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Going round and round


 Activity 3

Take a round of the field, and come back, beginning and ending at P. You end up with your face in the same direction, as it was at the commencement of the journey.
Going Round
How many degrees have you turned in the process?  How many degrees do you turn, if the field was not so circular? In fields like in Figure \ref{round}, what is your answer?
Non-Circular Round
Mischievous round takers will be disqualified, making rounds like in following figure.
Mischievous Rounds
Going Round a Triangle
Let us go round a triangle. Take any triangle ABC (on the field \ldots or on paper). Choose any point P on the segment BC. Go from P up to C, turn in the direction of A and travel (or trace) along CA, then along AB, and finally along BP. Stop at P.

1. How much (how many degrees) have you turned around, starting from P and ending at P? Which way was your face at the beginning and which way it is, when you end? (too trivial for a hint and better not be asked to average students)?

2. Oh \ldots degrees have I turned around! When did these turning happen? Let us repeat the travel again.
Measuring the turn
We were going in the direction of BC, the first turning took place at C. Let us measure it, call it X degrees. (Going BC and CC' way, there would have been no turning or zero degree turning.) There are only two more turnings at A  and at B, call the measure of the turning be Y and Z degrees.  These are the only turnings, so X + Y + Z must be the measure of the turning from P to P via C and B.

3. Observe the linear pairs

Getting the right measurements
X + X'  =\qquad  ; \ Y + Y'  =\qquad ; \ Z + Z'  =\qquad . So X'+Y'+Z'= .
Conclusion:

 Write the above conclusion in the form of a theorem.  Measurements  X', Y', Z' were for angles at points A, B, C,  respectively. How to add them?
 The only way is to transfer them all at one point, say C. The measure Y' at A, is obtainable by line initiating at C parallel to AB (in the direction of AB), similarly transfer the measure Z' at C. What is the transferred  X' + Y' + Z'?

4. Suppose, we repeat our experiment with a quadrilateral  ABCD. Pick up a point P on the periphery and go around.




Let us repeat the earlier questions: how many degrees have you turned around, starting from P and ending at P? When did the turnings take place? And so

  X+Y+Z+W =?.
As before, question \begin{eqnarray*} X+X'  &= & ?\\   Y+Y' & = & ?\\ Z+Z' & = &? \\   W+W' & = &? \end{eqnarray*}
And so X'+Y'+Z'+W'=?.

 It should not be difficult to consider the above process for any polygon with n  vertices: A, B, C, \ldots. Is it not obvious that
\begin{eqnarray*}  X+Y+Z+ \ldots & = & 4\ \mbox{right angles}\\  X+X'+Y+Y'+Z+Z'+ \ldots & = & n\ \mbox{times two right angles}. \end{eqnarray*}

 Therefore, \begin{eqnarray*} X'+Y'+Z' + \ldots & = & 2n\  \mbox{right angles} -2 \times 2\ \mbox{right angles} \\ & =  &  (n-2)\ \mbox{times}\ 2\ \mbox{right angles}\\ &  = & (n-2) \times 180^o. \end{eqnarray*}

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