Rest of Geom Chapter1 Part2

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We have been talking about a good definition with lines and triangles as examples. The book says, "A good definition must be reversible" using a table and Grandma will do her best to present it to the impaired. Definition (on this left side) and Reverse of the "definition' (on right) 'Collinear points' are points that lie Points that lie on the same line on the same line. are 'collinear points'. A 'right angle' is an angle whose An angle whose measure is is 90 degrees. 90 degrees is a 'right angle'. A 'line segment' is a set of points. A set of points is a 'line segment'. "The first two 'definitions' are reversible since the reverse of the 'definition' is a true statement. The reverse of the third "definition" is false since the points may be scattered as in Figure 1.9" (For the impaired the dots are all over in different spots with no connected line.) Therefore it is not a good definition. "The reverse of a definition will prove useful in our later work when trying to establish geometric properties of lines ,segments, angles, and figures. For example, a 'midpoint' of a segment may be defined as a point that divides a segment into two segments of equal length.' In Figure 1.10 it shows that is a point on a line that looks like it is in the middle on the line with AB. Using the 'reverse' of the definition of a midpoint as the point that divides a segment into segments of equal length is true if in using the 'reverse' the line AM= the line MB. If that is true in which as we can see it probably is then point M must be the midpoint of AB. "As another illustration, we may define an 'even' integer as an integer that leaves a remainder of 0 when divided by 2. How can we prove that a particular integer is an even number? Simple--we use the reverse of the definition to show that, when the integer is divided by 2, the remainder is 0. If this is true(which it probably is) then the integer must be an even number." Then we move to Initial Postulates the book says," In building a geometric system, not everything can be proved since there must be some basic assumptions, called 'postulates' (or 'axioms), that are needed as a beginning." Two postulates are if two points A and B 'determine' a line. A second line cannot be drawn through it because it would have to curve as shown. (Continued)