Geometry: Triangle Inequality and Angle-Side Relationship



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

Related Topics: More Geometry Lessons

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.

triangle

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.

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 > ss < 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.



Triangle inequality theorem
Intuition behind the triangle inequality theorem



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.

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.

        AC<AB<BC

Answer: AC<AB<BC



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



Angle side relationships in Triangles







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