Geometry

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Geometry

• Chapter 2 Reasoning and Proof

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What you will learn today

• Make conjectures based on inductive reasoning
• Find counterexamples
• Create conjunctions and disjunctions
• Determine truth values of conjunctions and
  disjunctions

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2.1 Inductive Reasoning and Conjecture

• A conjecture is an educated guess based on known
  information.
• Example
• Inductive reasoning is reasoning that uses a
  number of specific examples to arrive at a
  generalization or prediction.

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Example One

• For points P, Q, and R, PQ 9, QR 15, and PR
  12. Make a conjecture and draw a figure to
  illustrate your conjecture.

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You Do It

• For points L, M, and N, LM 20, MN 6, and LN
  14. Make a conjecture and draw a figure to
  illustrate your conjecture.

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Counterexample

• Conjectures are based on several observations
  that are mostly true.
• It only takes one false example to prove a
  conjecture is not true.
• The false example is called a counterexample.

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Example Two

• Determine whether each conjecture is true or
  false. Give a counter example for any false
  conjecture.
• Given m y 10, y 4
• Conclusion m 6
• Given noncollinear points R, S, and T
• Conclusion

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You Do It

• Determine whether each conjecture is true or
  false. Give a counterexample for any specific
  false conjecture.
• Given WXYZ is a rectangle
• Conclusion WX YZ and WZ XY
• Given JK KL LM JM
• Conclusion JKLM is a square

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2.2 Logic

• A statement is any sentence that is either true
  or false, but not both.
• Example p Today is Friday
• Where a statement is true or false is its truth
  value.
• Example p is true

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Logic

• The negation of a statement has the opposite
  meaning as well as an opposite truth value.
• Example p Today is not Friday
• Truth value of p is false
• Two or more statements can be joined to form a
  compound statements.

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Compound Statements

• A conjunction is a compound statement formed by
  joining two or more statements with the word and.
  
• Symbols p and q ? p q
• Both statements have to be true for the
  conjunction to be true.
• Example
• p Raleigh is a city in NC.
• q Raleigh is the capital of NC.
• p q Raleigh is a city in NC and Raleigh is the
  capital of NC.

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Example One

• Use the following statements to write a compound
  statement for each conjunction. Then find its
  truth value.
• p January 1 is the first day of the year
• q -5 11 -6
• r A triangle has three sides
• p q
• r q
• q and r

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You Do It

• Use the following statements to write a compound
  statement for each conjunction. Then find its
  truth value.
• p One foot is 14 inches
• q September has 30 days
• r A plane is defined by three noncollinear
  points
• p q
• r p
• q r

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Disjunction

• A disjunction is a compound statement formed by
  joining two or more statements with word or.
• Symbols p or q ? p v q
• Only one statement has to be true for the
  disjunction to be true
• Example
• p Raleigh is a city in NC
• q Raleigh is the capital of NC
• p v q Raleigh is a city in NC or Raleigh is the
  capital of NC

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Example Two

• Use the following statements to write a compound
  statement for each disjunction. Then find its
  truth value.
• p 100 5 20
• q The length of a radius of a circle is twice
  the length of its diameter
• p or q
• p v q

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You Do It

• Use the following statements to write a compound
  statement for each disjunction. Then find its
  truth value.
• p is proper notation for line AB
• q centimeters are metric units
• r 9 is a prime number
• p v q
• q v r

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Venn Diagrams

• Conjunctions can be illustrated with Venn diagrams

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Example Three
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You Do It
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Classwork

• Complete the following assignment and turn it in
  when you are finished (this way you dont have
  homework over the weekend ?)
• Worksheet
• Both sides all problems
• Will be graded for accuracy

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Warm - Up

• Make a conjecture about the following
• A, B, and C are points. AB 2, BC 4, and AC
  3
• Determine whether the following is true or false.
  Give a counterexample if the statement is false
• Given Points A, B, and C are collinear.
• Conclusion AB BC AC
• Given
• Conclusion
• Create the compound statement and determine the
  truth value.
• p 10 8 18
• q A rectangle has 3 sides
• p q
• p v q

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2.3 Conditional Statements

• Analyze statements in the if then form.
• Write the converse, inverse, and contrapositive
  of if then statements.
• Write and understand biconditional statements.
• Get 1500 cash back when you buy a new car.
• Free cell phone with every one year service
  enrollment.

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• The statements on the previous slide are examples
  of conditional statements.
• A conditional statement is a statement that can
  be written in the if-then form.
• Example
• Get 1500 cash back when you buy a new car.
• If you buy a car, then you get 1500 cash back.

