Mathematical Arguments and Triangle Geometry

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Mathematical Arguments and Triangle Geometry

• Chapter 2

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Deductive Reasoning

• A process
• Demonstrates that if certain statements are true
  
• Then other statements shown to follow logically
• Statements assumed true
• The hypothesis
• Conclusion
• Arrived at by a chain of implications

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Deductive Reasoning

• Statements of an argument
• Deductive sentence
• Closed statement
• can be either true or false
• Open statement
• contains a variable truth value determined once
  variable specified

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Deductive Reasoning

• Statements open? closed? true? false?
• All cars are blue.
• The car is red.
• Yesterday was Sunday.
• Rectangles have four interior angles.
• Construct the perpendicular bisector.

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Deductive Reasoning

• Nonstatement cannot take on a truth value
• Construct an angle bisector.
• May be interrogative sentence
• Is ?ABC a right triangle?
• May be oxymoron

The statement inthis box is false
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Rules of Logic

• Use logical operators
• and, or
• Evaluate truth of logical combinations
• P and Q

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Rules of Logic

• Combining with or
• P or Q

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Rules of Logic

• Negating a statement
• not P

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Conditional Statements

• Implication P implies Q if P
  then Q

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Conditional Statements

• Vivianis TheoremIF a point P is interior toan
  equilateral triangle THEN the sum of the lengths
  of the perpendiculars from P to the sides of the
  triangle is equal to the altitude.

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Conditional Statements

• What would make the hypothesis false?
• With false hypothesis, it still might be possible
  for the lengths to equal the altitude

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Conditional Statements

• Consider a false conditional statement
• IF two segments are diagonals of a trapezoidTHEN
  the diagonals bisect each other
• How can we rewrite this as a true statement

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Conditional Statements

• Where is this on the truth table?
• We want the opposite
• IF two segments are diagonals of a trapezoidTHEN
  the diagonals do not bisect each other

TRUE statement
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Conditional Statements

• Given P ? Q
• The converse statement is Q ? P
• Hypothesis and conclusion interchanged
• Consider truth tables
• Reversed

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Conditional Statements

• Given P ? Q
• The contrapositive statement is ?Q ? ?P
• Note they have the same truth table result
• This can be useful in proofs

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Conditional Statements

• Cevas theorem
• If lines CZ, BY, and XA are concurrentThen
• State the converse, the contrapositive

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Conditional Statements

• Cevas theorem a biconditional statement
• Both statement and converse are true
• Note two separate proofs are required
• Lines CZ, BY, and XA are concurrent IFF

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Mathematical Arguments

• Developing a robust proof
• Write a clear statement of your conjecture
• It must be a conditional statement
• Proof must demonstrate that your conclusions
  follow from specified conditions
• Draw diagrams to demonstrate role of your
  hypotheses

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Mathematical Arguments

• Goal of a robust proof
• develop a valid argument
• use rules of logic correctly
• each step must follow logically from previous
• Once conjecture proven then it is a theorem

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Mathematical Arguments

• Rules of logic give strategy for proofs
• Modus ponens P ? Q
• Syllogism P ? Q, Q ? R, R ? Sthen P ? S
• Modus tollens P ? Q and ? Qthen ? P -- this
  is an indirect proof

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Universal Existential Quantifiers

• Open statement has a variable
• Two ways to close the statement
• substitution
• quantification
• Substitution
• specify a value for the variablex 5 9
• value specified for x makes statement either true
  or false

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Universal Existential Quantifiers

• Quantification
• View the statement as a predicate or function
• Parameter of function is a value for the variable
• Function returns True or False

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Universal Existential Quantifiers

• Quantified statement
• All squares are rectangles
• Quantifier All
• Universe squares
• Must show every element of universe has the
  property of being a square
• Some rectangles are not squares
• Quantifier there exists
• Universe rectangles

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Universal Existential Quantifiers

• Venn diagrams useful in quantified statements
• Consider the definitionof a trapezoid
• A quadrilateral with a pair of parallel sides
• Could a parallelogram be a trapezoid according to
  this diagram?
• Write quantified statements based on this diagram

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Negating a Quantified Statement

• Useful in proofs
• Prove the contrapositive
• Prove a statement false
• Negation patterns for quantified statements

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Try It Out

• Negate these statements
• Every rectangle is a square
• Triangle XYZ is isosceles, or a pentagon is a
  five-sided plane figure
• For every shape A, there is a circle D such that
  D surrounds A
• Playfairs Postulate Given any line l, there is
  exactly one line m through P that is parallel to
  l (see page 41)

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Congruence Criteria for Triangles

• SAS If two sides and the included angle of one
  triangle are congruent to two sides and the
  included angle of another triangle, then the two
  triangles are congruent.
• We will accept this axiom without proof

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Angle-Side-Angle Congruence

• State the Angle-Side-Angle criterion for triangle
  congruence (dont look in the book)
• ASA If two angles and the included side of one
  triangle are congruent respectively to two angles
  and the included angle of another triangle, then
  the two triangles are congruent

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Angle-Side-Angle Congruence

• Proof
• Use negation
• Justify the steps in the proof on next slide

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ASA

• Assume AB? ?DE

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Orthocenter

• Recall Activity 1
• Theorem 2.4 The altitudes of a triangleare
  concurrent

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Centroid

• A median the line segment from the vertex to
  the midpoint of the opposite side
• Recall Activity 2

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Centroid

• Theorem 2.5 The three medians of a triangle are
  concurrent
• Proof
• Given ABC, medians ADand BE intersect at G
• Now consider midpointof AB, point F

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Centroid

• Draw lines EX and FY parallel to AD
• List the pairs ofsimilar triangles
• List congruent segments on side CB
• Why is G two-thirds of the way along median BE?

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Centroid

• Now draw medianCF, intersectingBE at G
• Draw parallels asbefore
• Note similar triangles and the fact that G is
  two-thirds the way along BE
• Thus G G and all three medians concurrent

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Incenter

• Consider the angle bisectors
• Recall Activity 3
• Theorem 2.6The angle bisectors of a triangle are
  concurrent

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Incenter

• Proof
• Consider angle bisectors for angles A and B with
  intersection point I
• Constructperpendicularsto W, X, Y
• What congruenttriangles do you see?
• How are the perpendiculars related?

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Incenter

• Now draw CI
• Why must it bisect angle C?
• Thus point I is concurrent to all three
  anglebisectors

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Incenter

• Point of concurrency called incenter
• Length of all three perpendiculars is equal
• Circle center at I, radius equal to perpendicular
  is incircle

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Circumcenter

• Recall Activity 4
• Theorem 2.7The three perpendicular bisectors of
  the sides of a triangle are concurrent.
• Point of concurrency called circumcenter
• Proof left as an exercise!

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Euler Line

• What conclusion did you draw from Activity 9?

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Euler Line

• Proof
• Find line through two of the points
• Show third point also on the line

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Euler Line

• Given OG throughcircumcenter, Oand centroid, G
• Consider X onOG with G between O and X
• Recall G is 2/3 of dist from A to D
• What similar triangles now exist?
• Parallel lines?
• Now G is 2/3 dist from X to O

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Euler Line

• X is on altitudefrom A
• Repeat argumentfor altitudes fromC and B
• So X the same point on those altitudes
• Distinct non parallel lines intersect at a unique
  point

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Preview of Coming Attractions

• Circle Geometry
• How many points to determine a circle?
• Given two points how many circles can be drawn
  through those two points

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Preview of Coming Attractions

• Given 3 noncolinear points how many distinct
  circles can be drawn through these points?
• How is the construction done?
• This circle is the circumcircle of triangle ABC

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Preview of Coming Attractions

• What about four points?
• What does it take to guarantee a circle that
  contains all four points?

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Nine-Point Circle (First Look)

• Recall the orthocenter, where altitudes meet
• Note feet of the altitudes
• Vertices for the pedaltriangle
• Circumcircle of pedal triangle
• Passes through feet of altitudes
• Passes through midpoints of sides of ABC
• Also some other interesting points try it

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Nine-Point Circle (First Look)

• Identify the different lines and points
• Check lengths of diameters

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Cevas Theorem

• A Cevian is a line segment fromthe vertex of a
  triangle to a pointon the opposite side
• Name examples of Cevians
• Cevas theorem for triangle ABC
• Given Cevians AX, BY, and CZ concurrent
• Then

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Cevas Theorem

• Proof
• Name similartriangles
• Specify resultingratios
• Now manipulate algebraically to arrive at product
  equal to 1

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Converse of Cevas Theorem

• State the converse of the theorem
• If
• Then the Cevians are concurrent
• Proving uses the contrapositive of the converse
• If the Cevians are not concurrent
• Then

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Menelaus Theorem

• Recall Activity 10

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Menelaus Theorem

• Consider that the ration AZ/ZB is negative
• are in opposite
  directions
• Theorem 2.8In triangle ABC with X online BC, Z
  on line AB, X, Y, Z collinear
• Then

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Mathematical Arguments and Triangle Geometry

• Chapter 2