2.1 Conditional Statements

Goals

•Recognize a conditional statement

•Write postulates about points, lines and planes

Recognizing Conditional Statements

Conditional Statements If-Then Statements

If a number is divisible by both 2 and 3 then it is divisible by 6.

HYPOTHESIS CONCLUSION

If a polygon has four sides then it is a quadrilateral.

If a number greater than two is even, then it is not prime.

I will dry the dishes if you wash them.

Recognizing Conditional Statements

Conditional statements can be True or False

• To show a conditional statement is true, you must present an argument to show true in all cases.

• To show conditional statement is false, you only have to have a single counterexample.

Recognizing Conditional Statements

Example:

Write a counterexample:

If a number is odd, then it is divisible by 3

Recognizing Conditional Statements

IF two angles are supplementary, THEN the sum of their angles is 180 degrees.

IF you are 5 feet tall, THEN are also 60 inches tall.

Example 1 State the hypothesis and conclusion for each statement.

Recognizing Conditional Statements

IF two angles are adjacent, THEN they have a common vertex.

Three noncollinear points are coplanar IF they lie on the same plane.

Example 1 State the hypothesis and conclusion for each statement.

Example 2

Recognizing Conditional Statements

Rewrite in if-then form

All monkeys have tails.

Vertical angles are congruent.

Example 2

Recognizing Conditional Statements

Rewrite in if-then form

Supplementary angles have measures whose sum is 180°.

Practice is cancelled if it rains.

Recognizing Conditional Statements

The CONVERSE of a conditional statement is formed by interchanging the hypothesis and conclusion. 

conditional statement

If x – y is positive then x > y .

converse

If x > y then x – y is positive.

Recognizing Conditional Statements

If the CONVERSE statement is true the converse may or may not be true.

conditional statement

If two angles are adjacent they share a common side.

Recognizing Conditional Statements

1. IF two angles are adjacent, THEN they have a common vertex. CONVERSE - IF two angles have a common vertex, THEN they are adjacent. 2. IF two angles are supplementary, THEN the sum of their angles is 180 degrees. CONVERSE - IF two angles have a sum of 180 degrees, THEN they are supplementary. 3. IF you are 5 feet tall, THEN are also 60 inches tall. CONVERSE - IF you are 60 inches tall, THEN are also 5 feet tall.

Recognizing Conditional Statements

The denial of a statement is called a NEGATION. 

RST is an obtuse angle.

Intersecting lines are coplanar.

If we take a test today we do not have homework.

Recognizing Conditional Statements

Given a conditional statement, its INVERSE can be formed by negating both the hypothesis and conclusion. 

The inverse of a true statement is not necessarily true.  EXAMPLE Conditional statement: If the angle is 75 degrees, then it

is acute.

Inverse: If the angle is not 75 degrees, then it is not acute.

Recognizing Conditional Statements

If you have vertical angles, then they are congruent.

Example 3Find the inverse of the following statement. Is it True or False

Recognizing Conditional Statements

CONTRAPOSITIVE: Formed by negating the hypothesis and conclusion of the converse of the given conditional. 

When forming a contrapositive of a conditional it may be easier to write the converse first – then negate each part.

Example: Statement: If the angle is 75 degrees then it is acute .

Recognizing Conditional Statements

When two statements are both true or both false, they are called equivalent statements.

A conditional statement is equivalent to its contrapositive.

The inverse and converse of any conditional statement are equivalent.

Original If mA = 30°, then A is acute.

InverseIf mA 30°, then A is not acute.

Converse If A is acute then mA = 30°.

Contrapositive If A is not acute then mA 30°.

FT

If two angles are vertical, then they are congruent.

Recognizing Conditional Statements

Example 5:Write the contrapositive of the conditional statement

Using Point, Line and Plane Postulates

Postulate 5 Two Points - Line

                        Through any two points there is exactly one line.

(as an If-then statement)If there are two points, then there is exactly one line that contains them.

Using Point, Line and Plane Postulates

Postulate 6 Line - Two Points

                        

A line contains at least two points.

Postulate 7 Three Points - Plane

Using Point, Line and Plane Postulates

If two lines intersect, then their intersection is exactly one point.

Using Point, Line and Plane Postulates

Postulate 8 Three Points - Plane

                         

     

Through any three non collinear points there is exactly one plane. 

(or as an If-then statement)If there are three non collinear points, then there is exactly one plane that contains them.

Using Point, Line and Plane Postulates

Postulate 9 Plane - Three Points

            

                   

A plane contains at least three non collinear points.

Using Point, Line and Plane Postulates

Postulate 10 Two Points - Line - Plane

If two points lie in a plane, then the entire line containing those two points lies in the plane.

Using Point, Line and Plane Postulates

Postulate 11 Plane Intersection - Line

If two planes intersect, then their intersection is a line.

                                          

Using Point, Line and Plane Postulates

Example:

Decide whether the statement is True or False. If False, give a counterexample.

Three points are always contained in a line.

Homework

2.1 10-46 mult of 3