mathematical arguments and triangle geometry chapter 3
TRANSCRIPT
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Mathematical Arguments and Triangle Geometry
Chapter 3
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Coming Attractions
• Given P Q Converse is Q P Contrapositive is Q P
• Proof strategies Direct Counterexample
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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 The proposition
• Open statement contains a variable – truth value determined
once variable specified The predicate
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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
The statement inthis box is false
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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 variable
x + 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
, ( )x S P x
, ( )x R P x
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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
P Q Q P P Q P Q
, ( ) , ( )x P x x P x
, ( ) , ( )x P x x P x
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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 Playfair’s Postulate:
Given any line , there is exactly one line m through P that is parallel to (see page 41)
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Proof and Disproof
• Start by being clear about assumptions Euclid’s postulates are implicit
• Clearly state conjecture/theorem What are givens, the hypothesis What is conclusion
P Q
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Proof and Disproof
• Direct proof Work logically forward Step by step Reach logical (and desired) conclusion
• Use Syllogism If P Q and Q R and R S are
statements in a proof Then we can conclude P S
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Proof and Disproof
• Counterexample in a proof All hypotheses hold But discover an example where conclusion
does not
• This demonstrates the conjecture to be false
• Counterexample suggests Alter the hypotheses … or … Change the conclusion
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Step-By-Step Proofs
• Each line of proof Presents new idea, concept Together with previous steps produces new
result
• Text suggests Write each line of proof as complete sentence Clearly justify the step
• Geogebra diagrams are visual demonstrations
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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 (don’t 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
x DE AB DX
ABC DXF
C XFD
But given C EFD
AB DX DE
ABC DEF
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Incenter
• Consider the angle bisectors
• Recall Activity 6
• Theorem 3.4The 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• Construct
perpendicularsto 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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Viviani’s Theorem
• IF 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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Viviani’s Theorem
• 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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Converse of Viviani’s Theorem
• IF the sum of the lengths of the perpendiculars from P to the sides of the triangle is equal to the altitude THEN a point P is interior toan equilateral triangle
• Create a counterexample to this converse
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Contrapositive
• Recall Given P Q Contrapositive is Q P
• These two statements are equivalent They mean the same thing They have the same truth tables
• Contrapositive a valuable tool Use for creating indirect proofs
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Orthocenter
• Recall Activity 4
• Theorem 3.8 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 Activites
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Centroid
• Theorem 3.9 The three medians of a triangle are concurrent
• Proof Given ABC, medians AD
and BE intersect at G Now consider midpoint
of 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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Circumcenter
• Recall Activities
• Theorem 3.10The 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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Ceva’s Theorem
• A Cevian is a line segment fromthe vertex of a triangle to a pointon the opposite side Name examples of Cevians
• Ceva’s theorem for triangle ABC Given Cevians AX, BY, and CZ concurrent Then
1AZ BX CY
ZB XC YA
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Ceva’s Theorem
Proof
• Name similartriangles
• Specify resultingratios
• Now manipulate algebraically to arrive at product equal to 1
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Converse of Ceva’s 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
1AZ BX CY
ZB XC YA
1AZ BX CY
ZB XC YA
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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 pedal
triangle
• 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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Mathematical Arguments and Triangle Geometry
Chapter 3