Presentation on theme: "Bell Work 9/18/12 State the angle relationship and solve for the variable 1)2) 3) Find the distance and midpoint of AB, where A is at (2, -1) and B is."— Presentation transcript:
Bell Work 9/18/12 State the angle relationship and solve for the variable 1)2) 3) Find the distance and midpoint of AB, where A is at (2, -1) and B is at (5, -11) 4) In your own words, what is proof? How can we prove things in geometry? 4y y
Outcomes I will be able to: 1) Recognize and analyze a conditional statement 2) Write postulates about points, lines, and planes using conditional statements 3) Begin to have an understanding of geometric proof
Agenda 1) Bell Work 2) Outcomes 3) Angle Relationship Partner Activity 4) White Board Logic Activity 5) Conditional Statement Notes 6) Conditional Statement White Board 7) Conditional Statement Notes Continued 8) Video clip on Converse Statements
Logic Activity Exercise1: Use logical reasoning skills to answer the following questions. Five athletes were returning from a cross-country race. From the following information, can you tell how Amber, Becky, and Danielle placed in the race? – Amber was not last. – Amber came in after Erin. – Danielle was not first. – Courtney placed third. – Erin placed second.
Logic Activity Exercise 2. Four friends left one slice of pizza in the kitchen and went into the next room to play games. During the next half hour, each friend left the room for a few minutes and then returned. At the end of the hour, all four went back into the kitchen and found that the last slice of pizza was gone. Use the following statements to figure out who ate it. Only one of the following statements is true. – Ryan: "Kyle ate it.” – Kyle: "Chris ate it.“ – James: "Who me? Can't be.“ – Chris: "Kyle is lying when he says I ate it."
Logic Activity Starting with the word cat, how can we end up with the word dog by only changing one letter at time? We must make sure the new word created each time is an actual word. C – A – T ---> B – A – T B – A – T ---> B – A – G B – A – G ---> B – O – G B – O – G ---> D – O – G On your white boards, attempt to do the same thing starting with the word “give” and try to get the word “math”
Conditional Statements Conditional Statement- a statement that has two parts, a hypothesis and a conclusion. It can be written in the form If (p), then (q). The hypothesis is the part (p) of a conditional statement following the word if. The conclusion is the part (q) of a conditional statement following the word then. Example: If it is noon in Georgia, then it is 9 AM in California Hypothesis: If it is noon in Georgia Conclusion: then it is 9 AM in California (p) (q)
If/Then Statements Are all conditional statements always true? Example: If you have a math class, then you have Mrs. Stall. On your white boards, determine whether the following If/Then statements are true or false
If/Then Statements 1) If you are in Indianapolis, then you are in Indiana. 2) If you are in Indiana, then you are in Indianapolis. 3) If you are a teenager, then you are in high school. 4) If you have a sister, then you are not an only child. 5) If your dad has blue eyes, then you have blue eyes.
Rewriting a sentence in If/Then form Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. Example: Vertical angles are congruent. If two angles are vertical, then they are congruent.
Re-Writing Statements If/Then Form a. Two points are collinear if they lie on the same line. b. All mammals breathe oxygen c. A number divisible by 9 is also divisible by 3 a. If two points lie on the same line, then they are collinear. b. If an animal is a mammal, then it breathes oxygen. c. If a number is divisible by 9, then it is divisible by 3.
Conditional Statements Are Conditional Statements always true? No, as we saw earlier, some conditional statements can have counterexamples Conditional statements can either be true or false. It is false only when the hypothesis is true and the conclusion is false. To show that a conditional statement is true, you must present an argument that the conclusion follows for all cases that fulfill the hypothesis. To show that a conditional statement is false, describe a single counterexample that shows the statement is not always true. Example: If x² = 16, then x = 4. Is there any other way for this to be correct? Counterexample: x = -4
Converse of a Conditional Statement Converse – a conditional statement in which the hypothesis and the conclusion are switched. Example: Conditional: If you see lightning, then you hear thunder. Converse: If you hear thunder, then you see lightning. Example: Write the converse of the following conditional statement. Statement: If two segments are congruent, then they have the same length.
Exit Ticket Converse Practice Write each statement in the “if – then” form. Then write its Converse. Label each Statement True or False. 1. A number divisible by 25 is also divisible by A person over the age of 16 is eligible for a driver’s license. 3. “Where all think alike, no-one thinks much.” -Walter Lipman 4. Two angles have the same measure if they are congruent. 5. If you go to Herron High School, you are required to wear a uniform.
Negation of a Conditional Statement Negation – A statement that is formed by writing the negative of a statement Example: Statement: If m ∠ A = 30°, then ∠ A is acute. Negation: If m ∠ A ≠ 30°, then ∠A is not acute. The symbol for negation is ~
Inverse of a Conditional Statement Inverse: A statement formed by negating the hypothesis and conclusion of a conditional statement. Example: Conditional - If there is snow on the ground, then flowers are not in bloom. Inverse - If there is no snow on the ground, then flowers are in bloom.
Contrapositive of a conditional Contrapositive – When the hypothesis and the conclusion of the converse are negated ***Always find the converse first before finding the contrapositive. Conditional: If there is snow on the ground, then flowers are not in bloom Converse: If flowers are not in bloom, then there is snow on the ground Contrapositive: If flowers are in bloom, then there is not snow on the ground
Inverse, Converse, Contrapositive 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°.
Conditional Statements When two statements are both true or both false, they are called equivalent statements. A conditional statement is equivalent to its contrapositive. Similarly, the inverse and converse of any conditional statement are equivalent, as shown in the table above.
Point, Line and Plane Postulates a. Postulate 5. Through any two points there is exactly one line. There is exactly one line (line n) that passes through the points A and B. b. Postulate 6. A line contains at least two points: Line n contains at least two points. For instance, line n contains the points A and B. c. Postulate 7: If two lines intersect, then their intersection is exactly one point. Lines m and n intersect at point A. d. Postulate 8: Through any three noncollinear points there exists exactly one plane Points A, B, C are contained in plane P
Point, Line and Plane Postulates e. Postulate 9: A plane contains at least three noncollinear points. Plane P contains at least three noncollinear points, A, B, and C. f. Postulate 10: If two points lie in a plane, then the line containing them lies in the same plane Points A and B lie in plane P. So, line n, which contains points A and B, also lies in plane P. g. Postulate 11: If two planes intersect, then their intersection is a line. Planes P and Q intersect. So, they intersect in a line, labeled m
Example Example: Rewrite Postulate 5 in if-then form. Then find its inverse, converse, and contrapositive. If-then form: Inverse: Converse: Contrapositive:
Geometric Proof Watch the following clip from Alice and Wonderland Think about the difference between, “saying what you mean” and “meaning what you say.” Is there difference? &feature=player_embedded
Given the following conditional statement, write the inverse, converse, and contrapositive statements Conditional – If you have Mr. McGrew, then you have a math class Inverse = Converse = Contrapositive = Exit Quiz