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If then statement

• An if then statement is written in the for if
  p, then q.
• The phrase immediately following the word if is
  called the hypothesis
• The phrase immediately following the word then is
  called the conclusion
• p ? q, read if p then q, or p implies q.

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Example 1

• Identify the hypothesis and conclusion for each
  statement.
• If points A, B, and C lie on line m, then they
  are collinear.
• The Tigers will play in the tournament if they
  win their next game.

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Your Turn

• Identify the hypothesis and conclusion of each
  statement.
• If a polygon has 6 sides, then it is a hexagon.
• Tamika will advance to the next level of play if
  she completes the maze in her computer game.

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Writing statements in if then form

• Some statements are conditional but are not in if
  then form.
• It is easier to identify the hypothesis and
  conclusion before writing the sentence in if
  then form
• Example
• All apes love bananas
• Hypothesis An animal is an ape
• Conclusion It loves bananas
• If then If an animal is an ape, then it loves
  bananas

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Example 2

• Identify the hypothesis and conclusion of each
  statement. Then write each statement in the if
  then form.
• An angle with a measure greater than 90 is an
  obtuse angle.
• Perpendicular lines intersect.

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Your Turn

• Identify the hypothesis and conclusion of each
  statement. Then write each statement in the if
  then form.
• Distance is positive.
• A five sided polygon is a pentagon.

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Truth Value

• All cases of conditional statements are true
  except where the hypothesis is true and the
  conclusion is false.

p q p ? q
T T T
T F F
F T T
F F T
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Example 3

• Determine the truth value of the following
  statement for each set of conditions
• If you get 100 on your test, then your teacher
  will give you an A.
• You get 100 your teacher gives you an A
• True
• You get 100 your teacher gives you a B
• False
• You get 98 your teacher gives you an A
• True
• You get 85 your teacher gives you a B
• True

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Your Turn

• Determine the truth value of the following
  statement for each set of conditions
• If Parker rests for 10 days, his ankle will heal.
• Parker rests for 10 days, and he still has a hurt
  ankle
• False
• Parker rests for 3 days, and he still has a hurt
  ankle
• True
• Parker rests for 10 days, and he does not have a
  hurt ankle anymore
• True
• Parker rests for 7 days, and he does not have a
  hurt ankle anymore
• True

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Related Conditionals
Statement Formed by Symbols Examples
Conditional Given hypothesis and conclusion p ? q If two angles have the same measure, then they are congruent.
Converse Switch hypothesis and conclusion q ? p If two angles are congruent, then they have the same measure
Inverse Negate both the hypothesis and conclusion p ? q If two angles they do not have the same measure, then are not congruent.
Contrapositive Negate both and switch the hypothesis and conclusion q ? p If two angles are not congruent, then they do not have the same measure.
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• You can not determine any relationship between a
  conditional and the converse and inverse as far
  as truth value.
• However, the following is true
• The truth value of the conditional and
  contrapositive will always be the same
• The truth value of the converse and the inverse
  will always be the same
• Statements with the same truth values are said to
  be logically equivalent
• Conditional and Contrapositive are logically
  equivalent
• Converse and Inverse are logically equivalent

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Example 3

• Write the conditional, converse, inverse, and
  contrapositive of the statement Linear pairs of
  angles are supplementary. Determine whether each
  statement is true or false. If a statement is
  false, give a counterexample.

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Your Turn

• Write the conditional, converse, inverse, and
  contrapositive of the statement All squares are
  rectangles. Determine whether each statement is
  true or false. If the statement is false, give a
  counterexample.

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Biconditional Statement

• A biconditional statement is the conjunction of a
  conditional statement and its converse.
• (p ? q) (q ? p) is written p ? q, and read p if
  and only if q, can be abbreviated iff
• Both the conditional and the converse must be
  true for a biconditional to be true.

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Example 4

• Write each biconditional as a conditional and its
  converse. Then determine whether the
  biconditional is true or false. If false, give a
  counterexample.
• Two angles measures are complements if and only
  if their sum is 90.
• x gt 9 iff x gt 0

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Your Turn

• Write each biconditional as a conditional and its
  converse. Then determine whether the
  biconditional is true or false. If false, give a
  counterexample.
• A calculator will run if and only if it has
  batteries.
• 3x 4 30 iff x 7

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Homework

• Workbook
• Section 2.3
• 1 10 (all)
• You do not have any practice on the biconditional
  statement. Make sure you know how to create
  statements and know the truth value. It will be
  covered on your quiz tomorrow!

