NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and Its Properties

Triangle and Its Properties Class 7 Questions with Answers

What is a triangle?

A triangle is a simple closed figure made up of three sides, three vertices, and three angles.

What is a median of a triangle?

A median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.

Every triangle has three medians.

What is an altitude of a triangle?

An altitude is a perpendicular line segment drawn from a vertex to the opposite side or its extension.

Therefore, an altitude makes an angle of 90° with the side on which it falls.

What is the Angle Sum Property of a triangle?

The sum of the three interior angles of a triangle is always 180°.

For a triangle with angles AA, BB, and CC:

A+B+C=180∘A+B+C=180^\circ

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How do you find a missing angle in a triangle?

Use the Angle Sum Property.

For example, if two angles are 60° and 70°:

Third angle=180∘−(60∘+70∘)\text{Third angle}=180^\circ-(60^\circ+70^\circ) =50∘=50^\circ

Therefore, the missing angle is 50°.

What is the exterior angle property of a triangle?

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

For example, if the two opposite interior angles are 50° and 60°:

Exterior angle=50∘+60∘=110∘\text{Exterior angle}=50^\circ+60^\circ=110^\circ

What is the relationship between the sides of a triangle?

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

For sides aa, bb, and cc:

a+b>ca+b>c b+c>ab+c>a c+a>bc+a>b

This condition helps determine whether three given lengths can form a triangle.

Properties of Triangle Class 7

The important properties covered in this chapter include:

These properties are useful when solving triangle and its properties class 7 questions with answers.

Read More : NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles

Types of Triangles Class 7

Triangles can be classified according to their sides and angles.

Types of Triangles Based on Sides

Equilateral Triangle:
All three sides are equal. Each angle measures 60°.

Isosceles Triangle:
Two sides are equal, and the angles opposite those equal sides are also equal.

Scalene Triangle:
All three sides have different lengths.

Types of Triangles Based on Angles

Acute-Angled Triangle:
All three angles are less than 90°.

Right-Angled Triangle:
One angle is exactly 90°.

Obtuse-Angled Triangle:
One angle is greater than 90°.

Median and Altitude of a Triangle

Median and altitude are different even though both are drawn from a vertex.

A median is concerned with the midpoint, while an altitude is concerned with perpendicular distance.

Triangle and Its Properties Class 7 Extra Questions and Answers

Can three line segments of lengths 2 cm, 3 cm and 5 cm form a triangle?

No.

For a triangle, the sum of any two sides must be greater than the third side.

Here:

2+3=52+3=5

Since the sum is equal to the third side and not greater, these lengths cannot form a triangle.

Can sides of 4 cm, 5 cm and 6 cm form a triangle?

Yes.

4+5>64+5>6 5+6>45+6>4 6+4>56+4>5

All three conditions are satisfied, so these lengths can form a triangle.

If two angles of a triangle are 45° and 65°, find the third angle.

Using the Angle Sum Property:

Third angle=180∘−(45∘+65∘)\text{Third angle}=180^\circ-(45^\circ+65^\circ) =70∘=70^\circ

Therefore, the third angle is 70°.

If an exterior angle is 120° and one opposite interior angle is 50°, find the other opposite interior angle.

Using the exterior angle property:

120∘=50∘+x120^\circ=50^\circ+x x=70∘x=70^\circ

Therefore, the other interior angle is 70°.

Can a triangle have two right angles?

No.

Two right angles would already give:

90∘+90∘=180∘90^\circ+90^\circ=180^\circ

There would be no angle left for the third vertex. Therefore, a triangle cannot have two right angles.

