Inequalities in a Triangle: Definition, Examples, and Proofs

Inequalities in a Triangle: A triangle is a planar shape bordered by three lines in a plane. Consider three points A, B, and C that are not in a straight line. The line segments AB, BC, and CA are thus called sides, while the angles BAC, ABC, and ACB are referred to as angles of the triangle ABC.

The total of either of the triangle’s two sides is greater than the sum of the triangle’s three sides. Furthermore, the bigger angle has the longer side opposite it. Let us look at the inequality in a triangle and several theorems about it in this post.

What are Triangles?

Let \(A, B, C\) be three non-collinear points. Then the figure formed by three line segments \(AB, BC,\) and \(CA\) is called a triangle with vertices \(A, B\) and \(C\) denoted by \(\Delta ABC\)

A \(\Delta ABC\) has three sides, namely \(AB, BC,\) and \(CA\) and three angles, namely \(∠BAC, ∠ABC\) and \(\angle BCA\)

Congruence of Triangles

If two figures have exactly the same shape and size, they are said to be congruent.
For congruence, we use the symbol \( \cong .\)

Two plane figures are congruent if each, when superposed on the other, covers it exactly.
Two triangles are congruent if pairs of corresponding sides and corresponding angles are equal.

Thus, \(\Delta ABC \cong \Delta DEF\), If
\(A B=D E, B C=E F\) and \(CA = FD,\angle A = \angle D,\angle B = \angle E\) and \(\angle C = \angle F\)

It should be kept in mind that while superposing one triangle on another triangle that is congruent to it, we match the corresponding vertices. We have shown one such matching in the above figure. 

When we write \(\Delta ABC \cong \Delta DEF\) it means that \(A, B\) and \(C\) are matched with \(D, E\) and \(F\) respectively, and we write \(A↔D, B↔E\) and \(C↔F\) and therefore, \(\Delta ABC \leftrightarrow \Delta DEF\)

Clearly, in this case, \(AB=DE, BC=EF, CA=FD\) and \(∠A=∠D, ∠B=∠E\) and \(∠C=∠F\)
On the other hand, if we write \(\Delta ABC = \Delta EFD\), it would mean that \(A↔E, B↔F\) and \(C↔D.\)

In this case, \(AB=EF, BC=FD, CA=DE\) and \(∠A=∠E, ∠B=∠F, ∠C=∠D\)
So, in the congruence of triangles, the order of the letters is important.

The symbol \( \leftrightarrow \) stands for matching.

Inequalities in a Triangle

The term “inequality” means “not equal”. Let us consider an example. 
Consider a triangle \(ABC\) as shown in the below figure. It has three sides \(BC, CA\) and \(AB.\) Let us denote the sides opposite the vertices \(A, B, C\) by \(a, b, c\) respectively. That is, 
\(a=BC, b=CA\) and \(c=AB.\)

The sides of a triangle satisfy an important property as stated below: 
Property: The sum of any two sides of a triangle is greater than the third side. That is, in a triangle \(ABC,\) we have:
\(b + c > a,c + a > b\) and \(a + b > c\)
This important property of a triangle is known as triangle inequality.

To verify the above property, let us perform the following activity.

Activity to Check the Triangle Inequality Property

Draw any three triangles \({T_1},{T_2}\) and \({T_3}\). Label each one as \(\Delta ABC\) Measure, in each case, the three sides \(a=BC, b=CA\) and \(c=AB\)

From the above table, we will find that:

1. Each value of \(b+c-a\) is positive
2. Each value of \(c+a-b\) is positive
3. Each value of \(a+b-c\) is positive.

Now,
\(b+c-a\) is positive \( \Rightarrow b + c – a > 0 \Rightarrow b + c > a\)
\(c+a-b\) is positive \( \Rightarrow c + a – b > 0 \Rightarrow c + a > b\)
\(a+b-c\) is positive \( \Rightarrow a + b – c > 0 \Rightarrow a + b > c\)
Thus, we have confirmed the following geometrical truth: 

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

Remark 1: From the above discussion, we find that three line segments whose lengths are equal to three given numbers form the sides of a triangle if the sum of the lengths of every pair of two of these is greater than the length of the third. 

