Mathematical Arguments and Triangle Geometry

Chapter 3

Coming Attractions

• Given P Q Converse is Q P Contrapositive is Q P

• Proof strategies Direct Counterexample

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

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

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.

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

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

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

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

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

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

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)

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

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

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

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

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

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

Angle-Side-Angle Congruence

• Proof

• Use negation

• Justify the steps in the proof on next slide

ASA

• Assume AB DE

x DE AB DX

ABC DXF

C XFD

But given C EFD

AB DX DE

ABC DEF

Incenter

• Consider the angle bisectors

• Recall Activity 6

• Theorem 3.4The angle bisectors of a triangle are concurrent

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?

Incenter

• Now draw CI

• Why must it bisect angle C?

• Thus point I is concurrent to all three anglebisectors

Incenter

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

perpendicular is incircle

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.

Viviani’s Theorem

• What would make the hypothesis false?

• With false hypothesis, it still might be possible for the lengths to equal the altitude

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

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

Orthocenter

• Recall Activity 4

• Theorem 3.8 The altitudes of a triangleare concurrent

Centroid

• A median : the line segment from the vertex to the midpoint of the opposite side

• Recall Activites

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

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?

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

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!

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

Ceva’s Theorem

Proof

• Name similartriangles

• Specify resultingratios

• Now manipulate algebraically to arrive at product equal to 1

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

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

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

Preview of Coming Attractions

• What about four points? What does it take to guarantee a circle that

contains all four points?

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

Nine-Point Circle (First Look)

• Identify the different lines and points

• Check lengths of diameters

Mathematical Arguments and Triangle Geometry

Chapter 3