Triangle Inequality
The triangle inequality theorem is one of the important mathematical principles that is used across various branches of mathematics. In real life, civil engineers use the triangle inequality theorem since their area of work deals with surveying, transportation, and urban planning. The triangle inequality theorem helps them to calculate the unknown lengths and have a rough estimate of various dimensions. In this article, let's learn about the triangle inequality theorem and its proof using solved examples.
1.  What is Triangle Inequality? 
2.  How Does Triangle Inequality Work? 
3.  Triangle Inequality Proof 
4.  FAQs on Triangle Inequality 
What is Triangle Inequality?
The Triangle Inequality (theorem) says that in any triangle, the sum of any two sides must be greater than the third side. For example, consider the following ∆ABC:
According to the Triangle Inequality theorem:
 AB + BC must be greater than AC, or AB + BC > AC.
 AB + AC must be greater than BC, or AB + AC > BC
 BC + AC must be greater than AB, or BC + AC > AB.
How Does Triangle Inequality Work?
An easy way to understand how the triangle inequality theorem works in any ∆ABC is to imagine yourself walking along the sides of the triangle. If you have to go from A to B, for example, the shortest path will be segment AB. If you first go to C and then to B, the distance you cover, AC + CB, will surely be greater than AB. Alternatively, let's try and understand the Triangle Inequality theorem through construction. Suppose that you are given three lengths: x, y, and z. You are asked to construct a triangle with these sides. You proceed as follows:
First, you draw a segment AB of length z units.  
Next, keeping the tip of your compass at A, you draw an arc of length x units.  
Then, keeping the tip of your compass at B, you draw an arc of length y units, in a way so that it intersects the earlier arc.  
The point of intersection is your third vertex C. You join A to C and B to C, and thus you have your triangle.  
The question now is: can this always be done? Suppose that the length y was so small that your second arc could never intersect your first arc (which had a radius of x units), in this case, a triangle cannot be formed with these three lengths. 
Observe carefully that the two arcs will intersect only if the sum of the radii of the two arcs is greater than the distance between the centers of the arc. In other words, to be able to draw a triangle:
x + y must be greater than z
This means, for example, there can be no triangle with sides 2 units, 2 units, and 5 units, because:
2 + 2 < 5
This is how triangle inequality works.
Triangle Inequality Proof
Let us now discuss the Triangle Inequality proof. Consider the following triangle, ∆ABC:
We need to prove that AB + AC > BC.
Proof: Extend BA to point D such that AD = AC, and join C to D, as shown below:
We note that ∠ACD = ∠D, which means that in ∆ BCD, ∠BCD > ∠D. Sides opposite larger angles are larger, and thus: BD > BC
AB + AD > BC
AB + AC > BC (because AD = AC)
This completes our proof. We can additionally conclude that in a triangle:
 Since the sum of any two sides is greater than the third, then the difference of any two sides will be less than the third.
 The sum of any two sides must be greater than the third side.
 The side opposite to a larger angle is the longest side in the triangle.
Related Topics:
Check out these interesting articles to learn more about triangle inequality and its related topics.
Important Notes
Here is a list of a few points that should be remembered while studying triangle inequality:
 The Triangle Inequality theorem states that in any triangle, the sum of any two sides must be greater than the third side.
 In a triangle, two arcs will intersect only if the sum of the radii of the two arcs is greater than the distance between the centers of the arc.
 In a triangle, if the sum of any two sides is greater than the third, this means that the difference of any two sides will be less than the third.
Examples on Triangle Inequality

Example 1: Check whether it is possible to form a triangle with the following measures: 7 units, 4 units, and 5 units.
Solution:
Let us assign the values as: a = 4 units, b = 7 units, and c = 5 units. Now let us apply the triangle inequality theorem:
a + b > c
⇒ 4 + 7 > 5
⇒ 11> 5 ……. (this is true)
a + c > b
⇒ 4 + 5 > 7
⇒ 9 > 7…………. (this is true)
b + c > a
⇒7 + 5 > 4
⇒12 > 4 ……. (this is true)
Answer: Since all three conditions are true, it is possible to form a triangle with the given measurements: 7 units, 4 units, and 5 units.

Example 2: Peter has three measurements with him: 6 cm, 10 cm, and 17 cm. Will he be able to form a triangle with these three measurements?
Solution:
Let us assign the values as: a = 6 cm, b = 10 cm and c = 17 cm
As per the triangle inequality theorem, we have;
a + b > c
⇒ 6 + 10 > 17
⇒ 16 > 17 ………. (false, 17 is not less than 16)
a + c > b
⇒ 6 + 17 > 10
⇒ 23 > 10…………. (this is true)
b + c > a
10 + 17 > 6
27 > 6 ………. (this is true)
Answer: Since one of the conditions is false, Peter will not be able to form a triangle with these three measurements.
FAQs on Triangle Inequality
What is the Triangle Inequality Theorem?
As per the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
What are the Applications of Triangle Inequality?
The triangle inequality theorem is one of the most important mathematical principles that is used across various branches of mathematics. It is a useful tool for checking if a given set of three dimensions will form a triangle or not. In real life, mapping applications like Google Maps make use of triangle inequalities to calculate unknown distances between places.
How Can Three Equal Sides Form a Triangle as per Triangle Inequality?
When three equal sides form a triangle, they form an equilateral triangle, and it can work because when two side lengths are added together, they are larger than the third side.
What are the Symbols Used in Triangle Inequalities?
The math symbols used in triangle inequalities are: greater than (>), less than (<), greater than or equal (≥), less than or equal (≤), and the not equal symbol (≠).
What are the 3 Properties of the Triangle Inequality Theorem?
The 3 properties of the triangle inequality theorem are:
 If the sum of any two sides is greater than the third, then the difference of any two sides will be less than the third.
 The sum of any two sides must be greater than the third side.
 The side opposite to a larger angle is the longest side in the triangle.
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