Triangle Inequality

Key Concepts

• Relating side length and angle measure.
• Triangle inequality.
• Possible side lengths.

Introduction 

In the previous session, we learned about medians, centroid, altitudes, orthocenter, and special cases of an isosceles triangle. 

In this session, you will learn about inequalities in a triangle, relating side lengths and angle measures, triangle inequality, and possible side lengths in a triangle. 

Theorems 

Theorem 1

If one side of a triangle is longer than the other side, then the angle opposite the longer side is larger than the angle opposite the shorter side. 

Theorem 2 

If an angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. 

Example 1: 

Draw an acute-angled triangle and relate the side lengths and angle measures. 

Solution: 

Suppose a < b < c, 

The angle opposite to the side a is the smaller angle, 

The angle opposite to the side c is the larger angle. 

∠C > ∠B > ∠A 

Example 2: 

In an obtuse-angled triangle ABC, find the larger angle and longer side. 

Solution: 

In the obtuse-angled angle triangle ABC, it is clear that  

∠B is the obtuse angle, 

The obtuse angle is the largest in the obtuse-angled triangle. 

So, ∠B is the larger angle. 

From theorem 2,  

If an angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. 

The side opposite ∠B is the longer side. 

Theorem 3

If a side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. 

Proof: 

Given that PQ > PR 

Now we prove that ∠PRQ > ∠PQR 

Plot a point S on PQ such that PS = PR … (1) 

In isosceles triangle ∠PSR = ∠PRS … (2) 

∠PRQ = ∠PRS + ∠QRS … (3) 

∠PRQ > ∠PRS 

∠PRQ > ∠PSR … (4) (since from (2)) 

From the exterior angle theorem, ∠PSR = ∠PQR + ∠SRQ 

Therefore ∠PSR > ∠PQR …(5) 

From (4) and (5) 

Finally, we conclude ∠PRQ > ∠PQR 

Hence proved. 

Triangle Inequality Theorem 

Triangle Inequality 

Every group of three segments cannot form a triangle. 

It needs to fit certain relations. 

Example: 

If you draw the longest side as a base and connect the other two sides in the first and second figures, the triangle is not formed. This leads to the triangle inequality theorem. 

Triangle Inequality Theorem 

Theorem: 

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

AB + BC > AC, 

AC + AB > BC, 

AC + BC > AB.   

Possible Side Lengths 

Example: 

In a triangle, one side is of length 12 cm, another side is of length 5. Find the possible side lengths. 

Solution: 

Let the third side of the triangle be x 

Now x can be the smallest side or x can be the longest side. 

We use the properties of the triangle inequality. 

12 + 5 > x 

17 > x 

x + 5 > 12 

x > 12 – 5  

x > 7 

x > 7 and x < 17. 

The length of the third side is greater than 7 and less than 17.   

Real-Life Example  

In real life, the inequalities of a triangle are used mostly by civil engineers, as this involves so much in their work to find the unknown lengths of different dimensions. 

Example: 

In a construction field, a triangular shape ABC must be constructed (as shown in the image), the left side of the triangle is 12 feet and the right side of the triangle is 13 feet, the base is 26 feet, one angle is

45° and another angle is 54° what will be the other angle in that triangle? 

Solution: 

Given, that the left side of the triangle is 12 feet, and the right side of the triangle is 13 feet, the base is 26 feet, one angle is

45° and another angle is 54°. 

Since the sum of angles in a triangle = 180°

So,

180°−(45°+54°)

= 180° – 99° 

= 81°

The other angle is 81° 

Now the given theorem we have ∠BAC = 81° 

This will be at the top, which is at A, as the base is the longest side. 

Exercise

• Draw a triangle for side lengths 6, 8, and 10 units and mention the angles 30°, 60°, and 90°. Sketch the triangle and mark the angles.
• Mention the smaller side to the longer side from the given figure.

• If one side of a triangle is 11 cm and another side is 6 cm. Find the possible length of the third side.
• How can you say which angle is largest in a triangle?
• Mention the smaller angle to the larger angle from the given figure.

• Which group of lengths is used to form a triangle?

3, 4, 5

4, 2, 2

3, 2, 1

2, 2, 2

Q (7-10)

• From the above figure, what are the possible values of x?
• From the above figure, what are the possible values of PQ?
• From the above figure, what are the possible values of QR?
• From the above figure, what are the possible values of RP?

Concept Map

What we have learned

• Relating side lengths and angle measures.
• Triangle inequality.
• Possible side lengths.