Mathematical Arguments and Triangle Geometry Chapter 3

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

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

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

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.

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

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

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

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

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)

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

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

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 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

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

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

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?

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

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 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.

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 to an equilateral triangle Create a counterexample to this converse

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

Orthocenter Recall Activity 4 Theorem 3.8 The altitudes of a triangle are 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 of similar triangles List congruent segments on side CB Why is G two-thirds of the way along median BE?

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

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!

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

Ceva’s Theorem Proof Name similar triangles Specify resulting ratios 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

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