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Mathematical Arguments and Triangle Geometry

This chapter explores deductive reasoning and its application in triangle geometry. Learn about conditional statements, rules of logic, quantifiers, and how to develop robust mathematical arguments.

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Mathematical Arguments and Triangle Geometry

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  1. Mathematical Arguments and Triangle Geometry Chapter 2

  2. 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

  3. 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

  4. 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.

  5. 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

  6. Rules of Logic • Use logical operators • and, or • Evaluate truth of logical combinations • P and Q

  7. Rules of Logic • Combining with or • P or Q

  8. Rules of Logic • Negating a statement • not P

  9. Conditional Statements • Implication P implies Q if P then Q Possible to have either a true or a false conclusion If the hypothesis is false, an implication tells us nothing.

  10. Conditional Statements • Viviani’s 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.

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

  12. 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

  13. 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

  14. Conditional Statements • Given P  Q • The converse statement is Q  P • Hypothesis and conclusion interchanged • Consider truth tables • Reversed

  15. 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

  16. Conditional Statements • Ceva’s theorem • If lines CZ, BY, and XA are concurrentThen • State the converse, the contrapositive

  17. Conditional Statements • Ceva’s theorem – a biconditional statement • Both statement and converse are true • Note: two separate proofs are required • Lines CZ, BY, and XA are concurrentIFF

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. Negating a Quantified Statement • Useful in proofs • Prove the contrapositive • Prove a statement false • Negation patterns for quantified statements

  26. 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 l, there is exactly one line m through P that is parallel to l (see page 41)

  27. 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

  28. 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

  29. Angle-Side-Angle Congruence • Proof • Use negation • Justify the steps in the proof on next slide

  30. ASA • Assume ABDE

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

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

  33. 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

  34. 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?

  35. 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

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

  37. 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?

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

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

  40. 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!

  41. Euler Line • What conclusion did you draw from Activity 9?

  42. Euler Line Proof • Find line through two of the points • Show third point also on the line

  43. 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

  44. 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

  45. 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

  46. 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

  47. Preview of Coming Attractions • What about four points? • What does it take to guarantee a circle that contains all four points?

  48. 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

  49. Nine-Point Circle (First Look) • Identify the different lines and points • Check lengths of diameters

  50. 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

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