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Warm - Up

• Identify the hypothesis and conclusion of each
  statement
• If 2x 6 10, then x 2
• Write each statement in if then form
• Get a free visit with a one year fitness plan
• Vertical angles are congruent
• Write the converse, inverse, and contrapositive
  of each conditional statement. Determine whether
  each related conditional is true or false.
• All rectangles are quadrilaterals
• If you live in Dallas, then you live in Texas.

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Homework

• H 3x 4 -5 C x -3
• H you take a class in television broadcasting C
  you will film a sporting event
• If you do not remember the past, then you are
  condemned to repeat it.
• If two angles are adjacent, then they share a
  common vertex and a common side.
• True
• True
• True
• Converse If -8 gt 0 then (-8)2 gt 0 true
• Inverse If (-8)2 0, then -8 0 true
• Contrapositive If 8 0, then (-8)2 0
  false
• If you are a junior, then you wait on tables
• If you wait on tables, then you are a junior

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Quiz Time

• Please clear off your desk
• You will have plenty of time to complete your
  quiz
• When you are finished, please remain quiet until
  everyone else has finished.
• We will begin 2.4 Deductive Reasoning with the
  Law of Detachment and Law of Syllogism

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2.4 Deductive Reasoning

• Use the Law of Detachment
• Use the Law of Syllogism
• When you are ill, your doctor may prescribe an
  antibiotic to help you get better. Doctors may
  use a dose chart to determine the correct amount
  of medicine based on your weight.

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• The process that the doctors use to determine the
  amount of medicine a patient should take is
  called deductive reasoning.
• Deductive reasoning uses facts, rules,
  definitions, or properties to reach a logical
  conclusion.
• One way to do this is the Law of Detachment.

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Law of Detachment

• If p ? q is true and p is true, then q is also
  true.
• Example

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Example 1

• The following is a true conditional. Determine
  whether each conclusion is a valid based on the
  given information. Explain your reasoning.
• If a ray is an angle bisector, then it divides
  the angle into two congruent angles.

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Your Turn

• The following is a true conditional. determine
  whether each conclusion is a valid based on the
  given information. Explain your reasoning.
• If two segments are congruent and the second
  segment is congruent to a third segment, then the
  first segment is also congruent to the third.
• Given WX UV and UV RT.
• Conclusion WX RT.
• Given UV and WX RT.
• Conclusion WX UV and UV RT

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Law of Syllogism

• If p ?q and q ? r are true, then p ? r is also
  true.
• Another way to look at the transitive property.
• Some statements may not be written in conditional
  form. It may be easier to see the Law of
  Syllogism when statements are written in
  conditional form
• Example

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Example 2

• Use the Law of Syllogism to determine whether a
  valid conclusion can be reached from each set of
  statements.
• (1) If the symbol of a substance is Pb, then it
  is lead.
• (2) The atomic number of lead is 82.
• (1) Water can be represented by H2O.
• (2) Hydrogen (H) and Oxygen (O) are in the
  atmosphere.

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Your Turn

• Use the Law of Syllogism to determine whether a
  valid conclusion can be reached from each set of
  statements.
• (1) If Ashley attend the prom, she will go with
  Mark.
• (2) Mark is a 17 year old student.
• (1) If Mel and his date eat at the Peddler
  Steakhouse
• before going to the prom, they will miss
  the senior
• march.
• (2) The Peddler Steakhouse stays open until 10
  p.m.

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Example 3

• Determine whether the statement (3) follows from
  statements (1) and (2) by the Law of Detachment
  of the Law of Syllogism. If it does, state which
  law was used. If it does not, write invalid.
• (1) Vertical angles are congruent.
• (2) If two angles are congruent, then their
  measures are equal.
• (3) If two angles are vertical, then their
  measures are equal.

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• (1) If a figure is a square, then it is a
  polygon.
• (2) Figure A is a polygon.
• (3) Figure A is a square.

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Your Turn

• (1) If the sum of the squares of two sides of a
  triangle is equal to the square of the third
  side, then the triangle is a right triangle.
• (2) For ?XYZ, XY2 YZ2 XZ2
• (3) ?XYZ is a right triangle

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Your Turn (cont.)

• (1) If Ling wants to participate in the
  wrestling competition, he will have to meet an
  extra three times a week to practice.
• (2) If Ling adds anything extra to his weekly
  schedule, he cannot take karate lessons.
• (3) If Ling wants to participate in the
  wrestling competition, he cannot take karate
  lessons.

