Mathematical Arguments and Triangle Geometry Chapter 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

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

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

Rules of Logic P and Q Use logical operators and, or Evaluate truth of logical combinations P and Q

Rules of Logic Combining with or P or Q

Rules of Logic Negating a statement not P

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

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.

Conditional Statements What would make the hypothesis false? With false hypothesis, it still might be possible for the lengths to equal the altitude

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

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

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

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

Conditional Statements Ceva’s theorem If lines CZ, BY, and XA are concurrent Then State the converse, the contrapositive

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

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

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

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

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)

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

Orthocenter Recall Activity 1 Theorem 2.4 The altitudes of a triangle are concurrent

Centroid A median : the line segment from the vertex to the midpoint of the opposite side Recall Activity 2

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

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

Incenter Consider the angle bisectors Recall Activity 3 Theorem 2.6 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

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!

Euler Line What conclusion did you draw from Activity 9?

Euler Line Proof Find line through two of the points Show third point also on the line

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

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

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

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

Menelaus’ Theorem Recall Activity 10

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

Mathematical Arguments and Triangle Geometry Chapter 2