University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

235

Pages:

3
Author:

Karen Cantrell

Indirect Proof & Inequalities in One Triangle In your own words, write the meaning each vocabulary term. indirect proof when you prove a statement by assuming the opposite first and disproving that can't hovk How to Write an Indirect Proof (Proof by Contradiction) Step 1 Identify the statement you want to prove. Assume temporarily that this statement is false by assuming that its opposite is true. Step 2 Reason logically until you reach a contradiction. Step 3 Point out that the desired conclusion must be true because the contradiction proves the temporary assumption false. Theorems Theorem 6.9 Triangle Longer Side Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. B B 8 5 A C AB|> |BC| Smaller A so then m angle C>m angle A Theorem 6.10 Triangle Larger Angle Theorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. B B 50 deg 30° C A C A m angle A>m angle C so then BC > AB Theorem 6.11 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. C AB + BC > AC AC + BCl > AB AB + AC > BC ) Section 1: Indirect Proofs In Exercises 1-3, write the first step in an indirect proof of the statement. Not all the students in a given class can be above average. Assume that all the students in the class can be above average. No number equals another number divided by zero. Assume that there is a number that equals another number divided by zero. The square root of 2 is not equal to the quotient of any two integers. Assume that √2 is equal to the quotient of any two integers. Section 2: Contradictory Statements In Exercises 4 and 5, determine which two statements contradict each other. Explain your reasoning. A. ΔLMN is equilateral. B. LM = MN C. LM ≠ MN Explanation: Statements B and C contradict each other. In an equilateral triangle, all sides are equal. If LM = MN, then all sides must be equal. However, statement C states that LM ≠ MN, which contradicts the definition of an equilateral triangle. A. ΔABC is a right triangle. B. ∠A is acute. C. ∠C is obtuse. Explanation: Statements A and C contradict each other. In a right triangle, one angle is a right angle (90 degrees). If ∠C is obtuse (greater than 90 degrees), then the triangle cannot be a right triangle. Section 3: Ordering Angles in a Triangle In Exercises 6-8, list the angles of the given triangle from smallest to largest. Diagram: A triangle with sides labeled 10, 12, and 15. Answer: ∠C, ∠A, ∠B Diagram: A triangle with sides labeled 14, 15, and 16. Answer: ∠D, ∠E, ∠F Diagram: A triangle with sides labeled 14, 15, and 17. Answer: ∠H, ∠J, ∠G