Triangles

• An early Greek Mathematician, Euclid wrote a book called Elements
• He defined 5 basic rules or postulates or basic rules
  • A straight line segment can be drawn by joining any two points.
  • A straight line segment can be extended indefinitely in a straight line
  • Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.
  • All right triangles are equal.
  • If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
• The fifth postulate has caused quite a stir over the years.
  • It was thought that it was too complex, I could be proven from the other four.
  • There have been may different statements of it. All are equivalent.
    • Exactly one line can be drawn through any point not on a given line parallel to the given line.
    • The sum of the angles of a triangle is 180 degrees
    • If two parallel lines are cut by a transversal, ...
    • A circle, which contains all three vertexes, can be drawn around every triangle.
    • And many others.
  • We can use deductive reasoning to prove all of these are the same.
  • They have tried to prove that this is not necessary by trying to establish a geometry where Euclid's fifth postulate is not true.
    • This has given rise to Non-Euclidean geometry.
    • Which might actually describe space.
• Ok, so what do we need?
  • The sum of the angles of a triangle is 180 degrees.
  • Example 1: Problems 2, 4 and 6 page 552
• They name a bunch of triangles: (Look at the box on 548)
  • If all angles are less than 90, then the triangle is acute
  • If an angle is greater than 90, then the triangle is obtuse
  • If one angle is 90, then it is a right triangle.
  • If two sides have the same length, the triangle is isosceles
    • And the opposite angles have the same measure.
    • If all three angles are the same the triangle is equilateral
    • • And all sides are the same length.
    • Example 2: Problems 8 and 10
  • Similar triangles are triangles that have the same angles, but possibly different lengths of sides.
    • Matching angles are called corresponding angles
    • Corresponding angles are usually marked
    • The length of sides are in a ratio.
    • Example: 12, 14, 16

      12:
           x/10 = 12/15
           x = (12*10)/15 =  8
      
      14: 
         (5+x)/ 5 = 5 / 4 
           4(5+x) = 25
           20 +4x = 25
               4x = 5
        	  x = 5/4
      
      16: 
              4/5 = x/7.5
         4(7.5)/5 = x 
                6 = x
      	 

  • Pythagoras, among others, discovered that in a right triangle
    • A²+B² = C²
    • Where C is the length of the side opposite the right angle.
    • Example: 22, 24, 26

      22:
          c2 = 242 + 7 2
                        = 625
          c = 25
      
      24:
          c2 = 112 + 5 2
                        = 121 + 25 
      		  = 144
          c = 12
      
      26:
          b2 = c2 -a2
                        = 132 - 52
      		  = 169 -25
      		  = 144
      		b = 12
      	

  Homework

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