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Geometry Proofs Math 416 Time Frame Definition Congruent Triangles Axiom & Proofs Propositions Definitions Geometric Proofs The essence of pure mathematics The creative and artistic center of math The ability to explain in a detailed concise logical manner how a proposition (problem) is either true or false. Definitions (con’t) Detailed – hard facts Concise – short to the point Logical – set of rules based on reason A proof generally falls back to things that are either known, accepted or already proven. This is how we attain knowledge Enlightenment Gaining Knowledge Proposition Proposition Proposition Proposition Definition Axiom Thoerem Definitions Definition: You define something once you identify its essential characteristics For example, triangle – a two dimensional polygon with three sides Not Must Axiom Axioms: An obvious statement that is acceptable without proof For example, the shortest distance between two points is a straight line Propositions Propositions are statements that require proof Once proven they are called theorems For example Proof 3 1 2 STATEMENT AUTHORITIES <1 + <3 = 180° <2 + < 3 = 180° <1 = <2 = 180 DEFINITION DEFINITION ALGEBRA Theorums This proposition now becomes a theorem Hence, vertically opposite angle theorem Theorems can be used in a proof as an authority Definitions must use terms that are already defined Be reversible once you have the characteristics you have the object not give unnecessary information Examples #1 of Definitions Definition: A belingas is a shape with a dot on a vertex are belingas Which of the following is a belingas? Example #2 of a Definition Stencil #1 Which of the following is a Gatu? Are Gatus Definition: A Gatu is a shape with at least one curved side Axioms A statement not requiring proof A whole is equal to the sum of its part Completion C A B D < ABD = <ABD + <CBD • Any quantity can be replaced by another equal quantity Axioms Easiest thing to do is to assign numbers to letters… a=0;b=4;c=4;q=4 Replacement… If a + b = c AND b = q Then a + q = c The shortest distance between two points is a straight line Only one line can pass through the same two points Given a point and a direction, only one line with that direction can pass through the point Postulates Theorems we will not prove are called postulates specifically the congruence postulates Hypothesis: Given two triangles with corresponding sides equal we say CONC: Two triangles are congruent X A ABC B CY Z YZX By S S S Postulates Hypothesis: Given two triangles with two corresponding sides equal and the contained angle equal Conclusion: The two triangles are congruent X A ABC ° By SAS Y B ° C Z ZXY Postulates Hypothesis: Given two triangles with two corresponding angles equal and the contained side equal Conclusion: The two triangles are congruent A X O B O X CY X ABC Z ZXY By ASA Do #2 Theorems Once again we will not prove But you may be required to You should be able to Theorems The 90° completion theorem or the complementary angle theorem x HYP: Diagram y CONC < X + <Y = 90° The 180° Completion Theorem HYP Diagram x y CONC <x + <y = 180 Vertically Opposite Angle Theorem 1 4 3 2 Conclusion < 1 = < 2 < 3 = <4 Triangle Sum Theorem 1 2 3 Conclusion <1 + <2 + <3 = 180° Isosceles Triangle Theorem 1 Conclusion <1 = <2 2 Given an isosceles triangle, the angles opposite the equal sides are equal Isosceles Triangle Theorem Converse A B Conclusion AB = AC Given an isosceles triangle, the sides opposite the equal angles are equal C Note: The converse is true also to prove // lines Parallel Line Theorem 1 Sometimes called Corresponding angles 3 a c Conclusion <4 = < a <3<b 2 4 b d <1 < a <2 = <b <3 = < c <4 = <d <3 + <a = 180° <4 + <b = 180° Parallelogram Theorem and Converse D A x B C In a parallelogram opposite sides are equal, opposite angles are equal and the diagonals bisect each other Conclusion: AD = BC Opposite AB = DC Sides < BAD = <DCB Opposite Angles < ABC = < ADC BX = XD Diagonals AX = XC Bisected Triangle Parallel Similarity Theorem A B C Conc ABC ˜ D E ADE Do #3 Test Question If ABC ˜ XYZ and then < XYZ is 50°, how much is angle ABC? 50° Vertically opposite angles is an example of a a) Theorum b) axiom c) definition d) postulate Pythagoras Theorem A b c Given a right angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides HYP: Diagram B a C CONC: b2 = a2 + c2 Pythagoras Examples Solve for x x2 = 62 + 82 x x2 = 36 + 64 6 x = 10 8 Solve for x 202 = x2 + x2 400 = 2x2 200 = x2 14.14 = x2 x = 14.14 20 x x The 30-60-90 Theorem A 60° c B The side opposite the 30° angle is half the hypotenuse. b HYP: Diagram 30° C CONC: c = ½ b OR b = 2c The 30-60-90 Theorem Converse A b B 2b If the hypotenuse is twice the length of one of the legs, the angle opposite the leg is 30° C HYP: Diagram CONC: <ACB = 30° 30-60-90 Examples (2x)2=x2+196 6 4x2=x2+196 Opposite the 30° 30° It is half the x hypotenuse 14 x = 12 30° 3x2=196 x2= 65.33 x = 8.08 x Exam Question D A Hyp: Diagram Conc: < ABC = < ADC B C Construction AC Exam Questions Con’t Fill in the missing authorities Statement Authorities < DAC = <ACB // Line Theorum < DCA = <BAC // Line Theorum AC = AC Reflex Thus DAC <ABC = < ADC BCA ASA Definition Prove the following A B C HYP: diagram Statement < BAD=<ACD < ABC = <ABD ABD˜ CBA Authorities HYP Reflex AA AB = BD = AD DEFN D CB BA CA AB2 = BC • BD Cross Multipln CONC: AB2 = BC • BD Do #5 & 6 Tips for Success Always work on what you know The more facts you put into a question the closer you will get to the answer Extend the lines Exam Questions & Practice We will do more examples on the board together… P262, p266, 267, 268, p272, 274 Study Guide Test