Topics
4. Triangles
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Geometry

4.03 Triangle inequalities

Lesson
Practice

Triangle inequalities

Exploration

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  1. Describe the relative size of the angle that is opposite the longest side.
  2. Describe the relative size of the angle that is opposite the shortest side.
  3. If you know the measure of the angles, but not the sides, how could you order the sides by length?
  4. If you know the lengths of the sides, but not the measure of the angles, how could you order the angles by size?

For a triangle to exist (be considered valid), the length of each side must be within a range that is determined by the lengths of the other two sides.

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 labeled a, b and c.

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

  • a+b>c
  • a+c>b
  • b+c>a

We can make a connection between the relative size of each angle and its opposite side length in a triangle. The smallest angle will be opposite the shortest side, while the largest angle will be opposite the longest side of the triangle.

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 Angles 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.

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.

Worked Solution
Create a strategy

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

Apply the idea

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.

Reflect and check

When testing whether a set of sides satisfies the triangle inequality theorem or not, it is sufficient to only check that the combined length of the two shortest sides is greater than the longest side.

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.
Worked Solution
Create a strategy

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.

Apply the idea

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

For the triangle in the figure, order the sides from shortest to longest.

Triangle A B C with angle A measuring 72 degrees, angle B measuring 52 degrees, and angle C measuring 55 degrees.
Worked Solution
Create a strategy

We can use the fact that the longest side is opposite the largest angle and the shortest side is opposite the smallest angle to order the sides from shortest to longest.

Apply the idea

Since \angle B is the smallest angle (52\degree), \overline{AC} will be the shortest side. \angle C is the next smallest angle (55\degree), so \overline{AB} will be the next shortest side. Finally, \angle A is the largest angle (72\degree), so \overline{BC} will be the longest side.

This means that the sides, written in order from shortest to longest, are AC < AB < BC.

Reflect and check

Recall that when we are referring to the side of the triangle, we draw a line over the name of the side, \overline{AB}. But when referring to the length of the side, we do not draw a line over the side name, AB.

Example 4

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

Find the range of values for x.

Worked Solution
Create a strategy

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

Apply the idea

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

Since we cannot reasonably have side lengths that are zero or less, we will not consider the values from -6 to zero from the third inequality. And since the sum of any two sides must exceed the third side, any values of x from 1 to 6 would not form a valid triangle.

All three inequalities must be met in order to form a valid triangle.

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

Reflect and check

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.

Example 5

Minerva is hiking in Redwood National Park. She is camping in Berry Glenn and plans to hike up to Orick Hill. After the hike, she takes the trail down to Orick. Will she need to hike further than 2.5 \text{ miles}?

A map of the Redwood National Park showing 3 sites: Starting from the top of the map, Berry Glenn, Orick Hill, and Orick. A triangle is drawn using the three site as vertices. The distance between Berry Glenn and Orick is 2.5 miles and is the longest side of the triangle. The Orick Hill has an elevation of 863 feet. Speak to your teacher for more details.
Worked Solution
Create a strategy

Use the triangle inequality theorem to determine if Minerva needs to hike more than 2.5 \text{ miles}.

Apply the idea

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Therefore, Minerva will hike more than 2.5 \text{ miles} because her hike from Berry Glenn to Orick Hill to Orick must be greater than 2.5 \text{ miles}.

Idea summary

Triangles are valid if they satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

The smallest angle in a triangle will be opposite the shortest side, while the largest angle will be opposite the longest side of the triangle.

Outcomes

G.TR.1

The student will determine the relationships between the measures of angles and lengths of sides in triangles, including problems in context.

G.TR.1a

Given the lengths of three segments, determine whether a triangle could be formed.

G.TR.1b

Given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie.

G.TR.1c

Order the sides of a triangle by their lengths when given information about the measures of the angles.

G.TR.1d

Order the angles of a triangle by their measures when given information about the lengths of the sides.

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