Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 1: Ordering Triangle Side Lengths and Angle Measures

Write the angles in order from smallest to largest.

The angles from smallest to largest are F, H and G.

The shortest side is , so the smallest angle is F.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 2: Ordering Triangle Side Lengths and Angle Measures

Write the sides in order from shortest to longest.

mR = 180° – (60° + 72°) = 48°

The smallest angle is R, so the shortest side is .

The sides from shortest to longest are

48°

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 3:

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 4:

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

A triangle is formed by three segments, but not every set of three segments can form a triangle.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

NOTE: Just check that the sum of the two shorter sides is greater than the longest side.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 5: Applying the Triangle Inequality Theorem

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 5: Applying the Triangle Inequality Theorem

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 6: Finding Possible Side Lengths

The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.

Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.

Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.

x + 8 > 13

x > 5

8 + 13 > x

21 > x

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 7

The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

You can also use side lengths to classify a triangle as acute or obtuse.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle