Mathematical Arguments and Triangle Geometry

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