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Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Write indirect proofs.

Apply inequalities in one triangle.

CN#5: Objectives

indirect proofVocabulary

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

So far you have written proofs using direct reasoning.

True hypothesisàconclusion that was true.

In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or atheorem.

Helpful Hint

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Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 1: Writing an Indirect Proof

Step 1 Identify the conjecture to be proven.Given: a > 0

Step 2 Assume the opposite of the conclusion.

Write an indirect proof that if a > 0, then

Prove:

Assume

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 1 Continued

Step 3 Use direct reasoning to lead to a contradiction.

However, 1 > 0.

1 £ 0

Given, opposite of conclusion

Zero Prop. of Mult. Prop. of Inequality

Simplify.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Step 4 Conclude that the original conjecture is true.

Example 1 Continued

The assumption that is false.

Therefore

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Check It Out! Example 1

Write an indirect proof that a triangle cannot have two right angles.

Step 1 Identify the conjecture to be proven.Given: A triangle’s interior angles add up to 180�.

Prove: A triangle cannot have two right angles.

Step 2 Assume the opposite of the conclusion.

An angle has two right angles.

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Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Check It Out! Example 1 Continued

Step 3 Use direct reasoning to lead to a contradiction.

However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0�.

mÐ1 + mÐ2 + mÐ3 = 180�90� + 90� + mÐ3 = 180�

180� + mÐ3 = 180�

mÐ3 = 0�

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Step 4 Conclude that the original conjecture is true.

The assumption that a triangle can have two right angles is false.

Therefore a triangle cannot have two right angles.

Check It Out! Example 1 Continued

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 2A: Ordering Triangle Side Lengths and Angle Measures

Write the angles in order from smallest to largest.

The angles from smallest to largest are ÐF, ÐH and ÐG.

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

The longest side is , so the largest angle is ÐG.

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Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 2B: Ordering Triangle Side Lengths and Angle Measures

Write the sides in order from shortest to longest.mÐR = 180� – (60� + 72�) = 48�The smallest angle is ÐR, so the shortest side is .

The largest angle is ÐQ, so the longest side is .

The sides from shortest to longest are

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.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Check It Out! Example 3a

Tell whether a triangle can have sides with the given lengths. Explain.

8, 13, 21

No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

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Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Check It Out! Example 3b

Tell whether a triangle can have sides with the given lengths. Explain.

6.2, 7, 9

Yes—the sum of each pair of lengths is greater than the third side.

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Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Check It Out! Example 3c

Tell whether a triangle can have sides with the given lengths. Explain.

t – 2, 4t, t2 + 1, when t = 4

Step 1 Evaluate each expression when t = 4.

t – 24 – 2

2

t2 + 1 (4)2 + 1

17

4t4(4)16

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Check It Out! Example 3c Continued

Step 2 Compare the lengths.

Yes—the sum of each pair of lengths is greater than the third length.

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Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Check It Out! Example 4

The lengths of two sides of a triangle are 22 inches and 17 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 < 39. The length of the third side is greater than 5 inches and less than 39 inches.

x + 22 > 17

x > –5

x + 17 > 22

x > 5

22 + 17 > x

39 > x