2 Homework. Dr. Franz Rothe February 21, 2015 All3181\3181_spr15h2.tex

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1 Math 3181 Dr. Franz Rothe February 21, 2015 All3181\3181_spr15h2.tex Name: Homework has to be turned in this handout. For extra space, use the back pages, or blank pages between. The homework can be done in groups up to four students due March /11 2 Homework 1 Problem 2.1. Among the incidence planes drawn in the figure on page 1, there are examples for which neither the elliptic, nor the Euclidean parallel property hold, but Hilbert s Parallel Postulate about the uniqueness of parallels is still valid. State Hilbert s Parallel Postulate for plane geometry. Which are these examples? Figure 1: Seven examples for incidence planes. 1

2 Corollary 1 (About uniqueness of parallels). For any incidence plane the following statements are equivalent: (a) uniqueness of parallels, stated in Hilbert s Parallel Postulate for plane geometry, holds; (b) any third line intersecting one of any two parallel lines intersects the other one, too; (c) any third line parallel to one of any two parallel lines, is parallel to the other one, too; (d) being equal or parallel defines an equivalence relation among the lines. Figure 2: Enumerate the pencils for the two models (A) and (B). Problem 2.2. Given an incidence plane for which Hilbert s Parallel Postulate is valid. As mentioned in the lecture, and stated in Corollary 1: being equal or parallel defines an equivalence relation among the lines. Each one of the equivalence classes of equal or parallel lines is still called a pencil. But, contrary to the situation for an affine plane, not all pencils need to consist of the same number of lines. Determine and enumerate the pencils for the two models (A) and (B) drawn in the figure on page 2. Put curely brackets around each pencil! 2

3 Figure 3: Configuration of scissors. Definition 1 (Configuration of scissors). A scissor is called the system of two distinct intersecting or parallel lines, with two points on each one of them, and the quadrilateral zig-zaging between these doublets. A configuration of scissors consists of two such scissors lying between the same pair of lines, together with consistent bijections between the points, and the sides of the two scissors. An example is shown in the figure on page 3. Figure 4: The Theorem of scissors. 3

4 Theorem 1 (Theorem of scissors). If the two scissors from a configuration of scissors have three pairs of corresponding parallel sides, their fourth pair of corresponding sides are parallels, too. An illustration is given in the figure on page 3. 4

5 Theorem 2 (Little Theorem of scissors). If a configuration of scissors lies between two parallel lines, and the two scissors have three pairs of corresponding parallel sides, their fourth pair of corresponding sides are parallels, too. Problem 2.3. Draw an illustration for the Little Theorem of scissors. 5

6 Figure 5: Prove the Theorem of scissors. Theorem 3. For any affine plane, (a) validity of the Desargues Theorem implies validity of the Theorem of scissors. (b) validity of the Little Desargues Theorem implies validity of the Little Theorem of scissors. Problem 2.4. Give a proof of part (a). You may assume the Desargues Theorem together with its converse. Into the figure on page 6, put the extra lines for the proof in green, and mark and name the extra points needed. Write a paragraph for the proof. 6

7 Proposition 1 (Existence and uniqueness of segment transfer). Given a segment AB and given a ray r originating at point A, there exists a unique point B on the ray r such that AB = A B. Problem 2.5. Provide the answers and complete the reasoning. Question. Which part of this statement is among Hilbert s axioms? Which axiom is used? Question. How does the uniqueness of segment transfer follow? needed for that part? Which axioms are Figure 6: Uniqueness of segment transfer Proof of uniqueness. Assume the segment AB can be transferred to the ray r from A in two ways, such that both AB = A B and AB = A B. We choose a point C not on?. One obtains the congruences A B = A B, A C = A C, B A C = B A C We are using axiom (III.2), the fact that a segment is congruent to itself, and an angle is congruent to itself by axiom (III.4c). By the axiom? for SAS, this implies? By the? of angle transfer, as stated in axiom (III.4), this implies that the rays C B = C B are equal. Hence B = B is the unique intersection point of the two different rays r = A B and?. We have shown that AB = A B and AB = A B imply B = B. Thus segment transfer is unique. 7

8 Problem 2.6. Give a detailed definition what is meant by congruence of two generic triangles. Moreover, explain with exact notation what the congruence ABC = A B C means, illustrated with a colored figure. 8

9 Proposition 2 (Isosceles Triangle Proposition). [Euclid I.5, and Hilbert s theorem 11] In a triangle with two congruent sides, the opposite angles are congruent. An isosceles triangle has congruent base angles. Problem 2.7. Provide the answers and complete the reasoning. Question. Formulate the theorem with specific quantities from a triangle ABC like drawn in the figure on page 9. Figure 7: An isosceles triangle Proof. Assume that the sides AB = AC of triangle ABC are congruent. We need to show that the base angles β = ABC and γ = BCA are congruent. Define a second triangle A B C by setting 1 A := A, B := C, C := B To apply? congruence, we match corresponding pieces: (1) BAC = CAB since by axiom (III.4d), an angle is congruent to the angle with the sides reversed. As the points are defined, we know? Hence BAC = B A C. (2) AB = A B. Question. Explain why this holds. (3) Similarly, we show AC = A C : Indeed, AC = AB because we have assumed the triangle to be isosceles, and congruence is a symmetric relation, and AB = A C by construction. Hence AC = A C. Finally, we use axiom (III.5). Items (1)(2)(3) imply ABC = A B C = ACB. But this is just the claimed? 1 It does not matter that the second triangle is just on top of the first one. 9

10 Figure 8: How to get ASA congruence Proposition 3 (ASA Congruence). [Theorem 13 in Hilbert] Two triangles with a pair of congruent sides, and pairwise congruent adjacent angles are congruent. Problem 2.8. State the ASA congruence, with the notation from the figure above. Prove the theorem.

11 Figure 9: Transfer of a triangle. Proposition 4 (Transfer of a triangle). Any given triangle ABC can be transferred into any given half-plane H of any given ray r, such that one obtains a congruent triangle A B C lying in the prescribed half-plane, and the given ray is r = A B, emanates from vertex A and lies on the side A B. Problem 2.9. Use the notation from the figure above. Prove the Proposition 4, starting from the axioms and ASA-congruence. State clearly which axioms you use, and where one has to use the ASA-congruence. 11

12 Problem 2.. Give purely geometric definitions for supplementary angles and for vertical angles. (Do not use any measurements!) Problem Give purely geometric definitions for the notions : right angle, acute angle, and obtuse angle. (Use only comparison of angles, not any measurements!) 12