Transcript 2.1x - My Haiku

Bell Work 9/18/12
• State the angle relationship and solve for the
variable
• 1)
2)
4y+ 10
11y
• 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?
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
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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?
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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.
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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(q)is 9
(p)
AM in California
• Hypothesis: If it is noon in Georgia
• Conclusion: then it is 9 AM in California
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 5.
2. 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
Contrapositive
If ∠A is acute, then m∠A = 30°.
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?
http://www.youtube.com/watch?v=1oupIOmnLJs
&feature=player_embedded
Exit Quiz
• 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 =