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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 Example 1 State the hypothesis and conclusion for each statement. 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. Recognizing Conditional Statements Example 1 State the hypothesis and conclusion for each statement. IF two angles are adjacent, THEN they have a common vertex. Three noncollinear points are coplanar IF they lie on the same plane. Recognizing Conditional Statements Example 2 Rewrite in if-then form All monkeys have tails. Vertical angles are congruent. Recognizing Conditional Statements Example 2 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 Example 3 Find the inverse of the following statement. Is it True or False If you have vertical angles, then they are congruent. 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. Original If mA = 30°, then A is acute. Inverse If mA 30°, then A is not acute. Converse If A is acute then mA = 30°. F T Contrapositive If A is not acute then mA 30°. A conditional statement is equivalent to its contrapositive. The inverse and converse of any conditional statement are equivalent. Recognizing Conditional Statements Example 5: Write the contrapositive of the conditional statement If two angles are vertical, then they are congruent. 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. Using Point, Line and Plane Postulates Postulate 7 Three Points - Plane 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

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