Mathematics 133
Notes from October 26, 2020, lecture session

The recording for this lecture is available online as  
http://math.ucr.edu/~res/math133-2020/Oct26.mp4  .

 Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable.
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/geometrynotes02b.f13.pdf   The second  part of the second chapter of the course notes.   The lecture covered material in three sections of the notes.  The material from Section II.4 begins with  the Isosceles Triangle Theorem  (Theorem II.4.1), and you should understand the linear algebraic proofs through Theorem II.4.11 at least passively (actively indicates an even better command of the material, however).  In Section II.5  you should actively understand the material up to the subheading Synthetic characterization of affine transformations.  In Section III.1  you should actively understand the material through the proof of Proposition  III.1.3.    
http://math.ucr.edu/~res/math133-2020/week4/sections_3.1-3.2/geometrynotes03a.f13.pdf   The first  part of the third chapter of the course notes.   See the previous paragraph for the material in Section III.1  that was covered in the lecture.  MORE INFORMATION ABOUT THIS MATERIAL WILL BE INCLUDED IN THE SUMMARY FOR THE NEXT LECTURE.
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/pons_asinorum.pdf     A color drawing summarizing Euclid's proof for half of the Isosceles Triangle Theorem with a summary of the proof.   Pappus' subsequent proof from the fourth century A.D. is shorter but it uses a powerful idea:  A triangle can be congruent to itself in more than one way.
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/superposition.pdf     A verbal description for the intuitive geometric notion of superposition (moving one figure, without changing its size or shape, so that it lies directly on top of another). The logical issues connected to the notion are also discussed.  This should be read to indicate the need for the material on geometrical transformations in the notes.
https://www.youtube.com/watch?v=86iU3fypTd4 https://www.youtube.com/watch?v=kO72ul9b8MA https://www.youtube.com/watch?v=-xhU-aRifnQ   Videos from the lectures which illustrate the notion of superposition.
https://math.ucr.edu/~res/math133-2020/week4/sections_2.4-3.5/Whiteboard.oct26-1.png   Important properties of geometric isometries in Euclidean geometry.  
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/metgeom.pdf     Detailed  formulations and proofs for statements about isometries that appear in the notes.  You should become  acquanted with the main results, but it is not necessary to know the proofs.
https://www.khanacademy.org/math/geometry/hs-geo-transformations   An animated discussion of geometric transformations in a more detailed manner than the notes; in particular, particular examples are discussed.  The graphics are significantly more refined than in earlier Khan academy videos, and if you have trouble visualizing the material on transformations you definitely should view the videos in this series.
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/reflections.pdf   This file gives a linear-algebraic formula for the geometric notion of mirror image (reflection) with respect to a line in the plane; a proof for the formula is also given.   There is a similar formula for reflections with respect to planes in 3-space, but it will not be needed in this course.
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/congruence-independence.pdf   This is an addendum to the Appendix to Section II.4, which explains why one cannot prove the congruence axioms as logical consequences of the preceding ones.  Roughly speaking, the point of the Appendix is that without the congruence axioms we have no way of knowing whether linear and angle measurement are interconnected.   This is done by constructing a model for the preceding axioms in which the SAS congruence property is false.   As indicated in the preceding lecture, if the latter is true, then ASA and SSS are also true, and in fact it suffices to assume any one of the three properties.
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/playfair-proclus.pdf   A statement and drawing of the Euclidean Parallel Postulate.
https://math.ucr.edu/~res/math133-2020/week4/sections_2.4-3.5/Whiteboard.oct26-2.png   Drawing to stress that parallel lines are coplanar; nonintersecting lines in 3-space which are not coplanar are called skew lines.  As the picture suggests, certain pairs of edges for a rectangular box are skew lines.
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/skew-lines.pdf   A proof that the shortest distance between two skew lines is realized along a common perpendicular.  The argument in this document uses multivariable calculus, but it is also possible to prove this using more elementary methods.  
http://math.ucr.edu/~res/math133-2020/week4/sections_2.4-2.5/synthetic-3Daffine.pdf   This document indicates that the 3-dimensional incidence axioms plus the Euclidean Parallel Postulate have significant logical consequences.   This theme is developed further in the Appendix to Section II.5.   The contents of the latter will not be covered in this course.