Class 7 Maths The Triangle and its Properties Notes

Introduction:

The chapter “The Triangle and its Properties” in Class 7 Maths focuses on the various types of triangles and their properties. Triangles are one of the most basic and fundamental shapes in geometry. Understanding triangles is crucial for students, as it forms the basis for many other complex geometrical concepts. In this chapter, students learn about different types of triangles, the sum of angles in a triangle, and the properties related to sides and angles.

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Key Concepts Covered:

1. What is a Triangle?

• A triangle is a closed figure with three sides and three angles. The sides of a triangle are straight lines, and the sum of the angles inside a triangle is always 180°.

Example: A triangle with vertices $A$A, $B$B, and $C$C is denoted as $\triangle ABC$△ABC.

2. Types of Triangles:

Triangles can be classified based on their sides or angles.

Based on Sides:

• Equilateral Triangle: All three sides are equal, and all three angles are equal (each angle is 60°).
  • Example: $\triangle ABC$△ABC where $AB = BC = CA$AB=BC=CA.
• Isosceles Triangle: Two sides are equal, and the angles opposite these sides are also equal.
  • Example: $\triangle ABC$△ABC where $AB = AC$AB=AC.
• Scalene Triangle: All three sides are of different lengths, and all three angles are different.
  • Example: $\triangle ABC$△ABC where $AB \neq BC \neq CA$AB=BC=CA.

Based on Angles:

• Acute Triangle: All three angles are less than 90°.
  • Example: $\triangle ABC$△ABC where $\angle A, \angle B, \angle C < 90^\circ$∠A,∠B,∠C<90∘.
• Right-Angled Triangle: One angle is exactly 90°.
  • Example: $\triangle ABC$△ABC where $\angle C = 90^\circ$∠C=90∘.
• Obtuse Triangle: One angle is greater than 90°.
  • Example: $\triangle ABC$△ABC where $\angle A > 90^\circ$∠A>90∘.

3. Sum of Angles in a Triangle:

• The sum of the interior angles of any triangle is always 180°.

Example:
In $\triangle ABC$△ABC, if $\angle A = 50^\circ$∠A=50∘ and $\angle B = 60^\circ$∠B=60∘, then $\angle C$∠C can be calculated as:$$\angle C = 180^\circ – (50^\circ + 60^\circ) = 70^\circ$$∠C=180∘−(50∘+60∘)=70∘

4. Exterior Angle Property of a Triangle:

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

Example:
In $\triangle ABC$△ABC, if $\angle D$∠D is an exterior angle at vertex $C$C, then:$$\angle D = \angle A + \angle B$$∠D=∠A+∠B

5. Pythagoras Theorem (For Right-Angled Triangles):

• The Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Formula:$$c^2 = a^2 + b^2$$c2=a2+b2

Where:

• $c$c is the length of the hypotenuse.
• $a$a and $b$b are the lengths of the other two sides.

Example:
In a right-angled triangle, if $a = 3 \, \text{cm}$a=3cm and $b = 4 \, \text{cm}$b=4cm, then the length of the hypotenuse $c$c is:$$c^2 = 3^2 + 4^2 = 9 + 16 = 25 \quad \Rightarrow \quad c = 5 \, \text{cm}$$c2=32+42=9+16=25⇒c=5cm

6. Properties of Triangles:

• Sum of Two Sides: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
  • Example: In $\triangle ABC$△ABC, if $AB = 5 \, \text{cm}$AB=5cm, $BC = 7 \, \text{cm}$BC=7cm, and $CA = 10 \, \text{cm}$CA=10cm, then:
  $$AB + BC > CA, \quad BC + CA > AB, \quad CA + AB > BC$$AB+BC>CA,BC+CA>AB,CA+AB>BC
• Inequality Theorem: The difference between the lengths of any two sides of a triangle is always smaller than the length of the third side.
  • Example: In $\triangle ABC$△ABC, $|AB – BC| < CA$∣AB−BC∣<CA, $|BC – CA| < AB$∣BC−CA∣<AB, $|CA – AB| < BC$∣CA−AB∣<BC.

7. Important Theorems:

• Angle Sum Property: The sum of the three interior angles of a triangle is always 180°.
• Exterior Angle Theorem: The exterior angle is equal to the sum of the two opposite interior angles.

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Important Questions with Answers:

1. What is the sum of the interior angles of a triangle?
   • Answer: The sum of the interior angles of a triangle is always 180°.
2. How many types of triangles are there based on sides?
   • Answer: There are three types of triangles based on sides: Equilateral, Isosceles, and Scalene.
3. What is the Pythagoras Theorem?
   • Answer: The Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: $c^2 = a^2 + b^2$c2=a2+b2.
4. What is the exterior angle property of a triangle?
   • Answer: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
5. What is an acute triangle?
   • Answer: An acute triangle is a triangle where all three angles are less than 90°.
6. What is the difference between a right-angled and an obtuse triangle?
   • Answer: A right-angled triangle has one angle equal to 90°, while an obtuse triangle has one angle greater than 90°.
7. State the triangle inequality theorem.
   • Answer: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.