Points, Lines, Planes and Angles

• A point
  • Represented as a dot
  • Has no length, width or height (thickness)
• A line
  • connects two points along the shortest possible path.
  • Has no width or thickness, but does have length.
• A plane
  • Has width and height, but not thickness.
  • Is a sheet of paper.
• A line is is normally drawn with two points, each labeled with a letter.
• A ray
• A line segment
• An angle is formed when two line segments intersect.
  • Angles are measured in degrees or radians.
  • Degrees are between 0 and 360
  • Radians are between 0 and 2π
  • Angles are labeled with the letters of the points. (ABC, B vertex)
• Example 1: Problem 2 and 4 page 544

  Since we are dealing with numbers on a clock, each change is 1/12 of 360.
  
  2. From 12 to 4   4*1/12 * 360 = 120
  4. From 1 to 7  (6 numbers)  6*360/12 = 180
     

• A protractor is a tool used to measure angles.
• You can measure the angle by aligning one side with 0, and reading where the other side crosses the protractor.
• Example 2: Numbers 6, 8,10

  6: 130 degrees
  8: 180 - 130 = 50 degrees
  10: 180 - 90 = 90 degrees
     

• A right angle is an angle who's measure is 90 degrees
• Draw one, with the little square at the angle.
• If two angles make upa a right angle, their sum is 90.
• Two angles that sum to 90 are complementary angles.
• Example 3: Problem 12

  90 - 32 = 58 degrees
     

• A line makes an angle of 180 degrees.
• A ray exiting from a line forms two supplementary angles.
• Supplementary angels are two angles who's sum is 180
• Example 4: Problem 14

  180 - 13 = 167
     

• Example 5: Problem 16

  52 degrees
     Compliment  90 - 52 - 38
     Supplement  180 - 52 = 128
     

• Example 6: Problem 22

  let x be the complimentary angle.
  
  x + 56 + x  = 90
  2x + 56 = 90
  2x = 34
  x = 17
     

• When two lines intersect, they form two pairs of vertical angles.
• Vertical angles are equal.
• Why?

           (a,b,c,d) around the circle.
  	 a+b = 180
  	 b+c = 180
  	 a+b = b+c
  	 a = c
     

• Example 7: Problem 26 and 28

  26: angle a (the existing angle) and 2 are vertical, 
     angle 2 is 133 degrees
     angle a and 1 (and 3) are supplementary
       angle 1 = 180 - 133 = 47 degrees.    
  
  28: Angle 1 and a are complimentary, so angle1 = 90 - 30 = 60
      Angle 3 and 1 are vertical, angle 3 = 60
      Angle 2 is 90, (vertical to the right angle)
     

• Parallel lines are lines which do not intersect.
• If a line crosses two parallel lines, it is called a transversal.
• If two parallel lines are cut by a transversal
  • Alternate interior angles are equal
  • Alternate exterior angles are equal
  • Corresponding angles are equal.
  • See page 543
• Last example, 30 and 32

  30
     2 5 and 6 are 54
     1 3 4 7 are 180-54 = 126
  
  32
     2 and a are complimentary angles 
     2 is 90 - 40 = 50 
     1 is 90 (vertical angles 
     3 is 1 + 4 = 140
  
     

Homework