Thursday, February 21, 2013

Ugh, What Now?

     I presented this problem to my 9th grade honors geometry students, and we got stuck. While we were working through it there was a retired math teacher in the room as well who was also stumped. After class, I showed it to a few of my colleagues; also stumped. We figured out a few possible paths that might help, but we are trying to solve it strictly within the constraints I gave my students, using knowledge that they would be able to grasp.
     We've been working with all of the different types of angles, quadrilaterals, triangles, and all of their properties recently. We've stumbled upon regular figures and what we can do with them. The end goal is to be able to do this, but we've been going through similar problems as a class to lead up to that point.
Here's what I gave them today (please ignore my phone shadow):
     I gave my students a regular hexagon, drew in some lines and labeled some angles to be calculated. I love these problems because it takes all of the knowledge that they've gained over the past few weeks and puts it into one beautiful package. They have to know the types of angles, symmetry, properties of various figures, etc. to solve for all of these angles. It's awesome to watch their brains working and the excitement on their faces when they figure these out. Even students who ask, "Why do I need to know how to do this?" are still engaged and have a desire to figure it all out. But anyway...
     In this problem, the black numbers are the names of the angles and the blue numbers are their measures. If you look carefully, we've calculated all of the angles that I drew in, whether or not they're labeled. Except for angles 14, 15, 24, and 25 at the top.That one line that passes by all of them (connecting the upper-left vertex to the midpoint of the other side on the right) is really messing things up.
     To start the problem I gave the students no angle measures; they had to calculate all this by themselves by figuring out a possible starting point. It is a regular hexagon, and it is drawn to scale, but that's all the info they had. I did not give them any side lengths either since we were focused on just finding the angles. 
     Angles 14, 15, 24, and 25 really threw us off. We've tried extending some lines outside the hexagon, we've tried drawing extra lines in, and I even had a student line up a congruent hexagon along the side formed by angles 6 and 16 to see if that would help her. My next step is to give the sides a length and see if trig. will lead me to freedom, but I was hoping to get the answer using only angle measures. 
     Can you figure out the measures of angles 14, 15, 24, and 25 using only knowledge of angles, polygons, and symmetry? Or is more information required? If so, what else do you need? Any guidance would be greatly appreciated.
**Note: I did figure this out, but it required me to work with the side lengths, using law of sines and cosines. Theoretically, my students could do that if they recognize the fact that the length of the sides doesn't matter. I've taught them law of sines/cosines, and I have some really bright students that could possibly make that connection. However, I am still curious to see if there is a way to do it without referencing the sides and using only angle measures. I'm leaning towards 'no, it can't be done' but still not fully convinced.


  1. I played around a little bit in Geometer's Sketchpad and saw that the angles are horrible decimals. :)

    There could be some value in getting the students to give you equations using the angles, like angle 14 + angle 24 = 60 degrees. There could also be some value in estimation. I found that one angle had to be less than 15 degrees.

    My intuition says that it isn't possible to just use angles/polygons/symmetry. I wonder if there's a way to use side lengths (but not measurements) and a single trig function to get the angle? Something like side A must be 30% of side B so the angle X is arcsin of .30. Or something along those lines. Maybe just using 30-60-90 properties or 45-45-90 properties.

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