Triangle Inequality

Triangle Inequality Definition

We have studied the two-dimensional figure, triangle. A triangle is a polygon formed by the intersection of three lines.. A triangle has three vertices, three sides, and three angles.

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The above figure shows \[\triangle ABC\], here AB, BC, AC, are the sides of the triangle ABC. A, B, C are the vertex and ∠ A, ∠B, ∠ C are the three angles. Also we have studied the equality of sides, angles and triangles. But we also come across unequal figures, so is there any relation between unequal sides and unequal angles of a triangle? Let us study triangle inequality definition and triangle inequality theorem proof and triangle inequality formula.


What is Triangle Inequality?

We have studied the equality of sides and angles between two triangles or in a triangle. Sometimes we may come across unequal objects or figures.


In our instances of comparisons, we take into consideration every part of the object. Similarly, when we compare various parts of a triangle, we compare every part individually. So let us study what is the triangle inequality. For instance, take the line segment shown in the figure:

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From the above figure, we can see that line segment AB is smaller than the line segment PQ. The difference is clear, yet to know the difference and confirm our findings we measure both of them. After measuring both, we come to the conclusion that they are unequal. Likewise, see the figure of angles below:

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The angles AOB and POQ are unequal. We can clearly see that ∠ POQ is greater than ∠AOB. These unequal figures have a close relationship between unequal sides and unequal angles of a triangle. 


The term triangle inequality means unequal in their measures. 


Let us consider any triangle of length AB, BC, and AC of three sides of a triangle.  Then the triangle inequality definition or triangle inequality theorem states that

The sum of any two sides of a triangle is greater than or equal to the third side of a triangle.

i.e.,

AB + BC ≥ AC


Now let us understand the relation between the unequal sides and unequal angles of a triangle with the help of the triangle inequality theorems.


Triangle Inequality Theorem

Theorem 1: If two sides of a triangle are unequal, the longer side has a greater angle opposite to it.


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Triangle inequality theorem-proof:


Given: in \[\triangle ABC\] in which AC > AB

To prove :∠ ABC > ∠ ACB

Construction: Take a point D on AC such that AB = AD, Join BD.


Proof:

In \[\triangle ABC\], we have 

AB = AD...by construction

∠ ABD =  ∠ ADB….(1)......angle opposite to equal side are equal

Now, consider BCD, we find that  ∠ADB is the exterior angle of BCD and an exterior angle is always greater than the interior opposite angle.

therefore,

∠ADB >  ∠DCB

∠ ADB =  ∠ACB…(2)

From 1 and 2 we get

∠ABD =  ∠ ACB…..(3)

But  ∠ABD is a part of  ∠ABC 

So,  ∠ABC =  ∠ABD…..(4)

From 3 and 4 we get 

∠ABC > ∠ ACB……..thus proved


Theorem 2: (Converse of Theorem 1) in a triangle the greater angle has the longer side opposite to it.

Triangle inequality theorem proof

Given:  In \[\triangle ABC\] ∠ ABC > ∠ ACB

To prove: AC > AB

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Proof:

 In  \[\triangle ABC\], we have the following three possibilities

  1. AC =AB

  2. AC < AB 

  3.  AC > AB

Case 1 : When AC = AB

AC = AB

∠ ABC = ∠ ACB...angles equal to opposite sides are equal

This is contradiction

Since given that ∠ ABC > ∠ ACB

AC \[\neq\] AB

Case 2: when AC < AB

AC < AB

∠ ACB  > ∠ ABC……..[Longest side has the greater angle opposite to it]

This also contradicts the given hypothesis.

Thus we are left with the only one possibility, AC > AB, 

Hence AC > AB


Theorem 3: The sum of any two sides of a triangle is greater than the third side of a triangle.

Given: in \[\triangle ABC\]

To prove: AB + AC > BC,

AB + BC > AC and 

BC + AC > AB

Construction: produce side BA to D such that AD = AC. join CD


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Proof: 

in \[\triangle ACD\], we have 

AC = AD ...by construction

∠ ADC = ∠ ACD…..angle opposite to equal sides are equal

∠ ACD = ∠ ADC 

∠ BCA + ∠ ACD > ∠ ADC ……[ ∠ BCA + ∠ ACD > ∠ ACD]

∠ BCD > ∠ ADC

∠ BCD > ∠ BDC………………..[because ∠ ADC = ∠ BDC]

BD > BC …………………………[ side opposite to greater angle is larger]

BA + AD > BC

BA + AC > BC……………………[ since AC = AD by construction]

AB + AC > BC

Thus, AB + AC > BC 

Similarly, we can prove  AB + BC > AC and BC + AC > AB


Triangle Inequality Formula

In \[\triangle ABC\]

AB + AC > BC,

AB + BC > AC and 

BC + AC > AB


Solved Examples

Example 1 : if side lengths 8cm, 5cm and 7cm are given can it form a triangle?

Solution: If 8cm, 5cm and 7cm are the sides of the triangle, then they should satisfy the inequality theorem.AB + AC > BC,

AB + BC > AC and 

BC + AC > AB

Let AB = 8cm, BC = 5cm and AC = 7cm

Then

AB + BC > AC

8 + 5 >  7

⇒ 13 > 7

BC + AC > AB

5 + 7 > 8 

⇒ 12 > 8

AB + AC > BC,

8 + 7 > 5 

⇒ 15 > 5 

All the three conditions are satisfied, therefore a triangle can be formed with sides 8cm, 5cm and 7cm.


Quiz Time

  1. if side lengths 18cm, 5cm and 4cm are given can it form a triangle?

  2. if side lengths 7cm, 15cm and 9cm are given can it form a triangle?

FAQ (Frequently Asked Questions)

1. What are Congruent Triangles?

Answer:

Two geometrical figures having exactly the same shape and size are said to be congruent figures.

Two triangles are congruent to each other if one of them is superimposed on another such that they both cover each other completely.

Two triangles are said to be congruent if the sides and angles of one triangle are exactly equal to the corresponding sides and angles of the other triangle.

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From the figure, △ABC is congruent to △DEF  and it is written as △ABC  ≌ △DEF

In two congruent triangles, corresponding parts of corresponding angles generally written as ‘c.p.c.t’ are equal they are:

∠ A = ∠ D,

∠ B = ∠ E,

 ∠ C = ∠ F  

and

 AB = DE ,

 BC = EF ,

 AC = DF.