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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 2002. 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 Statement Example Symbolic Form You Read It Conditional If an angle is a straight angle, then its measure is 180 degrees. pq If p, then q Converse If the measure of an angle is 180 degrees, then it is a straight angle. qp If 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 p A biconditional combines p q and q p as q. Statement Example Symbolic Form Biconditional An angle is a p straight angle if and only if its measure is 180 degrees. q You Read It p 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 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 𝑎 𝑐 If a = b and c ≠ 0, then = 𝑏 𝑐 Reflexive Property a=a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Substitution Property If a = b, then b can replace a in any expression • Summary – Properties of Congruence Property Example Reflexive Property AB ≈ AB <A ≈ <A Symmetric Property If AB ≈ CD, then CD ≈ AB If <A ≈ <B, then <B ≈ <A Transitive Property If 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 3 4 2 <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° 3x° 75° •Homework #11 •Due •Page 112 – 113 ▫# 1 – 6 all ▫# 8 – 18 all