Unformatted text preview:

Parallels,Similarity,ProportionJohn T.BaldwinParallels, Similarity, ProportionJohn T. BaldwinOctober 29, 2007Parallels,Similarity,ProportionJohn T.BaldwinContextWe are working in the situation with Hilbert’s axiom groups I,II, III (incidence, order, and congruence).We have proved SAS, ASA, and ASA.Parallels,Similarity,ProportionJohn T.BaldwinBasic properties of lines and transversalEuclid I.27,I.28: IIf a transversal crosses a pair lines and:1 corresponding angles are equal, or2 alternate interior angles are equal, or3 the sum of two interior angles on the same side is equal totwo right anglesthen the lines are parallel.This does not require the parallel postulate; the actualargument is I.16; an exterior angle of a triangle is greater thaneither of the other two angles.Parallels,Similarity,ProportionJohn T.BaldwinExerciseShow using basic properties (including all straight angles areequal) that the three conditions of Euclid I.27,28 are equivalent.Parallels,Similarity,ProportionJohn T.BaldwinBasic properties of lines and transversal, IIEuclid I.27,I.28If a transversal crosses a pair lines and the lines are parallelthen:1 corresponding angles are equal, and2 alternate interior angles are equal, and3 the sum of two interior angles on the same side equal totwo right anglesThis does require the parallel postulate.Note that Euclid’s version of the 5th postulate just says 3)implies not parallel.Parallels,Similarity,ProportionJohn T.BaldwinParallelogramsEuclid I.34In any parallelogram the opposite sides and angles are equal.Moreover the diagonal splits the parallelogram into twocongruent triangles.Immediate from our results on parallelogram and thecongruence theorems.Parallels,Similarity,ProportionJohn T.BaldwinArea of Parallelograms and trianglesEuclid I.35, I.38Parallelograms on the same base and in the same parallels havethe same area.Triangles on the same base and in the same parallels have thesame area.Parallels,Similarity,ProportionJohn T.BaldwinCommensurabilityWe say two segments AB and CD are Commensurable if thereis a third segment EF and two natural numbers n and m suchthat AB can be covered by n disjoint copies of EF and CD canbe covered by m disjoint copies of EF .We writeABCD=mn.Parallels,Similarity,ProportionJohn T.BaldwinCommensurable TrianglesEuclid VI.1Triangles under the same height are to each other as theirbases.The proof requires that the areas of the triangles becommensurable.Parallels,Similarity,ProportionJohn T.BaldwinProportionality: base caseEuclid VI.2If a line is drawn parallel to the base of triangle thecorresponding sides of the two resulting triangles areproportional and conversely.This uses the previous result and so needs that the two parts ofeach side are commensurable. Euclid does not recognize thisexplicitly.Parallels,Similarity,ProportionJohn T.BaldwinAssumptionsto find any triangle with area 12 and fixed base of length 61 can call any side the base2 triangles can have one obtuse angle3 length is always positive (to allow placement on x = −44 Euclidean geometry5 properties of the real numbers6 area defined as usual7 ”important to consider acute, right, isosceles, equilateraland obtuse triangles.”8 A − bh/2Parallels,Similarity,ProportionJohn T.BaldwinNot mentioned:1 the parallel postulate2


View Full Document

UIC MATH 592 - Parallels, Similarity, Proportion

Download Parallels, Similarity, Proportion
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Parallels, Similarity, Proportion and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Parallels, Similarity, Proportion 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?