DRILL If A is (2, 5) and B is (-3, 8), show segment AB is parallel to segment CD if C is (-1, 4) and D is (-11, 10). What is the length of AB? Slope Formula Distance Formula

4.5 Indirect Reasoning Geometry Mr. Calise Objectives: Read and write an indirect proof Use the Hinge Theorem and its Converse to compare side lengths and angle measures.

Indirect Reasoning In indirect reasoning all possibilities are considered and then all but one are proved false. Therefore the remaining possibility is true.

Using Indirect Proof In this lesson, you will study indirect proofs. An indirect proof is a proof in which you prove that a statement is true by first assuming that its opposite is true. If this assumption leads to an impossibility, then you have proved that the original statement is true.

Guidelines for writing an Indirect Proof Identify the statement that you want to prove is false. Begin by assuming the statement is false; assume its opposite is true. Obtain statements that logically follow from your assumption. If you obtain a contradiction, then the original statement must be true.

Real World Example If Jaeleen spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25. Given: The cost of the two items is more than $50. Prove: At least one of the items costs more than $25. Begin by assuming that the opposite of what you want to prove is true. That is neither item costs more than $25.

Real World Example (cont.) This means that both items cost $25 or less. This, in turn means that the two items together cost $50 or less. Which contradicts the “given” statement, that the amount spent was more than $50. So therefore our assumption that neither items costs more than $25 must be incorrect. Therefore, at least one of the items costs more than $25.

Ex. 1: Using Indirect Proof Use an indirect proof to prove that a triangle cannot have more than one obtuse angle. SOLUTION: Given ► ∆ABC Prove ►∆ABC does not have more than one obtuse triangle

Ex. 1: Using Indirect Proof Step 1: Begin by assuming that ∆ABC does have more than one obtuse angle. mA > 90° and mB > 90° Assume ∆ABC has two obtuse angles. mA + mB > 180° Add the two given inequalities. Step 2: You know; however, that the sum of all the measures of all three angles is 180°. mA + mB +mC = 180° Triangle Sum Theorem mA + mB = 180° - mC Subtraction Property of Equality Step 3: So, you can substitute 180° - mC for mA + mB in mA + mB > 180° 180° - mC > 180° Substitution Property of Equality 0° > mC Simplify

IMPOSSIBLE WHICH IS WHAT WE WANT The last statement is not possible; angle measures in triangles cannot be negative. ►So, you can conclude that the original statement must be false. That is, ∆ABC cannot have more than one obtuse triangle.

Homework Pages 209 – 210 #’s 1 – 14 #’s 19 – 23