Mathematics 133
Notes from November 2, 2020, and 
November 4, 2020, lecture sessions

The recordings for these lecturee are available online as  
http://math.ucr.edu/~res/math133-2020/Nov02.mp4            and
http://math.ucr.edu/~res/math133-2020/Nov04.mp4  .
We have combined the summaries for these lectures in order to avoid falling too far behind in creating files.  Since the second lecture is a straightforward continuation of the first, this seems OK.  The files are mainly listed in chronological order.
 Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable.
http://math.ucr.edu/~res/math133-2020/week5/geometrynotes03a.f13.pdf   The first  part of the third chapter of the course notes.   This lectures began with  the subheading, Transversals, parallel lines and angle sums of triangles, in Section III.2,  and it covers the remainder of the file through Section III.3.  
http://math.ucr.edu/~res/math133-2020/week5/geometrynotes03b.f13.pdf   The second part of the third chapter of the course notes.   These lectures covered the material from the beginning of Section III.4 up to the previously mentioned Theorem III.4.8.
https://math.ucr.edu/~res/math133-2020/week4/sections_3.1-3.2/transversal-pairs.pdf   Drawing indicating the various pairs of alternate interior angles, and related angle pairs, associated to two lines cut by a transversal.  Review from the previous week.
https://www.mathsisfun.com/geometry/alternate-interior-angles.html  An interactive file on the same topic as the previous item.  Review from the previous week.
https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov02-1.png   Summary of the part of the Alternate Interior Angle Theorem (Proposition III.2.11) which relies upon the Euclidean Parallel Postulate.
http://math.ucr.edu/~res/math133-2020/week5/sum180.pdf   Another proof that the sum of the angle measures for a triangle is 180 degrees  (Theorem III.2.13).  This proof uses the diagram for the proof of the Exterior Angle Theorem  (Theorem III.2.1).  
https://math.ucr.edu/~res/math133-2020/week5/semicircle-thm.pdf             https://math.ucr.edu/~res/math133-2020/week5/circleright.pdf   Two proofs of a result attributed to Thales:  An angle inscribed in a semicircle is a right angle.  The first proof uses vector methods, while the second uses classical Greek methods.
http://math.ucr.edu/~res/math133-2020/week5/hard-easy-geometry-problem.pdf   If the problems thus far in the notes and exercises seem too easy, here is a widely circulated problem which is more challenging but can be solved using material developed up to this point.
https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov02-2.png https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov02-3.png  https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov02-4.png https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov02-5.png       Drawings to illustrate examples and non–examples of convex polygons (note that a convex polygon is  NOT  a convex set because it is hollow by definition!!).
http://math.ucr.edu/~res/math133-2020/week5/trapezoid.pdf  Drawing of a trapezoid.
https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov02-6.png https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov02-7.png       Drawing and commentary for the theorem stating that two parallel lines are everywhere equidistant (Proposition III.3.5).  The first depicts all possible cases, and the second stresses the importance of the result for justifying the definitions of coordinate geometry. 

This is where the lecture of November 2 ends and the lecture of November 4 begins.

https://math.ucr.edu/~res/math133-2020/week5/reg-pentagon.pdf   Drawing for the proof that a regular pentagon is a convex polygon as defined in the notes.  Similar arguments are valid for regular polygons with 6 or more sides.
https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov04-1.png   Drawing for the proof of Proposition III.3.7,  which states that a line passing through the center of a circle meets the latter in two (diametrically opposite) points.
https://math.ucr.edu/~res/math133-2020/week5/thm3.3.8.pdf   Alternate (and possibly more transparent) formulation of Theorem III.3.8,  which is about points on a line which meets a circle in two points but does not pass through the center of the circle.  
https://math.ucr.edu/~res/math133-2020/week5/solid-triangle-centroid.pdf   Explantion why the concurrence point for a triangle's medians is the center of mass for the associated solid triangle (assumming the latter has uniform density).  This will not be covered on quizzes or the final examination.  
https://math.ucr.edu/~res/math133-2020/week5/concurrence3D.pdf   Determination of the locus (= set) of all points in 3-space which are equidistant from three noncollinear points; this set is the line through the circumcenter which is perpendicular to the plane determined by the original three points.  This will not be covered on quizzes or the final examination. 
https://artofproblemsolving.com/wiki/index.php?title=Law_of_Sines  https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-inscribed-angles/a/inscribed-and-central-angles-proof   The first link derives a simple but important relationship between the common ratio in the Law of Sines for a triangle (Theorem III.2.9) and the radius of the circle containing the three vertices of that triangle (the circumcircle).  The second link describes a proof of a theorem about inscribed angles which is needed to derive the result in the first link.  Although the proof in the second link does not explicitly take into account some issues involving plane separation and related concepts, it can be modified to deal with such questions;  one reference for the latter is the textbook, Geometry, by E. E. Moise and F. L. Downs (unfortunately, since I do not have access to my office I cannot give a more precise reference within the book).   The material in these links should be read and at least passively understood.
https://math.ucr.edu/~res/math133-2020/week5/Whiteboard.nov04-2.png   Drawing for the Proposition III.4.5  and the definition of an angle bisector.
https://math.ucr.edu/~res/math133-2020/week5/thm3.4.6.pdf   Alternate (and possibly more transparent) formulation and proof sketch Theorem III.4.6,  which characterizes the locus (or set) of points on the bisector of an angle.