Chapter 2 Reasoning and Proof.

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Presentation transcript:

Chapter 2 Reasoning and Proof

Section 1 Conditional Statements

GOAL 1: Recognizing Conditional Statements In this lesson you will study a type of logical statement called a conditional statement. A conditional statement has two parts, a hypothesis and a conclusion. When the statement is written in if-then form, the “if” part contains the hypothesis and the “then” part contains the conclusion. Here is an example:

Example 1: Rewriting in If-Then Form Rewrite the conditional statement in if-then form. Two points are collinear if they lie on the same line. All sharks have a boneless skeleton. A number divisible by 9 is also divisible by 3.

Conditional statements can either be true or false Conditional statements can either be true or false. To show that a conditional statement is true, you must present an argument that the conclusion follows for all cases that fulfill the hypothesis. To show that a conditional statement is false, describe a single counterexample that shows the statement is not always true.

Example 2: Writing a Counterexample Write a counterexample to show that the following conditional statement is false. If x2 = 16, then x = 4.

The _____________________ of a conditional statement is formed by switching the hypothesis and conclusion. Here is an example. Statement: If you see lightning, then you hear thunder. Converse: If you hear thunder, then you see lightning.

Example 3: Writing the Converse of a Conditional Statement Write the converse of the following conditional statement. Statement: If two segments are congruent, then they have the same length. Converse:

A statement can be altered by ______________________, that is, by writing the negative of the statement. Here are some examples. When you negate the hypothesis and conclusion of a conditional statement, you form the _________________. When you negate the hypothesis and conclusion of the converse of a conditional statement, you form the ______________________.

When two statements are both true or both false, they are called ______________________________________. A conditional statement is equivalent to its contrapositive. Similarly, the inverse and converse of any conditional statement are equivalent. This is show in the table above.

Example 4: Writing an Inverse, Converse, and Contrapositive Write the (a) inverse, (b) converse, and (c) contrapositive of the statement. If there is snow on the ground, then flowers are not in bloom.

Goal 2: Using Point, Line, and Plane Postulates In Chapter 1, you studied four postulates. Ruler Postulate (Lesson 1.3, page 17) Segment Addition Postulate (Lesson 1.3, page 18) Protractor Postulate (Lesson 1.4, page 27) Angle Addition Postulate (Lesson 1.4, page 27) Remember that postulates are assumed to be true – they form the foundation on which other statements (called theorems) are built.

Example 5: Identifying Postulates Use the diagram at the right to give examples of Postulates 5 through 11.

Example 6: Rewriting a Postulate Rewrite Postulate 5 in if-then form. Write the inverse, converse, and contrapositive of Postulate 5. inverse: converse: contrapositive:

Example 7: Using Postulates and Counterexamples Decide whether the statement is true or false. If it is false, give a counterexample. A line can be in more than one plane. Four noncollinear points are always coplanar. Two nonintersecting lines can be noncoplanar.