Geo 2.1 condtional statements

CONDITIONAL STATEMENTSUnit 2. Section 2.1

- Conditional statements are used in every field of human endeavor.

- They are crucial to the search for truth - If you are going to be able to

adequately interpret and judge the statements you hear, you must understand the structure of Conditional statements.

- WARNING: Once you get good at these,

you may be amused by statements made by politicians and other public speakers.

Objectives

Students will be able to: Identify conditional statements and place

them in if-then form Identify the hypothesis and conclusion of

a conditional statement Convert conditional statements into their

other logical variants Identify and use truth relationships of

conditional statements State and use the point, line and plane

postulates of geometry

What is a Conditional Statement? A statement that can be written in

the format:If …., then ….

The part after “if” is called the hypothesis Do not confuse this with the word

“hypothesis” from science

The rest (after “then”) is the conclusion

- Often statements are not in if –then form, but to test them out scientifically, we must convert them

- In English, there are infinite ways to rephrase a conditional statement; however, we will cover the three most common variations here

Statements Not in If-Then Form

Standard Sentence

Split the subject and predicate. Add “If it is” or “If they are” to the

subject Add “then it” or “then they” to the

predicate Smooth out the grammar

Example: “All dogs go to heaven.” “If they are dogs, then they go to heaven.”

“Whenever” or “When”

Replace “whenever” or “when” with “if”

Add “then” after the comma

Example: “Whenever I see a seagull, I think of home.” If I see a seagull, then I think of home.

“If” at the end

Move the “if” clause to the beginning Add “then” after the “if” clause

Example: “I eat if I am hungry.” “If I am hungry, then I eat.”

-Often a conditional statement can be difficult to prove or unwieldy to use.- By using logical variations, we find forms easier to prove or use.

Logical Variations of the Conditional

Converse, Inverse, & Contrapositive Converse: formed by swapping the

hypothesis and the conclusion

Inverse: formed by negating the hypothesis and conclusion

Contrapositive: formed by both negating and swapping the hypothesis and conclusion

Equivalent Statements

If the Conditional is true (or false) then so is the Contrapositive and vice versa.

Similarly, if the Converse is true, then so is the Inverse and vice versa

If both the Conditional and its Converse are true, then they can be rewritten as a Biconditional statement (more next class)

Example 1

If you added 2+2, you got 4

CONVERSE: If you got 4, then you added

2+2.

INVERSE: If you did not add 2+2, you did not get 4.

CONTRAPOSITIVE: If you did not get 4, then you did not

add 2+2.

Example 2 (the word “not”) If you do not eat, you will be hungry

CONVERSE: If you are hungry, then you did

not eat.

INVERSE: If you ate, then you are not hungry.

CONTRAPOSITIVE: If you are not hungry, then you

ate.

Point, Line and Plane Postulates

Point, Line & Plane Postulates

5. Through any two points there exists exactly one line.6. A line contains at least two points.7. If two lines intersect, then their intersection is exactly

one point.8. Through any three noncollinear points there exists

exactly one plane.9. A plane contains at least three noncollinear points.10.If two points lie in a plane, then the line containing

them lies in the plane.11.If two planes intersect, then their intersection is a line.

Reference

McDougal Littell Geometry (2001), Section 2.1