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There are many 2D shapes that you study in geometry, such as square, rectangle, circle, rhombus, etc. A triangle is also a 2D shape having three sides, three vertices, and three angles. There are many triangular objects, we come across in our daily lives such as sandwiches, traffic signs, cloth hangers, a rack in billiards, sailing boats, roofs, etc.

Let’s understand what is triangle in geometry and learn about its shape.

## What is Triangle in Geometry?

A triangle is a simple polygon with three sides, three vertices, and three interior angles. It is one of the basic shapes in geometry. A triangle is denoted by the symbol $\triangle$. There are various types of triangles classified on the basis of the sides and angles. Some major concepts, such as Pythagoras theorem and trigonometry, are based on the properties of triangles.

## Parts of Triangle

A triangle consists of various parts. The three parts are angles, sides (or edges), and vertices (singular vertex). Every triangle has 3 angles, 3 sides, and 3 vertices.

**Side:**A side of the triangle is a straight line segment such that two sides meet at each vertex. In the above figure, three sides are $\text{AB}$, $\text{BC}$, and $\text{CA}$**Angle:**An angle is formed by two sides of the triangle, which meet at a common point, known as the vertex. In the above figure, three angles are $\angle \text{ABC}$, $\angle \text{BCA}$, and $\angle \text{CAB}$.**Vertex:**The meeting point of a pair of sides is called its vertex. Its plural is vertices. In the above figure, the three vertices are $\text{A}$, $\text{B}$, and $\text{C}$.

**Note:** The sum of three angles of a triangle is $180^{\circ}$. In the above figure, $\angle \text{ABC} + \angle \text{BCA} + \angle \text{CAB} = 180^{\circ}$.

## What are the Interior and Exterior Angles of the Triangle?

A triangle consists of interior and exterior angles. The Interior angle is defined as the angle formed between two adjacent sides of a triangle. An exterior angle is defined as the angle formed between a side of a triangle and an adjacent side extending outward.

**Interior Angle:** Interior angle of a triangle is an angle that is formed inside the triangular region. Whenever we refer to any angle of a triangle, we mean its interior angle. The interior angle is represented by the name of the vertex it lies or by a three letter where the middle letter is the vertex where the angle lies and the other two represent the other two vertices.

The sum of three interior angles of a triangle is $180^{\circ}$.

In a triangle in the above figure $\angle \text{A} + \angle \text{B} + \angle \text{C} = 180^{\circ}$.

**Exterior Angle:** When any side of a triangle is extended, the angle that is formed with this side and its adjacent side is called the exterior angle of a triangle. There are three exterior angles in a triangle. Each exterior angle forms a linear pair with its corresponding interior angle.

The sum of three exterior angles of a triangle is $360^{\circ}$.

In a triangle in the above figure $\text{Ext } \angle \text{A} + \text{Ext } \angle \text{B} + \text{Ext } \angle \text{C} = 360^{\circ}$.

**Note:** The sum of exterior angles of any polygon is $360^{\circ}$.

## Characteristics of Triangle

The following are the important characteristics of a triangle that helps you to distinguish it from other geometric figures.

- A triangle has three sides, three vertices, and three interior angles.
- The sum of three angles of a triangle is $180^{\circ}$. This property is called as angle sum property of triangles.
- The sum of any two sides of a triangle is greater than the length of the third side.
- The difference between any two sides of a triangle is less than the length of the third side.
- The side opposite to the greatest angle is the longest and vice-versa.
- The side opposite to the smallest angle is the smallest and vice-versa.
- The exterior angle of a triangle is always equal to the sum of the opposite two interior angles of a triangle. This property is called the exterior angle property of triangles.
- In the case of a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the two remaining sides, i.e, $\left( \text{Hypotenuse}^{2} = \text{Base}^{2}+ \text{Altitude}^2 \right)$. It is commonly known as Pythagoras theorem.

## Practice Problems

- Define the following terms
- Triangle
- Interior Angle
- Exterior Angle
- Vertex

- State True or False
- The sum of three angles of a triangle is $100^{\circ}$.
- The sum of three angles of a triangle is $180^{\circ}$.
- The sum of the interior angle and its corresponding exterior angle of a triangle is $180^{\circ}$.
- The sum of the interior angle and its corresponding exterior angle of a triangle is $360^{\circ}$.
- The sum of any two sides of a triangle is equal to the length of the third side.

## FAQs

### What is a triangle in Geometry?

A triangle is a simple polygon with three sides, three vertices, and three interior angles. It is one of the basic shapes in geometry. A triangle is denoted by the symbol $\triangle$.

### Why is it called a triangle?

The word triangle comes from the Latin word *triangulus*, “three-cornered” or “having three angles,” from the roots tri-, “three,” and angulus, “angle or corner.”

### What are the three properties of a triangle?

The three basic properties of a triangle are:

a) The sum of all the angles of a triangle is equal to $180^{\circ}$.

b) The sum of the length of the two sides of a triangle is greater than the length of the third side.

c) The difference between the two sides of a triangle is less than the length of the third side.

## Conclusion

A triangle is a simple polygon with three sides, three vertices, and three interior angles. In any triangle, the sum of three angles is $180^{\circ}$ and the sum of any two sides is greater than the length of the third side. Similarly, the difference between any two sides of a triangle is less than the length of the third side.

## Recommended Reading

- Types of Angles in Maths(Acute, Right, Obtuse, Straight & Reflex)
- What is an Angle in Geometry – Definition, Properties & Measurement
- How to Construct a Tangent to a Circle(With Steps & Pictures)
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