Chapter 6: Inequalities in Geometry 6.2 – Inverses and Contrapositives 6.3 – Indirect Proofs (proof by contradiction) 6.4 – Triangle Inequalities

6.3 – Indirect Proofs (Proofs by Contradiction) 12/12 Contradictions: 6.3 – Indirect Proofs (Proofs by Contradiction) 1) “You are an unique individual – just like everybody else.” 2) An accurate estimate. 3) Hiding in plain sight. 4) The right to bear arms leads to less gun violence.

6.3 – Indirect Proofs (Proofs by Contradiction) 12/12 Template: 6.3 – Indirect Proofs (Proofs by Contradiction) 1) Assume temporarily that the conclusion (trying to prove) is FALSE. 2) Then reason logically until you reach a contradiction of a known fact (the given). 3) Therefore, the assumption is false, so the conclusion (trying to prove) is TRUE.

6.3 – Indirect Proofs Example 1: Given: Mr. Cheng is in college. Prove: Mr. Cheng graduated high school or equivalent. Assume temporarily: Mr. Cheng did not graduate from high school. Then: Mr. Cheng would not be in college if he did not graduate from high school or equivalent. This contradicts the given that Mr. Cheng is in college. Therefore: The assumption is false, so Mr. Cheng did graduate from high school or equivalent.

6.3 – Indirect Proofs Example 2: Given: 2r + 3 ≠ 17 Prove: r ≠ 7 Assume temporarily: r = 7 Then: 2r + 3 = 2(7) + 3 = 14 + 3 = 17. This contradicts the given that 2r + 3 ≠ 17. Therefore: The assumption is false, so r ≠ 7.

Example 3: Given: quad ABCD with AB ≇ BC 6.3 – Indirect Proofs Example 3: Given: quad ABCD with AB ≇ BC Prove: quad ABCD is not a rhombus. Assume temporarily: quad ABCD is a rhombus. Then: all sides are ≅, so AB ≅ BC. This contradicts the given that AB ≇ BC. Therefore: The assumption is false, so quad ABCD is not a rhombus.

Textbook Practice Page 210: Classroom Exercises: #1-9