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*}