Leaving on A Jet Plane

Definition 5-7

A surface is that which has length and breadth only. Our first 2-D figure! If you are getting the hang of these we see that a surface has length and width (breadth) but no depth, so imagine a piece of paper or fabric. The edges of a surface are lines. Remember when we defined lines, then we said the ends of lines are points? Same idea here. We defined a surface now we define the "ends" of a surface, which are lines. Look are a piece of paper, does it end? Odds are it does so look at the ends (there should be four unless you also have scissors) and what do they look like? Lines? Even if you cut the paper, what does the new edge look like? A line (or a curve if your hand shakes when you cut paper). A plane surface is a surface which lies evenly with the straight lines on itself. Analogies! Line is to straight line as surface is to plane surface! A straight line is "straight" and a plane surface is "flat." Since we are doing Euclidean Geometry, we really only care about geometry on planes, not surfaces in general, which is good news. All the normal geometry terms and theorems you know, work on planes but they don't always work on general surfaces. For example, triangle sum theorem works on a plane but it does not work on the surface of a sphere. Since a piece of paper or a computer screen is just a plane, we can do Euclidean Geometry on those very easily.

Tool Practice

Below we have our padlet and we have added the plane!! Well, we always had planes, one, the white screen itself, so no new tools for this one we just defined the screen of the computer as our plane to work on, if we worked on paper then the paper would be the plane. Below that, manipulate the two other padlets and see why it matters that we are on a plane vs on a surface

Triangle Sum Theorem in a plane! Drag the points around and watch the equation...

Triangle Sum Theorem on a surface (sphere in this case) Drag the points around and watch the equation...