Properties of Triangle
The triangle is a closed polygon that has three angles, three sides, and three vertices. Based upon the length of sides and measure of angles, the triangles are classified into different types of triangles. Properties of a triangle help us to identify a triangle from a given set of figures easily and quickly.
In the beginning, we start from understanding the shape of triangles, their types, and properties, theorems based on them such as Pythagoras theorem, etc. Let us learn here some basic properties of triangles.
1.  Triangles: Classification by Type 
2.  Triangle Properties 
3.  Solved Examples 
4.  Practice Questions 
5.  FAQs 
Triangles: Classification by Type
Triangles can be classified into two broad categories based on their angles and sides. Based on angles, there are 3 types of triangles, and based on sides, there are 3 types of triangles.
Triangle Properties
Properties of a triangle help us to identify relationships between different sides and angles of a triangle. Some of the important properties of a triangle are listed below.
Property 1  Angle Sum Property
As per the angle sum property, the sum of the three interior angles of a triangle is always 180°.
In the given triangle, angle P + angle Q + angle R = 180°
Property 2  Triangle Inequality Property
As per the triangle inequality property, the sum of the length of the two sides of a triangle is greater than the third side.
In △ ABC and △ PQR
 a + b > c ( 6 + 4 > 3)
 c + a > b (3 + 4 > 6)
 c+ b > a ( 6 + 3 > 4)
Property 3  Pythagorean Theorem
As per the Pythagorean theorem, in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Mathematically, it can be expressed as Hypotenuse² = Base² + Altitude².
Property 4  Side opposite the greater angle is the longest side
In order to understand the side opposite the greater angle is the longest side property, let's take the belowgiven triangle into consideration.
In this triangle, angle B is the greatest angle. Thus, the side AC is the longest side.Property 5  Exterior Angle Property
As per the exterior angle property, the exterior angle of a triangle is always equal to the sum of the interior opposite angles.
In the given triangle, Exterior angle E1 = angle PQR + angle QRP
There are 3 exterior angles in a triangle and all these exterior angles add up to 360° for any polygon.
Property 6  Congruence Property
As per the Congruence Property, two triangles are said to be congruent if all their corresponding sides and angles are equal.
 angle XYZ = angle DEF
 angle YXZ = angle EDF
 angle YZX = angle EFD
 XY = DE
 XZ = DF
 YZ = EF
The basic triangle properties such as the area and perimeter of a triangle are given below.
 Area of a triangle: The total amount of space inside the triangle is called the area of a triangle. The area is measured in square units. The general formula for calculating the area of a triangle is Area (A) = (1/2) × Base × Height
 Perimeter: The perimeter of a triangle = sum of all its three sides

Heron's formula: Heron’s formula is used to calculate the area of a triangle if the lengths of all the sides are known and the height of the triangle is not known. First, we need to calculate the semiperimeter(s). For a triangle with sides p, q, and r, s = (p+q+r)/2, the area is given by; A = \(\sqrt{S(SP)(SQ)(SR)}\)
Properties of a Triangle Related Topics
Check out these interesting articles to know more about properties of a triangle and its related topics.
 Angles
 Equilateral
 Area of Triangle
 Pythagoras Theorem
 Similarity in Triangles
 SSS Criterion in Triangles
Important Notes
 The triangle is a closed polygon that has three angles, three sides, and three vertices.
 Sides and angles are very important aspects of a triangle. We can classify various types of triangles in math by combining sides and angles.
 The general formula for calculating the area of a triangle is Area (A) = (1/2) × Base × Height
 The perimeter of a triangle is equal to the sum of all three sides of the triangle.
Solved Examples

Example 1: Two angles of a triangle measure 75° and 60°. What will be the measure of its third angle?
Solution:
Measures of two angles of a triangle are 75° and 60°
Sum of the measures of two angles = 75° + 60° = 135°
Sum of all three angles of triangle = 180°
Therefore, the measure of the third angle = 180°  135° = 45°. 
Example 2: Tim wants to construct a triangle with the lengths of sides 5 cm, 4 cm, and 9 cm. Can he do it?
Solution:
The side lengths are 5 cm, 4 cm, 9 cm.
5 cm + 4 cm = 9 cm
Here the sum of the two smaller sides is equal to the third side. But as per the triangle inequality theorem, the sum of any two sides should be greater than the third side.
Hence, Tim won't be able to construct a triangle with sides 5 cm, 4 cm, and 9 cm. 
Example 3: A triangle has a dimension of 3 cm, 4 cm, and 5 cm, where the base is 4 cm and the altitude of the triangle is 3.2 cm. Calculate the area and perimeter of the triangle.
Solution:
Sides of the triangle are: x = 3 cm, y = 4 cm and z = 5 cm
Altitude = 3.2 cmUsing the area of the triangle formula, we know, Area = 1/2 × base × height
A = (1/2) × 4 × 3.2
A = 6.4 sq.cm.The perimeter of the triangle is given by P = x + y + z
P = 3 + 4 + 5
P = 12 cm
Therefore, the area and perimeter of the given triangle are 6.4 cm^{2} and 12 cm respectively.
FAQs on Properties of Triangle
What are the 5 Properties of a Triangle?
Five properties of a triangle are listed below:
 A triangle has three sides, three vertices, and three angles.
 The sum of the three interior angles of a triangle is always 180°.
 The sum of the length of two sides of a triangle is always greater than the length of the third side.
 A triangle with vertices P, Q, and R is denoted as △PQR.
 The area of a triangle is equal to half of the product of its base and height.
How many Types of Triangles are there in Maths?
There are basically six types of triangles. They are scalene triangles, isosceles triangles, equilateral triangles, acute triangles, obtuse triangles, and right triangles.
What is a RightAngle Triangle?
A triangle that has one of the interior angles as 90 degrees is a rightangled triangle.
What is the Sum of all 3 Angles of a Triangle?
The sum of the three interior angles of a triangle is always 180°.
What do all Triangles have in Common?
Triangles come in all shapes and sizes but regardless, all triangles have three sides and three angles.
What is the Area of a Triangle?
The area of a triangle is equal to half of the product of its base and height. It is the space enclosed by the perimeter of the triangle.
What is the Triangle Inequality Theorem?
Triangle inequality states that the sum of the length of any two sides of a triangle is always greater than the length of the third side.