Remark 2: In a triangle, the angle opposite the largest side is the largest. 

Triangle Inequality Theorem

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

To prove: \(\angle ABC > \angle BCA\)

Proof: Let \(AC > AB\) in \(\Delta ABC\)
In \(\Delta ABD,AB = AD\) (By construction)
\(∠1=∠2\) (angles opposite to equal sides are equal) ……(i)
In \(\Delta BCD\)
\(∠2>∠DCB\) (exterior angle is greater than one of the opposite interior angles)
\(∠2>∠ACB (∠ACB=∠DCB) …….(ii)\)

From (i) and (ii)

\(∠1>∠ACB ……(iii)\)
But \(∠1\) is a part of \(∠ABC\)
\( \Rightarrow \angle ABC > \angle 1 \ldots \ldots (iv)\)
Now, from (iii) and (iv), we get:
\(∠ABC>∠ACB \)
Hence proved.

Theorem 2: In a triangle, the greater angle has the longer side opposite to it. 

Given: \(\Delta ABC\) such that \(∠C>∠B.\)

To prove: \(AB>AC\)

Proof: Let us assume that \(AC>AB, ∠B>∠C\) (By theorem 1: If two sides of a triangle are unequal, the longer side has a greater angle opposite to it.)
Which contradicts our assumptions.
Hence \(AB>AC.\)
Hence proved.

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

Given: A \(\Delta ABC\)

To prove: \(AB+AC>BC\)

Construction: Extend \(AB\) to \(D\) such that \(AD=AC\)

Proof: In \(\Delta ACD\)
\(AD=AC\) (By construction)
Therefore, \(∠1=∠2\) (In a triangle, the greater angle has the longer side opposite to it.)

Now, \(∠BCD=∠BCA+∠1\)
\(=∠BCA+∠2  (∠1=∠2)\)
Now, in \(\Delta BCD\)
\(∠BCD>∠2 \)
If two sides of a triangle are unequal, the longer side has a greater angle opposite to it.
\( \Rightarrow BD > BC\)
\( \Rightarrow AB + AD > BC\quad (BD = AB + AD)\)
\( \Rightarrow AB + AC > BC\) (By construction \(AD=AC\))

Similarly, we can prove \(AB+BC>AC\) and \(AC+BC>AB.\)

Solved Examples – Inequalities in a Triangle

Q.1. State whether the below numbers could be the lengths of the sides of a triangle.
i. \(2,3,4\)
ii. \(4,5,3\)
iii. \(2.5,1.5,4\)
Ans: (i) Given: \(2, 3, 4\)
\(2+3>4, 2+4>3\) and \(3+4>2\)
That is, the sum of any two given numbers is greater than the third number.
So, \(2, 3, 4\) can be the lengths of the sides of a triangle.
(ii) Given: \(4, 5, 3\)
Here, \(4+5>3, 4+3>5\) and \(5+3>4\)
That is, the sum of any two given numbers is greater than the third number.
So, \(4, 5, 3\) can be the lengths of the sides of a triangle.
(iii) Given: \(2.5, 1.5, 4\)
Here, \(2.5+1.5\) is not greater than \(4 \Rightarrow 2.5 + 1.5 = 4\)
So, the given numbers \(2.5, 1.5, 4\) cannot be the lengths of the sides of a triangle.

Q.2. In the \(\Delta ABC,AB = 3\;{\rm{cm}},BC = 4\;{\rm{cm}}\) and \(AC = 5\;{\rm{cm}}\) Name the smallest and the largest angles of the triangle.

Ans: As the largest angle is always opposite to the largest side. Therefore, \(∠B\) is the largest angle, and the smallest angle is opposite to the smallest side. 
Therefore, \(∠C\) is the smallest angle. 

Q.3. \(P\) is a point in the interior of \(\Delta ABC\) as shown in the below figure. State which of the following statements are true (T) or false (F).
(i) \(AP+PB<AB\)
(ii) \(AP+PC>AC\)
(iii) \(BP+PC=BC\)

Ans: (i) False: We know that the sum of any two sides of a triangle is greater than the third side. It is not true for the given triangle.
(ii) True: We know that the sum of any two sides of a triangle is greater than the third side. It is true for the given triangle.
(iii) False: We know that the sum of any two sides of a triangle is greater than the third side. It is not true for the given triangle.

