vertices of a scalene triangle are different.

**b. Isosceles Triangle: **If any two sides of a triangle are equal in length and the third side has a different length, then the triangle is called an isosceles triangle. The angles opposite to the equal sides of an isosceles triangle also measure the same.

**c. Equilateral Triangle:** A triangle with all three sides equal in length is called an equilateral triangle. As the sum of the three angles of a triangle is always 180°, each angle of an equilateral triangle measures 60°.

### Types Of Triangles Based On Internal Angles

Based on the internal angles, there are three types of triangles:

a. Acute-Angled Triangle

b. Right-Angled Triangle

c. Obtuse-Angled Triangle

Let us understand each of these:

**a. Acute-Angled Triangle:** If all the three internal angles of a triangle measure less than 90°, then it is an acute-angled triangle. The three internal angles are called acute angles.

**b. Right-Angled Triangle:** If any one of the internal angles of a triangle measures 90°, it is a right-angled triangle. As the sum of the three angles of a triangle is 180°, the other two angles of a right-angled triangle will be less than 90° and hence, are acute angles.

Note that the side opposite to the right angle is called the hypotenuse of a right-angled triangle. According to Pythagoras’ Theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides of a right-angled triangle.

**c. Obtuse-Angled Triangle:** If one of the three sides of a triangle measure more than 90°, then it is an obtuse-angled triangle.

### Important Points Related To Types Of Triangles

The same triangle can belong to two different types – one type based on sides and the other based on internal angles. For example:

- If two angles of a triangle measure 45° each and the third side measures 90°, it is:
**i.**a**right-angled triangle**as one angle measures 90°,**ii.**an**isosceles triangle**as the two sides opposite to the angles measuring 45° each will be equal in length.

These triangles are called**right-angled isosceles triangles**. - If one angle of a triangle measures 90° and the other two angles are unequal, then the triangle is:
**i.**a**right-angled triangle**as one angle measures 90°,**ii.**a**scalene triangle**as the three angles measure differently, thereby, making the three sides different in length.

These triangles are called**right-angled scalene triangles**.

Let us now move into the different properties of triangles.

### Properties Of Triangles

Let us consider the following triangle:

Some of the important triangle properties are as under:

- a. The sum of all three internal angles of a triangle is 180°, i.e. ∠A + ∠B + ∠C = 180°.
- b. The sum of all three external angles of a triangle is 360°, i.e. ∠A’ + ∠B’ + ∠C’ = 360°.
- c. The measure of an external angle of a triangle is equal to the sum of the measures of the two internal opposite angles. For example, ∠A’ = ∠B + ∠C.
- d. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. For example, AB + BC > CA.
- e. The difference between the lengths of any two sides of a triangle is less than the length of the third side. For example, AC – AB < BC.
- f. The side opposite to the smallest internal angle of a triangle is the smallest side of the triangle.
- g. The side opposite to the largest internal angle of a triangle is the longest side of the triangle.
- h. The height of a triangle is equal to the length of the line segment drawn perpendicular to any side of the triangle (called the base of the triangle) from the vertex opposite to the side.

### Formulas Of Triangles

Some of the important formulas of a triangle are as under:

a. Area of a triangle: 1/2 X Base X Altitude (Height)

b. Perimeter of a triangle, p: a + b + c, where a, b, and c are the lengths of the three sides of the triangle.

c. Perimeter of an isosceles triangle, p1: a + 2b, where b is the length of each of the equal sides.

d. Perimeter of an equilateral triangle, p2: 3a, where a is the length of each side of the triangle.

e. Semi-perimeter of a triangle, s: (p/2) = (a + b +c)/2

f. Area of a triangle as per Heron’s Formula: √s (s – a)(s – b)(s – c)

g. For a right-angled triangle, (Base)^{2} + (Height)^{2} = (Hypotenuse)^{2}

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So, now you know everything about the properties of triangles. We hope this detailed article helps you.

*If you have any queries, feel free to ask in the comment section below. We will get back to you at the earliest.*

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