Unit 2: Deductive Reasoning Pre-AP Geometry 1 Unit 2: Deductive Reasoning 1

2.1 If-then statements, converse, and biconditional statements Pre-AP Geometry 1 Unit 2 2.1 If-then statements, converse, and biconditional statements p. 33 in text 2

Conditional Statements A statement with two parts (hypothesis and conclusion) Also known as Conditionals If-then form A way of writing a conditional statement that clearly showcases the hypothesis and conclusion p→q Hypothesis- The “if” part of a conditional statement Represented by the letter “p” Conclusion The “then” part of a conditional statement Represented by the letter “q” 3

Conditional Statements Examples of Conditional Statements If today is Saturday, then tomorrow is Sunday. If it’s a triangle, then it has a right angle. If x2 = 4, then x = 2. If you clean your room, then you can go to the mall. If p, then q. The first statement is true. The second statement is false, triangles do not have to have a right angle. This was put together with a subject (a triangle) and predicate (it has a right angle) The third statement is false, x could equal -2. 4

Conditional Statements Example 1 Circle the hypothesis and underline the conclusion in each conditional statement If you are in Geometry 1, then you will learn about the building blocks of geometry If two points lie on the same line, then they are collinear If a figure is a plane, then it is defined by 3 distinct points Statements do not have to be true. The last one is clearly false. 5

Conditional Statements Example 2 Rewrite each statement in if…then form A line contains at least two points When two planes intersect their intersection is a line Two angles that add to 90° are complementary If a figure is a line, then it contains at least two points If two planes intersect, then their intersection is a line. If two angles add to equal 90°, then they are complementary. 6

Conditional Statements Counterexample An example that proves that a given statement is false Write a counterexample If x2 = 9, then x = 3 7

Conditional Statements Example 3 Determine if the following statements are true or false. If false, give a counterexample. If x + 1 = 0, then x = -1 If a polygon has six sides, then it is a decagon. If the angles are a linear pair, then the sum of the measure of the angles is 90º. 8

Conditional Statements Converse Formed by switching the if and the then part. Original If you like green, then you will love my new shirt. If you love my new shirt, then you like green. 9

Biconditional Statements Can be rewritten with “If and only if” Only occurs when the statement and the converse of the statement are both true. A biconditional can be split into a conditional and its converse. p if and only if q All definitions can be written as biconditional statements 10

Example Give the converse of the statement. If the converse and the statement are both true, then rewrite as a biconditional statement If it is Thanksgiving, then there is no school. If an angle measures 90º, then it is a right angle.

Quiz- Get out a piece of paper and answer the following questions: Underline the hypothesis and circle the conclusion. Then, write the converse of the statement. If the converse and the statement are true, rewrite as a biconditional statement. If not, give a counterexample. 1. If a number is divisible by 10, then it is divisible by 5. 2. If today is Friday, then tomorrow is Saturday. 3. If segment DE is congruent to segment EF, then E is the midpoint of segment DF.

Assignment Lesson 2.1 P. 35 #2-30 even

2.2: Properties from Algebra p. 37 Pre-AP Geometry 1 Unit 2 2.2: Properties from Algebra p. 37 p. 33 in text 15

Properties of equality Addition property If a = b, then a + c = b + c Subtraction property If a = b, then a – c = b – c Multiplication property If a = b, then ac = bc Division property If a = b, then 16

Reasoning with Properties from Algebra Reflexive property For any real number a, a = a Symmetric property If a=b, then b = a If Transitive Property If a = b and b = c, then a = c If ∠D ∠E and ∠E ∠F, then ∠D ∠F Substitution property If a = b, then a can be substituted for b in any equation or expression Distributive property 2(x + y) = 2x + 2y

Two-column proof A way of organizing a proof in which the statements are made in the left column and the reasons (justification) is in the right column Given: Information that is given as fact in the problem.

Reasoning with Properties from Algebra Example 1 Solve 6x – 5 = 2x + 3 and write a reason for each step Statement Reason 6x – 5 = 2x + 3 Given 4x – 5 = 3 4x = 8 x = 2 Subtraction property of equality Addition property of equality Division property of equality

Reasoning with Properties from Algebra Example 2 2(x – 3) = 6x + 6 Given 20

Reasoning with Properties from Algebra Determine if the equations are valid or invalid, and state which algebraic property is applied (x + 2)(x + 2) = x2 + 4 x3x3 = x6 -(x + y) = x – y Invalid. Valid Invalid 21

Quiz Complete proof # 11 and # 12 on p. 40 Homework: p. 41 # 3 - 15