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Classwork

• Worksheet
• Section 2.4
• 1 7

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Homework

• Workbook
• Section 2.4
• 1 7 (all)

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Warm - Up

• Use the Law of Syllogism to determine whether a
  valid conclusion can be reached from the set of
  statements.
• If it rains, then the field will be muddy.
• If the field is muddy, then the game will be
  cancelled
• Determine whether the Law of Detachment or the
  Law of Syllogism was used to find the conclusion.
• If it snows outside, you will wear your winter
  coat.
• It is snowing outside
• You wear your winter coat.

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Homework

1. Valid
2. Invalid
3. If two angles form a linear pair, then the sum of
   their measures is 180
4. If a hurricane is Category 5, then trees, shrubs,
   and signs are blown down.
5. Yes Law of Detachment
6. Invalid
7. If a virus is a parasite, then it harms its host.

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2.5 Postulates and Paragraph Proofs

• Identify and use basic postulates about points,
  lines, and planes.
• Write paragraph proofs.

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• Postulate a statement that describes a
  fundamental relationship between the basic terms
  of geometry.
• Postulates are accepted as true.
• The ideas of points, lines, and planes from
  Chapter 1 are considered postulates.

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Postulates

• 2.1 Through any two points, there is exactly one
  line
• 2.2 Through any three points not on the same
  line, there is exactly one plane.

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Example One

• Jesse is setting up a network for his mothers
  business. There are five computers in her
  office. He wants to connect each computer to
  every other computer so that if one computer
  fails, the others are still connected. How many
  connections does Jesse need to make?

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Your Turn

• Some snow crystals are shaped like regular
  hexagons. How many lines must be drawn to
  interconnect all vertices of a hexagonal snow
  crystal?
• Donna is setting up a network for her company.
  There are 7 computers in her office. She wants to
  connect each computer to every other computer so
  that if one computer fails, the others are still
  connected. How many connections does Donna have
  to make?

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More Postulates

• 2.3 A line contains at least two points.
• 2.4 A plane contains at least three points not
  on the same line.

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More Postulates

• 2.5 If two points lie in a plane, then the
  entire line containing those points lies in that
  plane.
• 2.6 If two lines intersect, then their
  intersection is exactly one point.
• 2.7 If two planes intersect, then their
  intersection is a line.

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Example Two

• Determine whether each statement is always,
  sometimes, or never true. Explain.
• If points A, B, and C lie in plane M, then they
  are collinear.
• Sometimes.
• There is exactly one plane that contains
  noncollinear points P, Q, and R.
• Always.
• There are at least two lines through points M and
  N.
• Never.

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Your Turn

• Determine whether each statement is always,
  sometimes, or never true. Explain.
• The intersection of plane M and plane N is a
  point
• Never.
• If A and B lie in plane W, then line AB lies in
  plane
• Always.
• Segment TR lines in plane M.
• Sometimes.

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Paragraph Proofs

• Undefined terms, definitions, postulates, and
  algebraic properties of equality are used to
  prove that other statements or conjectures are
  true.
• Example of this is a theorem statement that has
  been shown to be true.

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Paragraph Proofs

• Proof a logical argument in which each
  statement is supported by a statement that is
  accepted as true.
• Paragraph Proof (Informal Proof) you write a
  paragraph to explain why a conjecture for a given
  statement is true.

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Five essential parts of a good proof

• State the theorem or conjecture to be proven
• List the given information
• If possible, draw a diagram to illustrate the
  given information
• State what is to be proved
• Develop a system of deductive reasoning

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Example Three

• Given that M is the midpoint of segment PQ, write
  a paragraph proof to show that PM is congruent to
  MQ.

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Your Turn

• Given line AC intersecting line CD, write a
  paragraph proof to show that A, C, and D
  determine a plane.

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One More Example

• Given that is the angle bisector of ?CAD, write
  a paragraph proof to show that ?CAB ? ?DAB.

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Midpoint Theorem

• Theorem 2.8 If M is the midpoint of segment AB,
  then AM is congruent to MB.

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2.6 Algebraic Proof

• Use algebra to write two column proofs
• Use properties of equality in geometry.
• Lawyers develop their cases using logical
  arguments based on evidence to lead a jury to a
  conclusion favorable to their case.

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Algebraic Proof

• You have learned to use properties of equality to
  solve equations and verify relationships.
• Algebra is a system of mathematics with sets of
  numbers, operations, and properties that allow
  you to perform algebraic operations.

