Complete Guide For Class 7 Math Chapter 6: The Triangle and Its Properties

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The chapter on Triangles and Its Properties introduces key concepts essential for comprehending various geometric shapes and their interrelations. Students will learn to identify and categorize different types of triangles, including equilateral, isosceles, scalene, acute, right, and obtuse triangles. This chapter highlights the significance of the angle sum property, which asserts that the interior angles of a triangle always add up to 180 degrees, as well as the Pythagorean theorem, which is vital for right triangles. Furthermore, students will investigate concepts such as medians, altitudes, and perpendicular bisectors. Mastering these concepts is crucial for addressing more complex geometric problems in advanced mathematics and applying these skills in practical scenarios, such as construction, design, and engineering.

The Triangle and Its Properties

The chapter Triangle and its Properties starts with the definition of a triangle which is:

Definition of Triangle

A triangle is a simple closed curve made up of three line segments. It has

• Three  line segments: AB, BC, CA
• Three angles: ABC, BAC, AC 
• Three vertices: A, B, C

Elements of a Triangle: The three angles and the three sides of a triangle are called the elements of a triangle.

Now let’s understand the Medians & Altitudes of a Triangle

Median of a Triangle

According to the chapter Triangle and Its Properties, A median is a line segment joining a vertex to the mid-point of the opposite side. Every triangle has three medians.

• AN is the median of side BC.
• CL is the median of side AB.
• BM is the median of side AC.

Altitude of a Triangle

According to the chapter Triangle and its Properties, an altitude is a line segment drawn perpendicular from a vertex to the opposite side. Each triangle has three altitudes.

• AM is the altitude of side BC.
• BN is the altitude of side AC.
• CO is the altitude of side AB.

Classification of Triangles

In the chapter Triangle and its Properties, it is mentioned that triangles can be classified on various grounds. These Include:

1. Based on Sides

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

2. Based on Angles

• Acute Angled Triangle: All angles are less than 90°.
• Right Angled Triangle: One angle is exactly 90°.
• Obtuse Angled Triangle: One angle is greater than 90°.

Now let’s learn about Angle Sum Property

•  Angle Sum Property

The sum of the angles of a triangle is always equal to 180°. ∠A+∠B+∠C=180°

Special Types of Triangles

• Equilateral Triangle

            All sides are equal, and each angle measures 60°.

• Isosceles Triangle

           It has two sides of equal length, and the angles opposite these sides are also equal.

• Exterior Angle Property

The exterior angle of a triangle is equal to the sum of its two interior opposite angles. ∠ACD = ∠CAB+∠CBA.

Example of Exterior Angle Property

In the chapter Triangle and its Properties, there are various examples of Exterior Angle Properties, one of them is given below.

Example: Find the value of unknown angles using the property that the exterior angle equals the sum of the two interior opposite angles.

Inequality Property

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

Key Points

• AB + BC > AC
•  BC + CA > AB
• CA + AB > BC

Example:  If in triangle ABC, AM is the median of the triangle, Show that AB + BC + CA > 2 AM

Solution: In ABM, AB + BM > AM …(i) (sum of any two sides is greater than the third side). In ACM, AC + CM > AM …(ii) (sum of any two sides is greater than the third side).

Adding (i) and (ii),  we get AB + BM + AC + CM > AM + AM

 AB + (BM + CM ) + AC > 2AM

 AB + BC + AC > 2 AM 

Hence proved.

Pythagoras Theorem

Right-Angled Triangle: In a right-angled triangle, certain special names are given to its parts.

In a right-angled triangle: The Hypotenuse Square is equal to the sum of the base square and perpendicular square. 

 Hypotenuse² =  Base² + Perpendicular²

According to the given figure: c² = a² + b²

Examples Using Pythagoras’ Theorem

in the chapter Triangle and its properties, there are various examples of using the Pythagoras theorem. One of them is given below.

Example: The diagonals of a rhombus measure 16 cm and 30 cm. To find the perimeter of the rhombus, use the right triangle properties formed by the diagonals.

Solution: In a Rhombus the diagonals are perpendicular and bisect each other.

In right angled triangle AOB,                         

AB² = AO² + OB²

AB² = (8)² +(15)²

AB² = 64 + 225

AB² = 289

AB = 17

Hence the perimeter = 4 (side) = 4( 17 ) = 68 cm.

In conclusion, mastering the concepts in Chapter 6: The Triangle and Its Properties is vital for building a strong foundation in geometry. From understanding the basic properties of different types of triangles to applying the Pythagorean theorem, this chapter equips students with essential skills that are not only crucial for academic success but also have real-world applications in fields like construction, design, and engineering. With iPrep App’s engaging resources, including animated videos, practice questions, and comprehensive notes, you can reinforce your understanding of the chapter Triangle and its Properties and excel in your studies. Keep exploring, practicing, and perfecting your knowledge to unlock your full potential in mathematics!