Measuring Angles Without a Protractor

Date: 01/26/99 at 22:41:37
From: Maggie Jabakchourian
Subject: Measuring angles without using a PROTRACTOR.

Hi, 

My little sister is in the 6th grade, and her teacher gave her a 
question on measuring angles without using a protractor. We tried to 
figure it out by adding all the angles and subtracting the total from 
180 degrees. I was wondering if you might tell me another method of 
doing this by helping with this sample problem.                             

A trapezoid is drawn with four sides, labeled 1, 2, and 3. On the left 
top side of the trapezoid there is 60 degree angle shown. The problem 
states to find out the angles of 1, 2, and 3. How would we go about 
finding the answer? Any help would be greatly appreciated. Thank you.

Date: 01/27/99 at 12:29:34
From: Doctor Peterson
Subject: Re: Measuring angles without using a PROTRACTOR.

Hi, Maggie. There's a little missing in your description, since without 
extra information you can't figure out all the angles. I'm going to 
assume you have an isosceles trapezoid like this, where you know that 
sides B and D are the same length:

                        A
    +---------------------------------------+
     \60                                (3)/
      \                                   /
       \                                 /
      D \                               / B
         \                             /
          \                           /
           \(1)                   (2)/
            +-----------------------+
                        C

Your idea of subtracting from 180 would work in a triangle, but you 
need slightly different tricks here. You know two important things: in 
a trapezoid, sides A and C are parallel; in an isoceles trapezoid, not 
only are sides B and D equal, but angles 1 and 2 are equal. For 
parallel lines,

                \
    -------------+---------------
                  \a
                   \
                   b\c
    -----------------+-----------
                      \

angles a and b are equal, and angles b and c are supplementary (they 
add up to 180). From this you can get (1); from that you can get (2); 
and from that (or directly) you can get (3).

If the trapezoid is not isosceles, you can't tell what (2) and (3) are, 
because they could be anything:

                        A
    +-------------------------------+
     \60                         (3)|
      \                             |
       \                            |
      D \                           | B
         \                          |
          \                         |
           \(1)                  (2)|
            +-----------------------+
                        C

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/