Geometry Proofs Math 416 Time Frame n n

Geometry Proofs Math 416

Time Frame n n Definition Congruent Triangles Axiom & Proofs Propositions

Definitions n n 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) n n 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 Definition Axiom Thoerem

Definitions n n Definition: You define something once you identify its essential characteristics For example, triangle – a two dimensional polygon with three sides Not Must

Axiom n n Axioms: An obvious statement that is acceptable without proof For example, the shortest distance between two points is a straight line

Propositions n n n Propositions are statements that require proof Once proven they are called theorems For example Proof 3 1 2 STATEMENT AUTHORITIES <1 + <3 = 180° DEFINITION <2 + < 3 = 180° DEFINITION <1 = <2 = 180 ALGEBRA

Theorums n n 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 n n n 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 n n n n 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 n n n 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 n n 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 n n 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 n n n Once again we will not prove But you may be required to You should be able to

Theorems n The 90° completion theorem or the complementary angle theorem x n 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 n 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: b 2 = a 2 + c 2

Pythagoras Examples n Solve for x x 2 n = 62 + 82 x x 2 = 36 + 64 6 x = 10 8 Solve for x 202 = x 2 + x 2 400 = 2 x 2 200 = x 2 14. 14 = x 2 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 = 2 c

The 30 -60 -90 Theorem Converse A b B 2 b 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 (2 x)2=x 2+196 6 4 x 2=x 2+196 Opposite the 30° It is half the x hypotenuse 14 x = 12 30° 3 x 2=196 x 2= 65. 33 x = 8. 08 x

Exam Question D A Hyp: Diagram Conc: < ABC = < ADC B C Construction AC

Exam Questions Con’t n 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 AB 2 = BC • BD Cross Multipln CONC: AB 2 = BC • BD Do #5 & 6

Tips for Success n n n 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 n n We will do more examples on the board together… P 262, p 266, 267, 268, p 272, 274 Study Guide Test