Chapter 2 Section 2.1 – Conditional Statements Objectives: To recognize conditional statements To write converses of conditional statements

Conditional  another name for an “if-then” statement. ▫Ex: If you do your homework, then you will pass this class. Every conditional has two parts: ▫1. Hypothesis  part following the “if” ▫2. Conclusion  part following the “then”

Ex: Identify the hypothesis and conclusion in each statement. If the Angels won the 2002 World Series, then the Angels were world champions in If x – 38 = 3, then x = 41 If Kobe Bryant has the basketball, then he will shoot the ball everytime.

Ex: Write each sentence as a conditional statement. A rectangle has four right angles. A tiger is an animal. An integer that ends in 0 is divisible by 5.

Every conditional statement will have a truth value associated with it: either true or false. ▫A conditional is true if every time the hypothesis is true, the conclusion is also true. ▫A conditional is false if a counter-example can be found for which the hypothesis is true but the conclusion is false.

Ex: Show each conditional to be false by finding a counter-example. If it is February, then there are only 28 days in the month. If the name of a state contains the word New, then it borders the ocean.

Converse  occurs when the hypothesis and conclusion of a conditional statement are switched. Ex: Conditional If two lines intersect to form right angles, then they are perpendicular. Converse If two lines are perpendicular, then they intersect to form right angles.

It is important to see that just because the original conditional was true, does not mean the converse will also be true. Take the following for example: Conditional If a figure is a square, then it has four sides.True Converse If a figure has four sides, then it is a square.False

Summary – Conditional Statements/Converses StatementExampleSymbolic Form You Read It ConditionalIf an angle is a straight angle, then its measure is 180 degrees. p  qIf p, then q ConverseIf the measure of an angle is 180 degrees, then it is a straight angle. q  pIf q, then p

Homework #8 Due Page 83 – 84 ▫# 1 – 17 odd ▫# 23 – 31 odd

Section 2.2 – Biconditionals and Definitions Objectives: To write biconditionals To recognize good definitions

Biconditional  the statement created when a conditional and its converse are combined into a single statement with the phrase “if and only if” ▫This can only be done if both the conditional and the converse are true.

Ex: Take each conditional and write its converse. If both are true, then write a biconditional. If two angles have the same measure, then the angles are congruent. If three points are collinear, then they lie on the same line.

Summary – Biconditional Statements A biconditional combines p  q and q  p as p q. StatementExampleSymbolic Form You Read It BiconditionalAn angle is a straight angle if and only if its measure is 180 degrees. p qp if and only if q

Good Definition  a statement that can help you identify or classify an object. A good definition has three important components. ▫1. A good definition uses clearly understood terms. The terms should be commonly understood or already defined. ▫2. A good definition is precise. They will avoid such words as large, sort of, and almost. ▫3. A good definition is reversible. That means that you can write a good definition as a true biconditional.

Homework #9 Due Page 90 ▫# 1 – 23 odd

Section 2.3 – Deductive Reasoning Objectives: To use the Law of Detachment To use the Law of Syllogism

Deductive Reasoning (Logical Reasoning)  the process of reasoning logically from given statements to a conclusion. If the given statements are true, deductive reasoning will produce a true conclusion. Examples of Deductive Reasoning?

Property – Law of Detachment ▫If a conditional is true and its hypothesis is true, then its conclusion is true. ▫Symbolic form: If p  q is a true statement and p is true, then q is true.

Ex: What can be concluded about each given true statements? If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of AB. If a pitcher throws a complete game, then he should not pitch the next day. Jered Weaver is a pitcher who has just pitched a complete game.

Property – Law of Syllogism ▫If p  q and q  r are true statements, then p  r is a true statement. The Law of Syllogism allows us to state a conclusion from two true conditional statement when the conclusion of one statement is the hypothesis of the other statement.

Ex: Use the Law of Syllogism to draw a conclusion from the following true statements. If a number is prime, then it does not have repeated factors. If a number does not have repeated factors, then it is not a perfect square. If a number ends in 6, then it is divisible by 2. If a number ends in 4, then it is divisible by 2.

Homework #10 Due Page 96 – 97 ▫# 1 – 21 odd

Section 2.4 – Reasoning in Algebra Objectives: To connect reasoning in algebra and geometry.

Summary – Properties of Equality Addition PropertyIf a = b, then a + c = b + c Subtraction PropertyIf a = b, then a – c = b – c Multiplication PropertyIf a = b, then ac = bc Division Property Reflexive Propertya = a Symmetric PropertyIf a = b, then b = a Transitive PropertyIf a = b and b = c, then a = c Substitution PropertyIf a = b, then b can replace a in any expression

Summary – Properties of Congruence PropertyExample Reflexive PropertyAB ≈ AB <A ≈ <A Symmetric PropertyIf AB ≈ CD, then CD ≈ AB If <A ≈ <B, then <B ≈ <A Transitive PropertyIf AB ≈ CD and CD ≈ EF, then AB ≈ EF If <A ≈ <B and <B ≈ <C, then <A ≈ <C

Section 2.5 – Proving Angles Congruent Objectives: To prove and apply theorems about angles

Theorem  a statement proved true by deductive reasoning through a set of steps called a proof. In the proof of a theorem, a “Given” list shows you what you know from the hypothesis of the theorem. The “givens” are then used to prove the conclusion of a theorem.

Theorem 2.1 – Vertical Angles Theorem ▫All vertical angles are congruent <1 ≈ <2 and <3 ≈ <4

Theorem 2.2 – Congruent Supplements Theorem ▫If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Theorem 2.3 – Congruent Complements Theorem ▫If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Theorem 2.4 ▫All right angles are congruent. Theorem 2.5 ▫If two angles are congruent and supplementary, then each is a right angle.

Ex: Solve for x and y. y° 75° 3x°

Homework #11 Due Page 112 – 113 ▫# 1 – 6 all ▫# 8 – 18 all