Expert: Vishwas K Hajirnis - 4/29/2004

Question
Any help would be wonderful!  If the measure of an angle is 23 17'18" what is the measure of its supplement?

Another question I have is: How many diagonals does a decagon have?  Is it 20?

Answer
1. An angle and its supllimentary angle add up to 180
  degrees.
  Hence the supplimentary angle must be
  =180 - ( 23 17' 18" )
  =180 - ( 23 + 17' + 18" )
  =180 - 23 - 17' - 18"
  =179 + 1 - 23 - 17' - 18"
  (after replacing 180 by 179 + 1 )
  =179 - 23 + 1 - 17' - 18"
  =(179-23) + 59' + 1' - 17' - 18"
  (after replacing 1 degrees by 59' + 1' )
  =(156)+ 59' - 17' + 1' - 18"
  = 156 + (59'-17') + 60" - 18"
  (after replacing 1'by 60" )
  = 156 + 42' + 42"
  = 156 42' 42"

2. No the answer is not 20 it is 35
  If a polygon has n sides then it has
  n*(n-3)/2 diagonals.
  Since the given polygon is a decagon, n = 10.
  So we get 10 * 7 / 2 = 70 / 2 = 35
  ( Here '*' denotes multiplication )
  
  Alternatively, you can simply count the diagonals
  by starting at one of the vertices and go on
  counting diagonals from succesive vertices.
  You will get the answer as
  7 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 35
  
  Hope this is clear.