9_30

The class began with Brett proving Proposition 2.20. Although he had begun to prove it last time, it was suggested that the class try to redo the problem using different variables that the common ones given in the proofs. There was some confusion as A,B,C are used so often. With the change of variables, the proposition states: //Let the angle XYZ be an angle and T a point lying on the line XZ. Then T is in the interior of the angle XYZ IFF X*T*Z.// Since this is an IFF proposition, it is necessary to prove the proposition in 2 ways: 1)  // If T is in the interior of the angle XYZ, then X*T*Z. // 2)  // If X*T*X then T is in the interior of the angle XYZ. // Brett began proving the first part using definition 2.19 which states that if a point T is on the interior of an angle XYZ then: · Points T,Z are on the same side of the line YX  ·  Points T,X are on the same side of the line YZ Then using Lemma 2.7 he stated that since points T,Z are on the same side of the line YX and the line TZ intersects then line YX at point X, then T*Z*X or X*T*Z. This lemma was also used to show that since points T,X are on the same side of the line YZ and the line TX intersects the line YZ at point Z, then T*X*Z of Z*X*T. Thus he showed that: (T*Z*X OR X*T*Z) AND (T*X*Z OR Z*X*T) This can lead to several different outcomes. However, Brett simplified it using B-3 which states that given 3 distinct points X,Y,Z on the same line then only one can be between the other. Thus we can eliminate possibilities like T*Z*X and T*X*Z as this contradictory to B-3. It was concluded then that X*T*Z and Z*T*X is the only combination which satisfies B-3, and by B-1 X*T*Z = Z*T*X. Thus Brett had proved the first part of the proposition. To prove that if X*T*X then T is in the interior of the angle XYZ, Brett began by using Lemma 2.6 which states that if X*T*Z and the line YZ intersects the line XZ distinctly at some point Z, then the points X,T, are on the same side of the line YZ. This lemma was used again to show that if X*T*Z and the line YX intersects the line ZX distinctly at some point X, then the points Z,T, are on the same side of the line YX. Def 2.19 states that to show a point T is in the interior of an angle XYZ, then · Points X,T must be on the same side of the line YZ  ·  Points Z,T must be on the same side of the line YX Brett claimed that this is precisely what Lemma 2.6 stated that therefore we had completed the proof of proposition 2.20. When Brett began the proof he originally started by drawing a picture. This once again led to the question as to whether a picture should be drawn, as this sometimes lead to confusion or proofs based off pictures. Emina suggested trying to draw pictures in the Hyperbolic Upper Half Plane Model. Most of the class was unfamiliar with this model. This plane consists of semi-circles perpendicular to the x-axis and vertical lines in the first quadrant ** - why just first quadrant? . ** All the basic elements we were using (points, segments, rays, lines, etc) were also found in this model, however they look a little different. For example the angle ABC exists, but looks a little differently than the class was accustomed to. Emina challenged us to see if the propositions and lemmas we had proven held in this plane. She also introduced a new proposition and asked us to see if it was true in the Hyperbolic plane. //Given a point D in the interior of angle BAC, do there exist 2 points B’ on the ray AB and C’ on the ray AC such that B’*D*C’?// In a Euclidean Plane, this is always true, but In a Hyperbolic plane, it is not always true. This led us to look at parallel lines. In the model we had been using, two lines were parallel if they did not intersect. This definition holds in other models, but we found that other propositions did not remain true in different models. We looked at Euclid’s Parallel postulate and how it changed in different models. Euclid’s Parallel Postulate: Given a line l and a point p no on l, there is a UNIQUE line through p parallel to l. Spherical Parallel Postulate: Given a line l and a point p not on l, there is NO line through p that is parallel to l. Hyperbolic Parallel Postulate: Given a line l and a point p not on l, there are at least 2 lines through P parallel to l. We were given as assignment to go play around with hyperbolic planes to see how basic shapes we knew were transformed. The web site [|http://www.cs.unm.edu/~joel/NonEuclid/noneuclidean.html] allows us to do this. After we had made our discoveries we were assigned to write them in a short paper. Class continued with Lisa proving the first part of Proposition 2.21 which states: //If D is in the interior of an angle CAB then so is ever point on the ray AD except the point A.// Lisa began by letting x be a point on the ray AB which is not equal to the point A. If x is in the interior of the angle CAB, then by Definition 2.19 x and C must be on the same side of the line AB and that x and B are on the same side of the line AC. This is what Lisa wanted to prove. She used Proposition 2.15 to show that since A is a point on the line AC and points x and D are not on the line AC, then x and D are on the same side of the line AC. She used the proposition again to show that x and D are on the same side of the line AB. She then pointed out that since D is in the interior of the angle CAB we know the points D and C are on the same side of the line AB and the points D and B are on the same side of the line AC by Definition 2.19. Lisa then showed by B-4 that since D and C are on the same side of AB, and D and x are on the same side of AB, then x and C are on the same side of AB. Also, since D and B are on the same side of AC, and x and D are on the same side of AC, then x and B are on the same side of AC. Since this proposition is true for any x on the ray AD, Lisa had proved what she needed to. However, there were some students who were not sure if this was the simplest way to prove the proposition. This led us to a discussion that mathematics is not as clear-cut and straightforward as we often think it is. In proving propositions, as well as in all areas, we can be creative and solve problems in different ways. The idea that there is only one way to solve a problem is bogus, and we should be open to new ways of thinking. The class concluded with Kira proving the second part of proposition 2.21. Although she only got partway through the proof before class ended, the steps she took are recorded. The proposition states: // If T is in the interior of an angle XYZ then no point on the opposite ray YT is in the interior of the angle ZYX. // Kira began by using Definition 2.16 to show that since the ray YT had an opposite ray, which she renamed YW, then W*Y*T. She then used proposition 2.15 to show that every point on the ray YT except is on the same side of the line YX. She used this proposition again to show that every point on the ray YT except point Y is on the same side of the line YZ. She then noted that the line YT intersects line YZ at Y and the line YT intersects the line YX at point Y. She then used proposition 2.8 to show that T and W are on opposite sides of the line YZ and the line YX. This was when class ended, and the rest of the proof will be recorded in the next entry.