1. Mathematical Arguments and Triangle Geometry
Chapter 3

2. Coming Attractions Given P  Q Proof strategies Converse is Q  P
Contrapositive is  Q   P
Proof strategies
Direct
Counterexample

3. 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

4. Deductive Reasoning Statements of an argument Closed statement
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

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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

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

12. 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)

13. 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

14. Proof and Disproof Direct proof Use Syllogism 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

15. 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

16. Step-By-Step Proofs Each line of proof Text suggests
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

17. 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

18. 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

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

20. ASA
Assume AB DE

21. Incenter Consider the angle bisectors Recall Activity 6
Theorem 3.4 The angle bisectors of a triangle are concurrent

22. 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?

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

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

25. 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.

26. Viviani’s Theorem What would make the hypothesis false?
With false hypothesis, it still might be possible for the lengths to equal the altitude

27. 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 to an equilateral triangle
Create a counterexample to this converse

28. Contrapositive Recall Given P  Q These two statements are equivalent
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

29. Orthocenter Recall Activity 4
Theorem 3.8 The altitudes of a triangle are concurrent

30. Centroid
A median : the line segment from the vertex to the midpoint of the opposite side
Recall Activites

31. 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

32. 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?

33. 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

34. Circumcenter Recall Activities
Theorem 3.10 The three perpendicular bisectors of the sides of a triangle are concurrent.
Point of concurrency called circumcenter
Proof left as an exercise!

35. 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

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

37. 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

38. 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

39. 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

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

41. 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

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

43. Mathematical Arguments and Triangle Geometry
Chapter 3