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.
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Going Round |
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Non-Circular Round |
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Mischievous Rounds |
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Going Round a Triangle |
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.
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Measuring the turn |
3. Observe the linear pairs
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Getting the right measurements |
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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