Triangle and Its Properties Class 7 Worksheet with Answers

A. Fill in the blanks

1. A triangle has ______ sides.

2. The sum of the angles of a triangle is ______.

3. A line segment joining a vertex to the midpoint of the opposite side is called a ______.

4. An altitude forms an angle of ______ with the side.

5. A triangle with three equal sides is called an ______ triangle.

Answers:

1. three

2. 180°

3. median

4. 90°

5. equilateral

B. True or False

1. Every triangle has three sides.

2. The sum of the angles of a triangle is 360°.

3. An altitude is perpendicular to the opposite side or its extension.

4. An equilateral triangle has three equal sides.

5. Three sides of lengths 2 cm, 3 cm and 5 cm can form a triangle.

Answers:

1. True

2. False

3. True

4. True

5. False

C. Solve the following

1. Two angles of a triangle are 55° and 65°. Find the third angle.

180∘−(55∘+65∘)=60∘180^\circ-(55^\circ+65^\circ)=60^\circ

Answer: 60°

2. The angles of a triangle are 80°, 40° and x. Find x.

x=180∘−(80∘+40∘)x=180^\circ-(80^\circ+40^\circ) x=60∘x=60^\circ

Answer: 60°

3. Can 3 cm, 4 cm and 8 cm form a triangle?

No.

3+4=7<83+4=7<8

So, these lengths cannot form a triangle.

Read More : NCERT Solutions for Class 7 Maths Chapter 1 Integers

Triangle and Its Properties Class 7 Notes

Quick Notes on Medians

• A triangle has three medians.

• Each median joins a vertex to the midpoint of the opposite side.

• The three medians meet at a common point called the centroid.

Quick Notes on Altitudes

• A triangle has three altitudes.

• Each altitude is perpendicular to the opposite side or its extension.

• An altitude represents the perpendicular height from a vertex.

Quick Notes on Special Triangles

• Equilateral: Three equal sides and three equal angles.

• Isosceles: Two equal sides and two equal opposite angles.

• Scalene: Three unequal sides.

• Right-angled: One angle is 90°.

• Acute-angled: All angles are less than 90°.

• Obtuse-angled: One angle is greater than 90°.

How to Solve Triangle and Its Properties Questions

Follow these steps when solving numerical questions:

1. Read the given information carefully.

2. Identify the required angle or side.

3. Choose the correct triangle property.

4. Write the equation before calculating.

5. Solve the equation step by step.

6. Check whether the answer satisfies the triangle property.

For angle questions, the Angle Sum Property and exterior angle property are commonly used.

For questions involving three side lengths, check the triangle inequality before proceeding.

Triangle and Its Properties Class 7 Extra Practice

Question 1

One angle of a triangle is 90° and another is 35°. Find the third angle.

Answer:

180∘−(90∘+35∘)=55∘180^\circ-(90^\circ+35^\circ)=55^\circ

So, the third angle is 55°.

Question 2

An exterior angle of a triangle is 130°. One of its opposite interior angles is 75°. Find the other.

Answer:

130∘=75∘+x130^\circ=75^\circ+x x=55∘x=55^\circ

So, the other interior angle is 55°.

Question 3

Can 5 cm, 7 cm and 10 cm form a triangle?

Answer: Yes.

5+7>10,7+10>5,10+5>75+7>10,\quad 7+10>5,\quad 10+5>7

All three conditions are satisfied.

Question 4

A triangle has two equal sides. What type of triangle is it?

Answer: Isosceles triangle.

Question 5

A triangle has three equal sides. What type of triangle is it?

Answer: Equilateral triangle.

Quick Revision: The Triangle and Its Properties Class 7

• A triangle has three sides, three vertices and three angles.

• The sum of its interior angles is 180°.

• An exterior angle equals the sum of its two opposite interior angles.

• A median joins a vertex to the midpoint of the opposite side.

• An altitude is perpendicular to the opposite side or its extension.

• An equilateral triangle has three equal sides.

• An isosceles triangle has two equal sides.

• A scalene triangle has three unequal sides.

• A right-angled triangle has one 90° angle.

• The sum of any two sides of a triangle must be greater than the third side.

• The three medians meet at the centroid.

• The three altitudes meet at a point called the orthocentre.