Our consideration about ‘the problem on the sum of angles’ began in the following ‘Problem e’. Nishikawa et al. (2014) insisted that the ‘Problem e’ would be valuable teaching materials in junior and senior high school mathematics classes.
Problem e: Find the sum of angle α and β in the figure in which three congruent squares are placed side by side.
This article deepened Nishikawa et al. (2014) and its purposes are as follows:
(1) To make the similar problems of the ‘Problem e’ and to raise the value as the teaching materials of the problem more.
(2) To consider the ‘Problem e’ from the viewpoint of inversion and to explore the possibility as teaching materials.
In Section 2, we made the similar problems that had a different condition from the ‘Problem e’, and gave three kinds of elementary geometrical solutions in different ways of viewing sums of angles i.e. neighboring two angles, an external angle and expression of the product. As a result, it was revealed that these similar problems could become superior teaching materials.
Through the consideration in Section 2, we reached an idea that it was a general way to view the problem on sum of angles from the viewpoint of inversion. The idea is reflected in theorem 3.3 in Section 3 as the relationship of sum of angles with respect to a point on circle of inversion and a pair of points of the inversion. It would be said that the figure of theorem 3.3 is ‘the general figure of the sum of angles’. The result of theorem 3.3 is also viewed as a kind of extension of ‘The relationships between inscribed angles and its central angle in a circle’.
In Section 4, we considered a property in ‘the general figure of the sum of angles’. In Section 4.1, we made it clear that the figure was closely related to Apollonius’s circle and we came to the conclusion that these facts would be interesting teaching materials in high school mathematics classes. In Section 4.2, we obtained the proposition 4.4--4.7 by using the properties of inversion. The proposition 4.7 can be used in constructing one point of the inversion from the other.
There are many contents in this paper which give us a clear image of making students acquire some kind of mathematical ability. In this sense, we conducted content-study of ‘the problem on sum of angles’ for the development of a teaching material. In addition, the consideration of this paper clarified the core of the problem that everyone had overlooked by the elegant solution, by various answers to the similar problems and the generalization based on it. Therefore, we insist that the contents of this paper have values as a teaching material of teacher’s training for both pre-service and in-service teachers.
