A Tale of Three Intersecting Lines Class 7 Notes Maths Chapter 7

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Class 7 Maths Chapter 7 Notes A Tale of Three Intersecting Lines

Class 7 Maths Notes Chapter 7 – Class 7 A Tale of Three Intersecting Lines Notes

→ Using a compass helps in easily constructing triangles when side lengths are provided.

→ A triangle can be formed if each side is shorter than the sum of the other two sides – this is called the triangle inequality.

→ When a given set of side lengths satisfies the triangle inequality, a triangle can be constructed using them.

→ Triangles can be constructed when:

• Two sides and the included angle are known.
• Two angles and the included side are known.

→ The sum of the angles in any triangle is always 180°.

→ The altitude of a triangle is a perpendicular segment from a vertex to its opposite side.

→ Triangle types by sides:

• Equilateral: All sides are equal.
• Isosceles: Two sides are equal.
• Scalene: All sides are of different lengths.

→ Triangle types by angles:

• Acute-angled: All angles are less than 90°.
• Right-angled: One angle is 90°.
• Obtuse-angled: One angle is more than 90°.

Equilateral Triangles Class 7 Notes

A triangle is called an equilateral triangle when all three sides are equal in length and all three angles are equal. In an equilateral triangle, each angle measures 60 degrees. Because of these special properties, equilateral triangles are also known as regular triangles. They are a perfect example of symmetry in geometry.

• All sides are of equal length.
• All angles are equal (each is 60°).

Question 1.
Construct a triangle where each side is 5 cm long.
Solution:
This triangle is equilateral because:

• All three sides are equal: 5 cm
• All three angles will be 60°
• It looks the same from all three corners.

Construction:

1. Draw a line segment of 5 cm and name it AB.
2. Open an arc of 5 cm using a compass.
3. From points A and B, draw the arcs to intersect each other.
4. The point where they meet each other is point C.
5. Join A to C and B to C.
6. ABC is the required triangle.

 

Name of the triangle: Triangle ABC
Sides: AB = BC = CA = 5 cm
Angles: ∠A = ∠B = ∠C = 60°

Constructing a Triangle When its Sides are Given Class 7 Notes

In geometry, constructing a triangle means drawing it accurately using a ruler and compass. When the lengths of all three sides of a triangle are given, we can use a method called SSS Construction (Side-Side-Side). This method helps us draw a triangle with the exact measurements of the sides provided. It is important to remember that the sum of the lengths of any two sides must be greater than the third side. This rule is called the triangle inequality rule, and it ensures that a triangle can actually be formed.

Construction of Triangles when some Sides and Angles are Given Class 7 Notes

1. SAS Construction (Side-Angle-Side)
In this case, we are given two sides and the included angle (the angle between the two given sides).
Steps to Construct (SAS):

1. Draw the first side using a ruler.
2. At one end of this side, use a protractor to draw the given angle.
3. From the angle arm, measure the second side using a ruler and mark the point.
4. Join the endpoint to the free end of the first side to complete the triangle.

Question 1.
Construct triangle ABC where AB = 5 cm, ∠ABC = 60°, and BC = 4 cm.
Solution:
Given AB = 5 cm, ∠ABC = 60°, BC = 4 cm
Steps of Construction (SAS):

1. Draw a line segment AB = 5 cm using a ruler.
2. Place the protractor at point B and draw an angle ∠ABC = 60°.
3. On the arm of the angle, measure BC = 4 cm and mark a point C.
4. Join point C to point A to complete the triangle ABC.

2. AAS Construction (Angle-Angle-Side)
In this case, you are given two angles and one side that is not between the two angles.
Steps to Construct (AAS):

1. Draw the given side using a ruler.
2. At each end of the side, use a protractor to draw the given angles.
3. Extend the angle anus until they meet. The point where they meet is the third vertex of the triangle.

Question 1.
Construct triangle XYZ where YZ = 6 cm, ∠Y = 50°, and ∠Z = 60°.
Solution:
Given: YZ = 6 cm, ∠Y = 50°, ∠Z = 60°
Steps of Construction (AAS):

1. Draw a line segment YZ = 6 cm using a ruler.
2. At point Y, use a protractor to draw ∠Y = 50°.
3. At point Z, use a protractor to draw ∠Z = 60°.
4. Extend both angle arms. Let them meet at point X.
5. Join X to Y and X to Z. Triangle XYZ is formed.

Angle Sum Property Class 7 Notes

Angle Sum Property of a Triangle
The Angle Sum Property states that: The sum of the three interior angles of any triangle is always 180°. This rule is true for all types of triangles—whether it is a scalene, isosceles, or equilateral triangle.

Example:
In triangle ABC: ∠A = 50°, ∠B = 60°, and ∠C = 70°
Then, ∠A + ∠B + ∠C = 50° + 60° + 70° = 180°
So, the triangle follows the Angle Sum Property.

Constructions Related to Altitudes of Triangles Class 7 Notes

In our daily life, we often talk about the height of objects, like a person, a tree, or a building. Similarly, in geometry, we can talk about the height of a triangle. The height of a triangle from a vertex is the perpendicular distance from that vertex to the opposite side (or the line containing the opposite side). This perpendicular line segment is called an altitude.

A triangle has three altitudes, one from each vertex. These altitudes may lie inside or outside the triangle, depending on its type (acute, right, or obtuse). Constructing these altitudes helps us understand the triangle’s structure better and is an important skill in geometric drawing.
In this section, you will learn how to:

• • Identify the altitude from a vertex to the opposite side.
• • Use a miler and a set square to construct altitudes accurately.

Steps to Construct an Altitude from a Vertex to the Opposite Side

Question 1.
Construct the altitude from vertex A to side BC in triangle ABC.

Solution:
Step-by-Step Method:

1. Draw any triangle and label the vertices A, B, and C. Take side BC as the base.
2. Place the Set Square on BC. Align one arm of the right angle of the set square along the side BC (the base).
3. Slide the Set Square to Point A. Keeping one arm along BC, slide the set square until the other arm passes through Point A.
4. Draw the Perpendicular Line. Using the edge of the set square that passes through A, draw a straight line from A to BC. This line should meet BC (or its extension) at a right angle (90°).
5. Label the foot of the perpendicular. Mark the point where the perpendicular from A meets BC as D. Now, AD is the altitude from vertex A to base BC.

Types of Triangles Class 7 Notes

In geometry, triangles are one of the most fundamental shapes. A triangle is a three-sided polygon with three angles. The classification of triangles can be done in two main ways – based on the length of their sides and based on their angle measures.
On the sides, triangles can be:

• Equilateral: All sides are equal.
• Isosceles: Two sides are equal.
• Scalene: All sides are of different lengths.

By angles, triangles can be:

• Acute-angled: All angles are less than 90°.
• Right-angled: One angle is exactly 90°.
• Obtuse-angled: One angle is more than 90°.