Chapter 2 Deductive Reasoning

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Presentation transcript:

Chapter 2 Deductive Reasoning

Chapter 2 Deductive Reasoning page 32 Essential Question Can you justify the conclusion of a conditional statement?

Lesson 2-1 If-Then Statements (page 33) Essential Question Can you justify the conclusion of a conditional statement?

If-Then Statement: a statement whose basic form is If p, then q. Also know as conditional statements or conditionals . Example: If it is snowing, then it is cold. Conditional statements are either true or false .

HYPOTHESIS (p) a statement that follows the “ If ” in a “If-then” statement. If it is snowing, then it is cold. Example: It is snowing.

CONCLUSION (q) a statement that follows the “ then ” in a “If-then” statement. If it is snowing, then it is cold. Example: It is cold.

Forms for Conditional Statements: If it is snowing, then it is cold. It is snowing implies it is cold. It is snowing only if it is cold. It is cold if it is snowing. Circle one: TRUE or FALSE If p, then q. p implies q. p only if q. q if p.

If, then, implies, and only if are NOT part of the NOTE: If, then, implies, and only if are NOT part of the hypothesis or the conclusion.

Example: If angle X is acute, then the measure of angle X equals 60 degrees. ∠X is acute implies the m∠ X = 60º . ∠X is acute only if the m∠ X = 60º . The m∠ X = 60º if ∠X is acute . Circle one: TRUE or FALSE

CONVERSE (of a conditional statement) is formed by interchanging the hypothesis and the conclusion. Converses are either true or false .

If it is cold , then it is snowing . Example: CONDITIONAL … If it is snowing, then it is cold. CONVERSE … If it is cold , then it is snowing . Circle one: TRUE or FALSE If p, then q. If q, then p.

Example: CONDITIONAL … If angle X is acute, then the measure of angle X equals 60 degrees. CONVERSE … If the m∠ X = 60º , then ∠X is acute . Circle one: TRUE or FALSE If p, then q. If q, then p.

Please take note … You can not assume a converse is true just because the original statement is true.

NOTE: only one counterexample is needed to prove a statement false! an example used to prove that an if-then statement is false . The hypothesis is true and the conclusion is false . NOTE: only one counterexample is needed to prove a statement false!

Example: Two angles are adjacent if they have a common vertex. CONDITIONAL … If 2∠’s have a common vertex , then they are adj. ∠’s . TRUE or FALSE Draw or write a counterexample …

Example: Two angles are adjacent if they have a common vertex. CONVERSE … If 2∠’s are adj. ∠’s , then they have a common vertex. TRUE or FALSE NO counterexample is necessary since it is TRUE.

BICONDITIONAL a statement that combines a conditional and its converse with the words “ if and only if ”. (iff)

then their measures are equal. Example: CONDITIONAL … If two angles are congruent, then their measures are equal. BICONDITIONAL … Two angles are congruent if and only if their measures are equal. p ⇒ q p ⟺ q

Can you justify the conclusion of a conditional statement? Assignment Written Exercises on page 35 RECOMMENDED: 3, 5, 7, 9. 30, 31 REQUIRED: 12, 15, 16, 18, 21, 24, 27 Can you justify the conclusion of a conditional statement?