6.1.1 Understand and use angle properties (1)
In this week's tasks we look, somewhat tangentially, at the notion of interior angle sum of polygon.
Monday: This first task is quite challenging. Readers might notice that the angles hover around 180˚. Do they balance out?
A nice feature of the task is that there are lots of ways of solving it. Here are some responses, posted on Twitter, to an earlier version of the task:
And this is the way I first thought of solving it, where an angle of 180˚ is formed at each of the points:
Tuesday: This provides a simpler version of Monday's task, which you might give to students who are stuck, or which can be used to throw light on the original task.
Again, the task can be solved in several ways, for example by drawing a line through B' parallel to the original line, or by drawing the line segment AC and focussing on the angles of the resulting triangle AB'C.
Wednesday: Here we provide a contrast to the angle sum property of triangles (namely, that the sum of the interior angles is 180˚), by having 'triangles' that behave differently. Why is the angle sum normally constant but not here?
Thursday: Here we add a vertex to a polygon by simply placing it on an existing side. This increases the number of sides by 1 and produces an extra interior angle of size 180˚.
Some students might feel that V is not a 'true' vertex, since it doesn't produce a 'corner'. However we could 'rectify' this by moving V out or in slightly, in the way point B was moved in Tuesday's task. What does this do to the angle sum?
Friday: A powerful way of finding the interior angle sum of a polygon (especially when it is regular) is to make use of the property that the exterior angle sum is 360˚, regardless of the polygon's number of sides. Is it still the case that the sum is 360˚ for the two stars?
