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# Properties of Triangle – Theorems & Examples

December 12, 2022

This post is also available in: हिन्दी (Hindi)

A triangle is a polygon with three angles, three sides, and three vertices. A triangle’s properties help us easily identify a triangle from a given set of figures.

Let’s understand the properties of triangles that are based on their sides and angles with examples.

## Properties of Triangle

The 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 as follows.

• Angle Sum Property
• Triangle Inequality Property
• Pythagoras Property
• Exterior Angle Property
• Congruence Property
• Similarity Property

### 1. Angle Sum Property

According to the angle sum property of a triangle, the sum of all three interior angles of a triangle is $180^{\circ}$. The angle sum property is used to find the measure of an unknown interior angle when the values of the other two angles are known.

The angle sum property formula for any polygon is expressed as, $\text{S} = \left(n − 2 \right) \times 180^{\circ}$, where $n$ represents the number of sides in the polygon. This property of a polygon states that the sum of the interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it.

In the case of a triangle, $n = 3$, therefore, the formula for triangle becomes $\text{S} = \left(3 − 2 \right) \times 180^{\circ} = 1 \times 180^{\circ} = 180^{\circ}$.

In the above figure, in $\triangle \text{LMN}$, $\angle \text{L} + \angle \text{M} + \angle \text{N} = 180^{\circ}$.

### 2. Triangle Inequality Property

The triangle inequality theorem states, “The sum of any two sides of a triangle is greater than its third side and the difference of any two sides of a triangle is less than its third side.” This theorem helps us to identify whether it is possible to draw a triangle with the given measurements or not without actually doing the construction.

For a triangle with length of sides $a$, $b$ and $c$,

• $a + b \gt c$
• $b + c \gt a$
• $c + a \gt b$
• $|a – b| \lt c$
• $|b – c| \lt a$
• $|c – a| \lt b$

### 3. Pythagoras Property

According to the Pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as $\left( \text{Hypotenuse} \right)^{2}= \left( \text{Base} \right)^{2} + \left( \text{Altitude} \right)^{2}$.

In the above figure, $\triangle \text{ABC}$ is a right-triangle, right-angled at $\text{B}$, where

• CA is Hypotenuse
• BC is Base
• AB is Altitude

Therefore, according to Pythagoras property, $\text{CA}^{2} = \text{BC}^{2} + \text{AB}^{2}$

### 4. Exterior Angle Property

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles of the triangle. The exterior angle of any interior angle and the interior angle form a linear pair.

In the above figure, the internal angles of the triangle are $\angle \text{A}$, $\angle \text{B}$, and $\angle \text{C}$, and their corresponding exterior angles are $\text{Ext} \angle \text{A}$, $\text{Ext} \angle \text{B}$, and $\text{Ext} \angle \text{C}$.

Note: The interior angle and its corresponding exterior angle form a linear pair.

### 5. Congruence Property

According to the congruence property, two triangles are said to be congruent(equal) if all their corresponding sides and angles are equal.

In the above figure, in $\triangle \text{ABC}$ and $\triangle \text{DEF}$, if $\text{AB} = \text{EF}$, $\text{BC} = \text{FD}$, $\text{CA} = \text{DE}$ and $\angle \text{C} =\angle \text{D}$, $\angle \text{A} =\angle \text{E}$, $\angle \text{B} =\angle \text{F}$, then we say that $\triangle \text{ABC}$ is congruent to $\triangle \text{DEF}$ or $\triangle \text{ABC} \cong \triangle \text{DEF}$.

### 6. Similarity Property

According to the similarity property, two triangles are said to be similar if all their corresponding angles are equal and the corresponding sides are in the same ratio.

In the above figure, in $\triangle \text{ABC}$ and $\triangle \text{DEF}$, if $\frac{\text{AB}}{\text{EF}} =\frac{\text{BC}}{\text{FD}} = \frac{\text{CA}}{\text{DE}}$ and $\angle \text{C} =\angle \text{D}$, $\angle \text{A} =\angle \text{E}$, $\angle \text{B} =\angle \text{F}$, then we say that $\triangle \text{ABC}$ is similar to $\triangle \text{DEF}$ or $\triangle \text{ABC} \sim \triangle \text{DEF}$.

## Practice Problems

Explain the following properties of a triangle

• Angle Sum Property
• Triangle Inequality Property
• Pythagoras Property
• Exterior Angle Property
• Congruence Property
• Similarity Property

## FAQs

### What are the basic properties of a triangle?

The basic properties of a triangle are
a) Angle Sum Property
b) Triangle Inequality Property
c) Pythagoras Property
d) Exterior Angle Property
e) Congruence Property
f) Similarity Property

### What is a right-angled triangle?

A triangle that has one of the interior angles as $90^{\circ}$ is called a right-angled triangle.

### What is the angle sum property of a triangle?

According to the angle sum property of a triangle, the sum of the interior angles of a triangle is always $180^{\circ}$. For example, if the three interior angles of a triangle are given as $\angle \text{A}$, $\angle \text{B}$, and $\angle \text{C}$, then this according to this property $\angle \text{A} + \angle \text{B} + \angle \text{C} = 180^{\circ}$.

## Conclusion

A three-sided figure commonly called a triangle has three vertices and three angles. All triangles exhibit five properties which are the angle sum property, triangle inequality property, exterior angle property, congruence property, and similarity property. The right triangles also exhibit Pythagoras property.