How much is the sum of the widths of the angles marked with dots in the figures on the left? Does it vary or is it always the same? Try to study the phenomenon with the following file for Cinderella (or with a similar file that the teacher has prepared with other software), through which you can study the figure however it is arranged (on the right one of the possible forms that you can get). Then try to demonstrate what you supposed with these tests. |
In the case of the figure at the top right it is easily obtained that the sum of the angles is 90+90+90+90 degrees, that is 360°. Let's see other cases.
By making the calcuations it is verified that the sum is always 360°, apart from rounding problems: e.g. in the case of the figure above on the right I get 49.2 + 55.2 + 76.4 + 52.7 + 43.6 + 83 = 360.1 (49.2 could actually have been 49.193… and, analogously, other rounded measures could have been bigger than the exact measures, giving rise to a sum that could get to form 1 tenth more; in other situations we could obtain values whose sum deviates from 360 by 1 or 2 tenths more or less).
Let's try to prove it, that is, to convince ourselves with a reasoning that applies not only in individual cases.
In the case of figures arranged like the one on the right, I can reason like this: I add the angles of three "external" triangles (180°·3) and remove the sum of the angles of the triangle in the middle: I get 180°·2 = 360°.
In the case of figures arranged like the one on the left, I observe that the unmarked angles of the three triangles involved are equal to the angles of the triangle in the center of the figure (in that they are opposite each other), and therefore they have a sum of 180°. Moreover, the three triangles involved have angles that are worth 1180°·3 in all. So the sum of the amplitudes of the marked angles is equal to 180°·3−180° = 180°·2 = 360°.