# The Sum of any Two Sides of a Triangle is Greater than the Third Side Theorem and Proof

Are you looking for the inequalities in triangles theorem in various sites? If yes, then use this page and learn the theorems. The Sum of any Two Sides of a Triangle is Greater than the Third Side is also known as the triangle inequality theorem. You can add any two sides in a triangle you will get the sum of two sides will be greater than the third side. Also check out the examples on proving  Sum of any Two Sides of a Triangle is Greater than the Third Side Theorem.

## Prove that the Sum of any Two Sides of a Triangle is greater than the Third Side

Theorem: The sum of two sides of a triangle is greater than the third side
Let PQR be a triangle.
To prove that : QP + PR > QR
Proof:
Produce QP to A,
Such that, PA = PR
∠ PAR = ∠ PRA
Since, By the diagram,
∠ ARQ > ∠ PRA
∠ ARQ > ∠ PAR
QA > PQ ( Because the sides opposite to the larger angle is larger and the sides opposite to the smaller angle is smaller )
QP + PA > QR
QP + PR > QR
Hence proved
Therefore it is proven that the Sum of any Two Sides of a Triangle is Greater than the Third Side

Do Check:

### Problems on the Sum of the Length of any Two Sides of a Triangle is greater than the Length of the Third Side

Example 1.
If 5cm, 9cm, and 3cm are the measures of a three-line segment, can it be used to draw a triangle?
Solution:
The triangle is formed by three line segments 5cm, 9cm, 3cm then it should satisfy the inequality theorem.
Check if the sum of two sides is greater than the third side
5 + 9 > 3 = 14 > 3 so it is true
9 + 3 > 5 = 12 > 5 so it is true
5 + 3 > 8 = 8 > 9 so it is false
Therefore the sides of the triangle do not satisfy the inequality theorem.
So we cannot construct a triangle with these three line segments.

Example 2.
Could a triangle have side lengths of 7cm, 8cm, and 6cm?
Solution:
If 7cm, 8cm, and 6cm are the sides of the triangle then they should satisfy the inequality theorem
Hence
7 + 8 > 6 = 15 > 6 so it is true
8 + 6 > 7 = 14 > 7 so it is true
7 + 6 > 8 = 13 > 8 so it is true
All the three conditions are satisfied therefore a triangle could have side lengths of 6cm, 7cm, 8cm.

Example 3.
If the two sides of a triangle are 3cm and 8cm. Find all the possible lengths of the third side.
Solution:
To find the possible values of the third side of the triangle we can use the formula.
A difference of two sides less than unknown side less than the sum of two sides
8 – 3 < x < 8 + 3
5 < x < 11
There could be any value for the third side between 5 and 11.

Example 4.
If the two sides of a triangle are 2cm and 4cm. Find all the possible lengths of the third side.
Solution:
To find the possible values of the third side of the triangle we can use the formula.
A difference of two sides less than unknown side less than the sum of two sides
4 – 2 < x < 4 + 2
2 < x < 6
There could be any value for the third side between 2 and 6.

Example 5.
Could a triangle have side lengths as 3cm, 4cm, and 2cm?
Solution:
If 3cm, 4cm, and 5cm are the sides of the triangle then they should satisfy the inequality theorem
Hence
3 + 4 > 2 = 7 > 2 so it is true
4 + 2 > 3 = 6 > 3 so it is true
3 + 2 > 4 = 5 > 4 so it is true
All the three conditions are satisfied therefore a triangle could have side lengths as 2cm, 3cm, cm.

### FAQs on the Sum of any Two Sides of a Triangle is Greater than the Third Side

1. Why is the sum of two sides of a triangle greater than the third?

The sum of the lengths of any two sides of a triangle must be greater than the third side.

2. Is the difference between the two sides of a triangle is less than the third side?

A triangle is a polygon with three sides such that the sum of any two sides is greater than the third side. Thus, we can say that the difference between the two sides is less than the third side.

3. Is the sum of any two angles of a triangle always greater than the third side?

The Sum of any two angles of a triangle is always greater than the third angle. The Sum of any two angles of a triangle is either greater than the third angle or smaller than the third angle.