Orchids Logo
Admissions 2025-26
9-999-431-999info@orchids.edu.in
Login
Doubt Connect
Orchids Logo
☰

Properties Of Triangle

Properties of Triangle

The properties of triangles help us understand the rules that govern the structure and shape of a triangle. A triangle is a closed figure with three sides, three angles, and three vertices. These properties help us analyze the relationships between the sides and angles, validate the shape, and solve problems related to area, perimeter, and classification.

Triangles are classified based on two main aspects:

By sides: Equilateral, Isosceles, Scalene
By angles: Acute, Right, Obtuse

Let us explore the 5 properties of triangle, along with key formulas, theorems, and solved examples.

 

Table of Contents

 

What Are the Properties of Triangles?

To understand the properties of triangles, it's important to first recognize that every triangle, regardless of type, follows some fundamental rules. These properties are common across all triangles unless stated otherwise and form the basis of triangle geometry in mathematics.

 

Different Types of Triangles

Triangles can be classified in two ways:

Based on sides:

  • Equilateral Triangle: All sides and angles are equal.

  • Isosceles Triangle: Two sides and two angles are equal.

  • Scalene Triangle: All sides and angles are different.

Based on angles:

  • Acute Triangle: All angles are less than 90°.

  • Right Triangle: One angle is exactly 90°.

  • Obtuse Triangle: One angle is more than 90°.

Understanding triangle types helps in applying the correct properties of triangle formula during problem-solving.

 

Properties of Triangle

Here are the 5 important properties of triangle that every student should know:

1. Angle Sum Property

The sum of the three interior angles in any triangle is always 180°.
Formula: ∠A + ∠B + ∠C = 180°

2. Triangle Inequality Property

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

If the sides are a, b, and c:

  • a + b > c

  • b + c > a

  • c + a > b

3. Pythagoras Property

This property applies only to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Formula: Hypotenuse² = Base² + Height²

4. Exterior Angle Property

The measure of an exterior angle of a triangle is equal to the sum of the interior opposite angles.
Formula: Exterior Angle = Interior Angle 1 + Interior Angle 2

5. The Longest Side is Opposite the Largest Angle

In a triangle, the side opposite the greatest angle is the longest side. This helps in estimating side lengths when angles are known.

 

Additional Properties of Triangles

Congruence Property

Two triangles are congruent if all their corresponding sides and angles are equal. This is validated using congruence rules such as SSS, SAS, ASA, and RHS.

Area of Triangle

The area of a triangle refers to the total space enclosed by its three sides.
Basic formula: Area = ½ × base × height

Perimeter of Triangle

The perimeter is the sum of all three sides of a triangle.
Formula: Perimeter = a + b + c

Heron’s Formula

If all sides are known and height is not available:
Let a, b, c be the sides and
s = (a + b + c)/2
Then,
Area = √[s(s – a)(s – b)(s – c)]

 

Important Notes on Properties of Triangle

  • All triangles have three sides, three angles, and three vertices.

  • The angle sum property and triangle inequality are applicable to all triangle types.

  • Right triangles specifically follow the Pythagoras theorem.

  • The congruence property helps in comparing triangles and proving geometric relationships.

Solved Examples

Example 1:
Two angles of a triangle are 75° and 60°. What is the third angle?
Solution:
Sum = 75° + 60° = 135°
Third angle = 180° – 135° = 45°

Example 2:
Can a triangle have sides 5 cm, 4 cm, and 9 cm?
Solution:
5 + 4 = 9 → Not greater than third side
Hence, triangle cannot be formed (violates triangle inequality).

Example 3:

 Find the perimeter of a triangle with sides 3 cm, 4 cm, and 5 cm.
Solution:
Perimeter = 3 + 4 + 5 = 12 cm

 

Conclusion

Understanding the properties of triangles is fundamental to geometry. Whether it's using formulas to find angle measures, applying the triangle inequality theorem to check constructibility, or using Heron’s formula for area, these concepts play a vital role in both academic and real-world mathematical applications.

Grasping the 5 properties of triangle empowers students to solve problems accurately and builds a strong foundation in geometric reasoning.

Related Links : 

Triangles : Learn all about types, properties, and formulas of triangles with step-by-step lessons and examples.

Area Of Triangle :  Understand formulas and solve problems quickly !

 

Frequently Asked Questions on Properties of Triangle

Q1. What are the 5 properties of a triangle?

Angle sum property, triangle inequality, Pythagoras theorem, exterior angle property, and the longest side opposite the largest angle.

Q2. What is the angle sum property of triangles?

 The sum of all three interior angles of a triangle is always 180 degrees.

Q3. What is the triangle inequality theorem?

 It states that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

Q4. What formulas are used to find the area of a triangle?

The basic formula is ½ × base × height. If all sides are known, Heron’s formula is used.

Q5. How do we classify triangles?

 By sides: Equilateral, Isosceles, Scalene
By angles: Acute, Right, Obtuse

 

Explore the complete guide on the properties of triangles at Orchids International and build a solid foundation in geometry with interactive lessons and expert explanations.

Share
Loading Footer...