Geometry: Triangle Inequality and Angle-Side Relationship

In this lesson, we will learn two commonly used inequality relationships in a triangle:

• the Triangle Inequality Theorem
• the Converse of the Triangle Inequality Theorem
• the Angle-Side Relationship for triangles

Triangle Inequality Theorem

The Triangle Inequality theorem states that

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

For the above triangle:

a + b > c
b + c > a
a + c > b

The Converse of the Triangle Inequality theorem states that

It is not possible to construct a triangle from three line segments if any of them is longer than the sum of the other two.

Example 1: Find the range of values for s for the given triangle.

Solution:

Step1: Using the triangle inequality theorem for the above triangle gives us three statements:

        s + 4 > 7 ⇒ s > 3
        s + 7 > 4 ⇒ s > –3    (not valid because lengths of sides must be positive)
        7 + 4 > s ⇒ s < 11

Step 2: Combining the two valid statements:

        3 < s < 11

Answer: The length of s is greater than 3 and less than 11

The following video states and investigates the triangle inequality theorem

The following video describes triangle inequality by trying to construct triangles with different length segments.

The following video shows the conditions required to draw a triangle to illustrate triangle inequality.

Angle-Side Relationship

The Angle-Side Relationship states that

In a triangle, the side opposite the larger angle is the longer side.

In a triangle, the angle opposite the longer side is the larger angle.

Example 1: Compare the lengths of the sides of the following triangle.

Solution:

Step1: We need to find the size of the third angle. The sum of all the angles in any triangle is 180º.

            ∠A + ∠B + ∠C = 180°
        ⇒ ∠A + 30° + 65° = 180°
        ⇒ ∠A = 180° - 95°
        ⇒ ∠A = 85°

Step 2: Looking at the relative sizes of the angles.

        ∠B < ∠C < ∠A

Step 3: Following the angle-side relationship we can order the sides accordingly. Remember it is the side opposite the angle.

Answer:

The following video will show more examples of the angle-side relationships in triangles

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