Class 9 Maths Chapter 7 Triangles

The chapter “Triangles” in Class 9 introduces students to one of the most important shapes in geometry. It explains the basic concepts of triangles, their sides and angles, and the rules that help us understand their properties. This chapter mainly focuses on triangle congruence and the conditions under which two triangles are exactly the same. Learning about triangles helps students build a strong foundation for solving geometric problems and understanding shapes used in everyday life and higher mathematics. 

So, we're about to break triangles down in the simplest way possible, no stress, no confusion. Just triangles made easy.

Triangle Notes for Class 9

This section covers the key points and important concepts in a clear and easy-to-understand way. These notes are helpful for quick revision and better understanding of the chapter. So, download the notes from below and start your preparation today.

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1. Class 9 Triangle Notes Introduction

A triangle is a closed figure made up of three sides and three angles. The points where the sides meet are called vertices and the line segments joining them are called sides. Triangles can be classified in two ways. 

By sides, they can be equilateral (all sides equal, all angles 60°), isosceles (two sides equal, two angles equal) or scalene (all sides and angles different). By angles, triangles can be acute (all angles less than 90°), right-angled (one angle exactly 90°) or obtuse (one angle more than 90°). 

Triangles are widely used in real life, such as in buildings, bridges and even in nature, making them an important shape to understand.

2. What is a Triangle? 

• A triangle is a closed figure that is formed by the three life segments.
• A triangle is a simple polygon with 3 sides and 3 interior angles. It is one of the basic shapes in geometry in which the 3 vertices are joined with each other and it is denoted by the symbol △. 
• Triangle has 3 sides, angles and vertices
• Example: Suppose Triangle XYZ has sides XY,YZ and YX .

3. Types of Triangle 

Triangles can be classified on the basis of their sides and angles. Let us understand the classification of triangles with the help of the table given below which shows the difference between different types of triangles on the basis of angles and sides.

Let us look into different types of triangles.

1. On the basis of sides of Triangle 

a) Scalene Triangle- In this  triangle sides are different from each other.

b) Isosceles Triangle - In this Triangle only two sides are equal.

c) Equilateral Triangle - In this triangle all sides of the triangle are equal.

2. On the basis of Angle

a) Acute Triangle- In this triangle all angles are 90 degrees.

b) Right Triangle - In this triangle only one angle is 90 degrees.

c) Obtuse Triangle - In this triangle one side is greater than 90 degrees.

4. Triangle Inequality Theorem Proof

The triangle inequality theorem describes the relationship between the three sides of a triangle. 

According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side.

In simple words, this theorem specifies that the shortest distance between two distinct points is always a straight line.

Proof: Consider a ∆ABC as shown below, with a, b and c as the side lengths.

The triangle inequality theorem states that:

• a < b + c,
• b < a + c,
• c < a + b

The sum of any two sides of a triangle is always greater than the third side.

5. Congruency of Triangles

Congruent triangles are triangles that are exactly equal in shape and size.

If ∆ABC ≅ ∆PQR, then:

• AB = PQ
• ∠A = ∠P, etc.

Criteria for Congruence:

1. SSS (Side-Side-Side)
2. SAS (Side-Angle-Side)
3. ASA (Angle-Side-Angle)
4. AAS (Angle-Angle-Side)
5. RHS (Right angle-Hypotenuse-Side)

Properties of Triangle: Following are some properties of triangles:

• Isosceles Triangle Theorem: Angles opposite to equal sides are equal.
• Equilateral Triangle: All angles are 60°.

Inequalities in a Triangle

• The longest side is opposite the largest angle.
• The smallest side is opposite the smallest angle.

Problems Related to Triangles

Let us look at some problems.

Q.1. If 4cm, 8cm and 2cm are the measures of three lines segment. Can it be used to draw a triangle?

Solution: The triangle is formed by three line segments 4cm, 8cm and 2cm, then it should satisfy the inequality theorem.

Hence, let us check if the sum of two sides is greater than the third side.

• 4 + 8 > 2 ⇒ 12 > 2 ⇒ True
• 8 + 2 > 4 ⇒ 10 > 4 ⇒ True
• 4 + 2 > 8 ⇒ 6 > 8 ⇒ False

Therefore, the sides of the triangle do not satisfy the inequality theorem. So, we cannot construct a triangle with these three line-segments.

Q.2: If the two sides of a triangle are 2 and 7. Find all the possible lengths of the third side.

Solution: To find the possible values of the third side of the triangle we can use the formula:

A difference of two sides< Unknown side < Sum of the two sides

• 7 -2 < x < 7 + 2
• 5 < x <9

There could be any value for the third side between 5 and 9.

6. Area of a Triangle

The area of a triangle is the space covered by the triangle. It is half the product of its base and altitude (height). It is always measured in square units, as it is two-dimensional. 

Observe the triangle ABC given below which shows the base and height of a triangle which are used to calculate the area of a triangle.

• Area of Triangle= ½ x BxH
• Area of ΔABC = 1/2 × BC × AD

Here, BC is the base and AD is the height of the triangle.

7. Important Notes on Triangle

• A triangle is a 3-sided closed shape.
• There are two important formulas related to triangles, i.e., the Heron's formula and Pythagoras theorem.
• The sum of the interior angles of a triangle is 180° and is expressed as ∠1 + ∠2 + ∠3 = 180°.

In conclusion, you now have the key to solve any question on triangles. Just remember the rules and focus on understanding the logic behind them. 

Don’t only memorize the concepts, practice them regularly. Draw diagrams, revise the types and properties, stay calm during exams and you’ll be able to ace this chapter with confidence.

FAQs

Q1. Why is the sum of the angles of a triangle always 180°?

Ans: This happens because a triangle is part of a plane and the angles along a straight line sum to 180°. By drawing a line parallel to one side, we can see that the interior angles of the triangle always add up to 180°.

Q2. What does the Triangle Inequality Theorem tell us?

Ans: It states that the sum of any two sides of a triangle is always greater than the third side. This ensures that the three sides can actually form a closed triangle.

Q3. How do we know which side is the longest or shortest in a triangle?

Ans: The largest side is always opposite the largest angle and the smallest side is opposite the smallest angle. This helps in comparing sides and angles easily.

Q4. What makes two triangles congruent?

Ans: Two triangles are congruent if their sides and angles match according to a congruence rule (SSS, SAS, ASA, AAS or RHS). Congruence means the triangles are exactly equal in shape and size.

Q5. Why is understanding the height (altitude) important for triangles?

Ans: The height is needed to calculate the area of a triangle. It is always perpendicular to the base and without it, we cannot correctly find the space inside the triangle.