The document NCERT Chapter Notes 6 - The Triangle and its Properties, Mathematics, Class 7 | EduRev Notes is a part of the Class 7 Course Mathematics (Maths) Class 7.

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**6. The Triangle and its Properties**

**Triangles**

A triangle is said to be equilateral if each one of its sides is of the same length and each of one its **angles measures**

**The Triangle and Its Properties**

A triangle is a closed figure made of **three line segments.** Every triangle has** three sides, three angles, and three vertices.** These are known as **the parts of a triangle.** The sides and the angles of every triangle differ from one another; therefore, they do not look alike.

**Triangles can be classified based on their sides and angles.**

â™¦ Based on their sides, there are equilateral, isosceles and scalene triangles.

â™¦ Based on their angles, there are acute, obtuse and right-angled triangles.

**Equilateral triangle: **A triangle in which all the sides are equal is called an equilateral triangle. All the three angles of an equilateral triangle are also equal, and **each measures 60Â°.**

**Isosceles triangle: **A triangle in which any two sides are equal is called an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are called the base angles, **and they are equal.**

**Scalene triangle: **A triangle in which no two sides are equal is** called an Scalene triangle.**

**Acute-angled triangle**: A triangle with all its angles less than** 90Â° is known as an acute-angled triangle.**

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**Obtuse-angled triangle: **A triangle with one of its angles more than 90Â° and less than 180Â° is known as an obtuse-angled triangle.

**Right-angled triangle:** A triangle with one of its angles equal to 90Â° is known as a right-angled triangle. The side opposite the 90Â° angle is called the hypotenuse, and is the longest side of the triangle.

**Mark the mid-point** of the side of a triangle, and join it to its opposite vertex. This line segment is called a median. It is defined as a line segment drawn from a vertex to the mid-point of the opposite side. You can draw three medians to a given triangle. The medianspass through a common point. Hence, the mediansof a triangle are concurrent. This point of concurrence is called the centroid, and is denoted by G. The centroidand medians of a triangle always lie inside the triangle. The centroid of a triangle divides the median in the **ratio 2:1.**

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**Altitude: **The altitude of a triangle is a line segment drawn from a vertex and is perpendicular to the opposite side. A triangle has three altitudes. The altitudes of a triangle are concurrent. The point of concurrence is called the orthocentre, and is denoted by O.

**The altitude and orthocentre of a triangle **need not lie inside the triangle.

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**Properties of Triangles**

**The sum** of the three angles in a triangle

**Angle sum property:**

**The sum of the three angles** in a triangle is equal to 180Â°

Eg: If âˆ A,âˆ B and âˆ C are the angles of a triangle, then

Suppose a line XY is parallel to side BC. AB is a transversal that cuts line XY and AB, at A and B, respectively. As the alternate interior angles are equal, .âˆ 1 =âˆ 4 Also, âˆ 2 = âˆ 5 ,âˆ 4 ,âˆ 3 and âˆ 5 form a linear pair, and their sum is 180Â°.

**Exterior angle property:**

An exterior angle of a triangle is equal to the sum of its opposite interior angles.

Eg: In the figure here, âˆ 4 is called the exterior angle to triangle ABC, and âˆ 4 = âˆ 1 + âˆ 2.

The sum of the lengths of any two sides of a triangle is greater than the third side. In triangle ABC,

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**BC + CA > AB CA + AB > BC**

In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are ca**lled its legs.**

**Pythagorean theorem:**

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs. a^{2} = b^{2} + c^{2}

**Converse:**

If the Pythagoras property holds, then the triangle must be right-angled. That is, if there is a triangle such that the sum of the squares on two of its sides is equal to the square of the third side, then it must be a right-angled triangle.

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210 videos|109 docs|45 tests

### Exterior Angle Property of a Triangle

- Video | 03:37 min
### Examples: Medians, Altitudes and Exterior angle sum property

- Video | 05:33 min
### Test: The Triangle And Its Properties - 3

- Test | 20 ques | 20 min
### What are Medians of a Triangle?

- Video | 01:34 min
### Learning Angle Sum Property of a Triangle by an Activity

- Video | 02:58 min

- What are Altitudes of a Triangle?
- Video | 01:26 min
- Test: The Triangle And Its Properties - 2
- Test | 20 ques | 20 min