Q.4. In the below figure, \(P\) is the point on the side \(BC\). Complete each of the following statements using the symbol \(‘=’,’ > \)‘or \(‘< ‘\)to make it true.

(i) \(AP\_\_\_\_\_AB + BP\)
(ii) \(AP\_\_\_\_\_AC + PC\)
(iii) \(AP\_\_\_\_\_\frac{1}{2}\left({AB + AC + BC} \right)\)
Ans: (i) In \(\Delta APB,AP < AB + BP\) because the sum of any two sides of a triangle is greater than the third side. 
(ii) In \(\Delta APC,AP < AC + PC\) because the sum of any two sides of a triangle is greater than the third side. 
(iii) \(AP < \frac{1}{2}(AB + AC + BC)\)
In \(\Delta APB\) and \(\Delta ACP\), we can write as
\(AP < AB + BP… (i)\) (Because the sum of any two sides of a triangle is greater than the third side) 
\(AP < AC + PC … (ii)\) (Because the sum of any two sides of a triangle is greater than the third side) 
On adding (i) and (ii), we have
\(AP + AP < AB + BP + AC + PC\)
\(2AP < AB + AC + BC(BC = BP + PC)\)
\(AP < \frac{1}{2}\left( {AB + AC + BC} \right)\)

Q.5. \(O\) is a point in the exterior of \(\Delta ABC\) What symbol \(‘>’,’<’\) or \(‘=’\) will you see to complete the statement \(OA + OB\_\_\_\_AB?\) Write two other similar statements and show that \(OA + OB + OC > \frac{1}{2}(AB + BC + CA)\)
Ans: We know that the sum of any two sides of a triangle is always greater than the third side, in
\(\Delta OAB\) we have
\(OA + OB > AB…..{\rm{ }}(i)\)
\(OB + OC > BC \ldots \ldots (ii)\)
\(OA + OC > CA \ldots (iii)\)
On adding equations (i), (ii) and (iii) we get: 
\(OA + OB + OB + OC + OA + OC > AB + BC + CA\)
\(2(OA + OB + OC) > AB + BC + CA\)
\(OA + OB + OC > (AB + BC + CA)/2\)
\(OA + OB + OC > \frac{1}{2}(AB + BC + CA)\)
Hence proved.

Summary

In this article, we have learnt the definition of triangle, congruency of triangles and inequalities of the triangle. We have also done some activities to check the inequality properties of triangles and learnt the triangle inequality theorems. Furthermore, we solved some example problems based on the inequalities of the triangle.

Frequently Asked Questions (FAQs) – Inequalities in a Triangle

Q.1. What is the congruency of triangles?
Ans: Two triangles are congruent if pairs of corresponding sides and corresponding angles are equal. For congruence, we use the symbol \( \cong .\)

Q.2. What is triangle inequality?
Ans: The term “triangle inequality” is meant for triangles. Let us take \(p,q,\) and \(r\) are the measures of the three sides of a triangle, in which no side is being larger than the side \(r,\) then the triangle inequality states that \(p+q>r\)
This states that the sum of any two sides of a triangle is greater than or equal to the third side.

Q.3. Will \(8, 5, 7\) be the lengths of the sides of a triangle?
Ans: Given \(8, 5, 7\)
\(8+5>7, 8+7>5\)  and \(7+5>8\)
That is, the sum of any two given numbers is greater than the third number.
So,\( 8, 5, 7\) can be the lengths of the sides of a triangle.

Q.4. How do you write a triangle inequality?
Ans: Any side of a triangle must be lesser than the other two sides added together. If a side is equal to the other two sides, it is not a triangle. If \(a,b\) and \(c\) are the sides of a triangle, then, \(b+c>a, a+c>b,  a+b>c\)

Q.5. How can we apply triangle inequality in real life?
Ans: Using the inequality of triangle theorem, an engineer can find a sensible range of values for any unknown distance. This can be very beneficial when finding a rough estimate of the amount of material required to build a structure with undetermined lengths.

Now you are provided with all the necessary information on inequalities in a triangle and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.