2.1 Conditional Statements Goals Recognize a conditional statement Write postulates about points, lines and planes.

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Presentation transcript:

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. 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. State the hypothesis and conclusion for each statement.

Recognizing Conditional Statements Rewrite in if-then form All monkeys have tails. Vertical angles are congruent.

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 3 Find 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 m  A = 30°, then  A is acute. Inverse If m  A  30°, then  A is not acute. Converse If  A is acute then m  A = 30°. Contrapositive If  A is not acute then m  A  30°. F T

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 mult of 3