1. Mathematical Arguments and Triangle Geometry
Chapter 2

2. Deductive Reasoning A process Statements assumed true Conclusion
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 Closed statement
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. The statement in this box is false
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 in this box is false

6. Rules of Logic P and Q 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
If the hypothesis is false, an implication tells us nothing.
Possible to have either a true or a false conclusion

10. Conditional Statements
Viviani’s Theorem IF a point P is interior to an 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 trapezoid THEN 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 trapezoid THEN 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 P Q
P  Q
Q  P T F

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 P Q
P  Q T F Q P
Q  P F T

16. Conditional Statements
Ceva’s theorem
If lines CZ, BY, and XA are concurrent Then
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 concurrent IFF

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  S then P  S
Modus tollens: P  Q and  Q then  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 variable x + 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 definition of 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 triangle are 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 AD and BE intersect at G
Now consider midpoint of AB, point F

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

35. Centroid Now draw median CF, intersecting BE at G’
Draw parallels as before
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 2.6 The angle bisectors of a triangle are concurrent

37. Incenter
Proof
Consider angle bisectors for angles A and B with intersection point I
Construct perpendiculars to W, X, Y
What congruent triangles 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 angle bisectors

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 2.7 The 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 through circumcenter, O and centroid, G
Consider X on OG 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 altitude from A
Repeat argument for altitudes from C 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 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

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 from the vertex of a triangle to a point on the opposite side
Name examples of Cevians
Ceva’s theorem for triangle ABC
Given Cevians AX, BY, and CZ concurrent
Then

51. Ceva’s Theorem Proof Name similar triangles Specify resulting ratios
Now manipulate algebraically to arrive at product equal to 1

52. 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

53. Menelaus’ Theorem
Recall Activity 10

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

55. Mathematical Arguments and Triangle Geometry
Chapter 2