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Chapter 5

Chapter 5. 5-5 indirect proofs. Objectives. Write indirect proofs. Apply inequalities in one triangle. Indirect Proofs.

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Chapter 5

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  1. Chapter 5 5-5 indirect proofs

  2. Objectives Write indirect proofs. Apply inequalities in one triangle.

  3. 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.

  4. Writing Indirect Proofs

  5. Assume 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.

  6. solution • Step 3 Use direct reasoning to lead to a contradiction. 1  0 However, 1 > 0.

  7. The assumption that is false. Therefore Solution • Step 4 Conclude that the original conjecture is true.

  8. 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. • m1 + m2 + m3 = 180°

  9. 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.

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

  11. Example#4 • Write the angles in order from smallest to largest.

  12. Example#5 • Write the sides in order from shortest to longest.

  13. Student guided practice • DO problems 2-5 in your book page 348

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

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

  16. 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.

  17. Example • Tell whether a triangle can have sides with the given lengths. Explain. 2.3, 3.1, 4.6

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

  19. 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.

  20. Student guided practice • Do problems 6-10 in your book page 348

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

  22. Closure • Today we learned about indirect proofs • Next class we a re going to learn about inequalities in two triangles

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