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In your textbook on page 94
Reflexive Property a a
Symmetric Property If a b, then b a
Transitive Property If a b and b c, then a c
Add/Subtract Property If a b, then a c b c and a c b c
Multiply/Divide Property If a b, then a c b c and a c b c
Substitution Property If a b, then a can replace b anywhere
Distributive Property a(b c) ab ac
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Example 1

• Solve 3(x 2) 42.

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Your Turn

• Solve 2(5 3a) 4(a 7) 92

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• Two column proof (formal proof) contains
  statements and reasons organized in two columns.
• Each step is called a statement and the
  properties that justify each step are called
  reasons.

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Example 2

• Write a two column proof.

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Your Turn

• Write a two column proof.

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Geometric Proof

• Geometry uses variables, numbers, and operations.
• Segment measures and angle measures are real
  numbers, so properties from algebra can be used
  to discuss their relationships.

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Example 3

• On a clock, the angle formed by the hands at 200
  is a 60º angle. If the angle formed at 200 is
  congruent to the angle formed at 1000, prove
  that the angle at 1000 is a 60º angle.

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Your Turn

• A starfish have five arms. If the length of arm
  1 is 22cm, and arm 1 is congruent to arm 2, and
  arm 2 is congruent to arm 3, prove that arm 3 has
  length of 22 cm.

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Homework

• Workbook
• Page 11
• 1 - 6
• Page 12
• 1 and 2

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Warm - Up

• Determine whether the following are sometimes,
  always, or never true
• Three points determine a plane
• Points G and H are in plane X. Any point
  collinear with G and H is in plane X.
• The intersection of two planes can be a point.
• Create a two column proof to prove the
  following
• Given 2x 7 ½x 1
• Prove x 4

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Homework
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Homework (cont.)
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2.7 Proving Segment Relationships

• Write proofs involving segment addition
• Write proofs involving segment congruence.

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Ruler Postulate

• The points on any line segment can be paired with
  real numbers so that, given any two points A and
  B on a line, A corresponds to zero, and B
  corresponds a positive real number.
• Meaning, you can measure a segment with a ruler.

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Segment Addition Postulate

• If B is between A and C, then AB BC AC.
• If AB BC AC, then B is between A and C.

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Example One

• Given PQ RS.
• Prove PR QS.

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Your Turn

• Given PR QS
• Prove PQ RS

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Segment Congruence

• Reflexive
• Symmetric
• Transitive

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Example Two

• Given
• Prove

102
Your Turn

• Given
• Prove

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2.8 Proving Angle Relationships

• Write proofs involving supplementary and
  complementary angles.
• Write proofs involving congruent and right angles.

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Protractor Postulate

• Given and a number r between 0 and 180,
  there is exactly one ray with endpoint A,
  extending on either side of , such that the
  measure of the angle formed is r.
• Meaning you can use a protractor to measure an
  angle.

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Angle Addition Postulate

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Example One

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Your Turn

• At 4 oclock, the angle between the hour and
  minute hands of a clock is 120º. If the second
  hand stops where it bisects the angle between the
  hour and minute hands, what are the measures of
  the angles between the minute and second hands
  and between the second and hour hands?

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Theorems

• Supplement Theorem If two angles form a linear
  pair, then they are supplementary angles.
• Complement Theorem If the noncommon sides of
  two adjacent angles form a right angle, then the
  angles are complementary angles.

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Example Two

• If angle 1 and angle 2 form a linear pair and
  angle 2 67, find the measurement of angle 1.

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Your Turn

• If angle 1 and angle 2 for a linear pair and
  angle 2 166, find the measurement of angle 1.

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Congruency of Angles

• Congruence of angles is reflexive, symmetric, and
  transitive.

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Theorems

• Angles supplementary to the same angle or to
  congruent angles are congruent.
• Angles complementary to the same angle or to
  congruent angles are congruent.

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Example Three

• Given
• Prove

114
Your Turn

115
Vertical Angle Theorem

• If two angles are vertical angles, then they are
  congruent.

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Example Four

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Your Turn

118
Right Angle Theorems

• Perpendicular lines intersect to form four right
  angles.
• All right angles are congruent.
• Perpendicular lines form congruent adjacent
  angles.

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Right Angle Theorems

• If two angles are congruent and supplementary,
  then each angle is a right angle.
• It two congruent angles form a linear pair, then
  they are right angles.

120
Classwork/Homework

• Worksheet
• Lesson 2.7
• Lesson 2.8