Chapter 5

5 - 5 I N D I R E C T P R O O FS

CHAPTER 5

OBJECTIVES

Write indirect proofs.

Apply inequalities in one triangle.

INDIRECT PROOFS

• So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion 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.

WRITING INDIRECT PROOFS

WRITING INDIRECT PROOFS

• Write an indirect proof that if a > 0, then 1/a >0 • Solution:• Step 1 Identify the conjecture to be proven.• Given: a > 0• Prove: 1/a >0

• Step 2 Assume the opposite of the conclusion.Assume

SOLUTION

• Step 3 Use direct reasoning to lead to a contradiction.

1 0

However, 1 > 0.

SOLUTION

• Step 4 Conclude that the original conjecture is true.

The assumption that is false.

Therefore

EXAMPLE#2

• 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.• Step 3 Use direct reasoning to lead to a

contradiction.• m1 + m2 + m3 = 180°

SOLUTION

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

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

EXAMPLE#4

• Write the angles in order from smallest to largest.

EXAMPLE#5

• Write the sides in order from shortest to longest.

STUDENT GUIDED PRACTICE

• DO problems 2-5 in your book page 348

TRIANGLES

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

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

EXAMPLE

• Tell whether a triangle can have sides with the given lengths. Explain.• 7, 10, 19

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

EXAMPLE

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

2.3, 3.1, 4.6

EXAMPLE

• Tell whether a triangle can have sides with the given lengths. Explain.• t – 2, 4t, t2 + 1, when t = 4

APPLICATIONS

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

STUDENT GUIDED PRACTICE

• Do problems 6-10 in your book page 348

HOMEWORK

• DO even problems from 16 -25 in your book page 248

CLOSURE

• Today we learned about indirect proofs • Next class we a re going to learn about

inequalities in two triangles