Measurement

1. Measurement MA418 – Spring 2010 McAllister

2. Ruler Postulate (p. 41) • Every line can be made into an exact copy of the real number line using a 1-1 correspondence. • Line segments can be associated with a number that we call its measure. • Line segments with the same measure are said to be congruent. • Lines and rays are infinite and can not be measured.

3. Protractor Postulate (p. 42) • If we place one ray of an angle at 0 degrees on a protractor and the vertex at the midpoint of the bottom edge of the protractor, then there is a 1-1 correspondence between all other rays that can serve as the second (terminal) side of the angle and the real numbers between 0 and 180 inclusive, as indicated by a protractor.

4. Two types of angle measurement systems • Degrees – minutes – seconds (fractional form) and decimal degrees. • There are 60 minutes in 1 degree and 60 seconds in 1 minute • so 32° 15’ 30” (32 degrees, 15 minutes, 30 seconds) is like [32 + 15/60 + 30/(60x60)] degrees. • We can convert this to decimal degrees by dividing out the fractions: so 32° 15’ 30” ≈ [32 + 0.25 + 0.0083] degrees

5. Now let’s go from decimal degrees to fractional form • Suppose we have 241.32 degrees. If we want this in degrees – minutes seconds, we convert the decimal fraction part of the number back into fraction form. • 0.32 x 60 = 19.2, so the 19 becomes the minutes and we convert the .2 to seconds. • 0.2 x 60 = 12 • So 241.32° = 241° 19’ 12”

6. Let’s practice on these examples • Convert from d-m-s to decimal form • 45° 16’ 43” • 137° 47” • Convert from decimal form to d-m-s • 96.125° • 101.027°