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5. Relationships in Triangles
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United States of AmericaTennessee
Geometry

5.06 Inequalities in one triangle

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Concept summary

Valid triangle

A set of three sides whose lengths satisfy the triangle inequality theorem.

Triangle inequality theorem

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

A triangle with sides labelled a, b and c.

From the triangle inequality theorem, we get the following inqualities:

  • a+b>c
  • a+c>b
  • b+c>a

Rearranging the inequalities can also give us lower bounds for the lengths of each side.

Opposite side-angle relationship theorem

If one 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.

Triangle A B C with the smaller side B C and angle A are colored green, and the longer side A C and angle B colored blue. The inequality, the measure of angle B is greater than measure of angle A, is shown.
Opposite angle-side relationship theorem

If one 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.

Triangle A B C with the smaller side B C and angle A are colored green, and the longer side A C and angle B colored blue. The inequality, A C is greater than B C, is shown.
Triangle A B C is drawn with sides of varying lengths. Side A C is the longest, side B C is the shortest. Side length A B is intermediate between A C and B C.

In this triangle, we can see that:

AC>AB>BC

Using the theorems above, we can say that:

m\angle B > m\angle C > m\angle A

Similarly, we could observe the order of the angle measures and determine the order of the side lengths.

Corollary to the triangle sum theorem

The acute angles of a right triangle are complementary

Right triangle with the 2 acute angles labeled 1 and 2. The equation, measure of angle 1 plus measure of angle 2 equals 90 degrees, is shown.

Since the hypotenuse of a right triangle is always the longest side and is opposite the right angle, the legs must be shorter and opposite acute angles using the side-angle relationship.

Worked examples

Example 1

Suppose that we have three sides of lengths 4, 7 and 12. Determine if these three sides can form a valid triangle.

Approach

We want to check whether the lengths of the three sides satisfies the triangle inequality theorem or not.

Solution

Compare the sum of each pair of sides to the third side.

  • 4+7<12
  • 4+12>7
  • 7+12>4

The side of length 12 is longer than the combined lengths of the other two sides, so we do not satisfy the triangle inequality theorem.

The three given sides cannot form a valid triangle.

Example 2

For the triangle in the figure, state its angles in order of ascending measure, from smallest to largest.

Triangle A B C with side A B of length 7, side A C of length 11, and side B C of length 12.

Approach

Since a longer side is opposite a larger angle, we can order the angles in ascending measure based on the order of ascending side lengths.

Solution

In ascending order, the sides lengths of the triangle are AB, AC, BC.

This means that the angles, written in order of ascending measure, are \angle C, \angle B, \angle A.

Example 3

A valid triangle has side lengths of 4, 10 and x.

Find the range of values for x.

Approach

We can determine the upper and lower bounds for x using the inequalities from the triangle inequality theorem.

Solution

Since we know that the given side lengths form a valid triangle, the following inequalities must be true:

  • 4+10>x
  • 4+x>10
  • 10+x>4

If we simplify each inequality, we get

  • 14>x
  • x>6
  • x>-6

From this, we get the range of values 6<x<14.

Reflection

Notice that, for a valid triangle, the possible range of the third side's length is between the sum and difference of the two given side lengths.

Outcomes

G.CO.C.9

Use definitions and theorems about triangles to solve problems and to justify relationships in geometric figures.

G.MP1

Make sense of problems and persevere in solving them.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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