Converse, Inverse, and Contrapositive Statements ( Read ) Geometry CK-12 Foundation PDF

Preview

Summary

Download Converse, Inverse, and Contrapositive Statements ( Read ) Geometry CK-12 Foundation PDF

Description

5/9/2019



Converse, Inverse, and Contrapositive Statements ( Read ) | Geometry | CK-12 Foundation

(/)

 Click Create Assignment to assign this modality to your LMS.

You are viewing an older version of this Read.



Go to the latest version. ()

Conditional statements drawn from an if-then statement.

Converse, Inverse, and Contrapositive Statements Converse, Inverse, and Contrapositive What if your sister told you, "If you do the dishes, then I will help you with your homework"? What's a statement that is logically equivalent to what your sister said?

Converse, Inverse, and Contrapositive Consider the statement: If the weather is nice, then I will wash the car. This can be rewritten using letters to represent the hypothesis and conclusion:

In addition to these positives, we can also write the negations, or “not”s of version of not

, is

and

. The symbolic

.

Using these negations and switching the order of

and

, we can create three more conditional

statements.

If we accept “If the weather is nice, then I’ll wash the car” as true, then the converse and inverse are not necessarily true. However, if we take the original statement to be true, then the contrapositive is also true. We say that the contrapositive is logically equivalent to the original if-then statement. It is

sometimes the case that a statement and its converse will both be true. These types of statements are called biconditional statements. So, is true and is true. It is written , with a double arrow to indicate that it does not matter if

or

is first. It is said, “

if and only if

”. Replace the “if-

then” with “if and only if” in the middle of the statement. “If and only if” can be abbreviated “iff.”

Converse, Inverse, and Contrapositive: Lesson (Geometry Concepts)

Finding the Converse, Inverse, and Contrapositivive 1. Use the statement: If

, then

.

a) Find the converse, inverse, and contrapositive. b) Determine if the statements from part a are true or false. If they are false, find a counterexample. The original statement is true.

2. Use the statement: If I am at Disneyland, then I am in California.

a) Find the converse, inverse, and contrapositive. b) Determine if the statements from part a are true or false. If they are false, find a counterexample. The original statement is true.

Notice for the inverse and converse we can use the same counterexample. This is because the inverse and converse are also logically equivalent.

The Converse, Contrapositive, and Inverse of an If-Then Statement

Determining True Statements within a Biconditional Statement The following is a true statement: if and only if

is an obtuse angle.

Determine the two true statements within this biconditional.

Statement 1: If Statement 2: If

, then

is an obtuse angle

is an obtuse angle, then

.

You should recognize this as the definition of an obtuse angle. All geometric definitions are biconditional statements.

Converse, Inverse, and Contrapositive: Examples (Geometry Concepts)

Earlier Problem Revisited The following information answers the question asked at the beginning of this Section: Your sister presented you with the if-then statement, "If you do the dishes, then I will help you with your homework." If we take the original statement to be true, then the contrapositive is also true. The following contrapositive statement is logically equivalent to the original if-then statement: "If I do not help you with your homework, then you will not do the dishes."

Examples Example 1 Use the statement: Any two points are collinear.

a) Find the converse, inverse, and contrapositive, and determine if the statements are true or false. If they are false, find a counterexamples First, change the statement into an “if-then” statement: If two points are on the same line, then they are collinear.

Example 2 2. Is

true? If not, find a counterexample. Is

true? If not, find a counterexample.Is

true? If not, find a counterexample.Is true? If not, find a counterexample.

Review For questions 1-4, use the statement: If

and

, then

is the midpoint of

1. If this is the converse, what is the original statement? Is it true? 2. If this is the original statement, what is the inverse? Is it true? 3. Find a counterexample of the statement. 4. Find the contrapositive of the original statement from #1. 5. What is the inverse of the inverse of ? HINT: Two wrongs make a right in math! 6. What is the one-word name for the converse of the inverse of an if-then statement? 7. What is the one-word name for the inverse of the converse of an if-then statement? 8. What is the contrapositive of the contrapositive of an if-then statement? For questions 9-12, determine the two true conditional statements from the given biconditional statements. 9. A U.S. citizen can vote if and only if he or she is 18 or more years old. 10. A whole number is prime if and only if it has exactly two distinct factors. 11. Points are collinear if and only if there is a line that contains the points. 12.

if and only if

.

13. a. Is

true? If not, find a counterexample.

.

b. Is c. Is

true? If not, find a counterexample. true? If not, find a counterexample.

d. Is

true? If not, find a counterexample.

14. a. Is

true? If not, find a counterexample.

b. Is

true? If not, find a counterexample.

c. Is

true? If not, find a counterexample.

d. Is true? If not, find a counterexample. 15. the measure of is a right angle a. Is

true? If not, find a counterexample.

b. Is

true? If not, find a counterexample.

c. Is d. Is 16.

true? If not, find a counterexample. true? If not, find a counterexample.

the measure of

is an acute angle

a. Is

true? If not, find a counterexample.

b. Is

true? If not, find a counterexample.

c. Is

true? If not, find a counterexample.

d. Is

true? If not, find a counterexample.

17. Write a conditional statement. Write the converse, inverse and contrapositive of your statement. Are they true or false? If they are false, write a counterexample. 18. Write a true biconditional statement. Separate it into the two true conditional statements.

Review (Answers) To view the Review answers, open this PDF file (https://www.ck12.org/flx/show/attachment/AnswerKey_CK-12-Chapter-02-Geometry-Concepts.pdf) and look for section 2.3.

Found a content error? Tell us (https://docs.google.com/forms/d/e/1FAIpQLSf0FzLyIImbJsJ2gVKD2wVRCfkXuGTXeZLPXlng4U2KZ0Er9g/v formkey=dHlaUFlvNnZrRzhPcWVWbzJ0ZGVFb1E6MQ&entry.1000002=https://www.ck12.org/geometry/con inverse-and-contrapositive-statements/lesson/Converse-Inverse-and-Contrapositive-GEOM/)

Vocabulary

Language: English▼

English

Term

Definition

biconditional statement

A statement is biconditional if the original conditional statement and the converse statement are both true.

Conditional Statement

A conditional statement (or 'if-then' statement) is a statement with a hypothesis followed by a conclusion.

Logically Equivalent

A statement is logically equivalent if the "if-then" statement and the contrapositive statement are both true.

premise

A premise is a starting statement that you use to make logical conclusions.

Show Details

Please wait... Please wait... (javscript:retur

se;)

Upload Failed

https://www.ck12.org/geometry/converse-inverse-and-contrapositive-statements/lesson/Converse-Inverse-and-Contrapositive-GEOM/



7/7...