Tuesday, November 25, 2014

A Jumble of Geometry

Recently in class, we began with some talking points - questions related to geometry to spark discussion. One of the questions asked, "From the diagram given below, you can find the measure of angle D":
After a brief look at the diagram, the answer was clear: no. The measure of angle D can not be found because the diagram does not tell have parallel lines - if this was given the answer would be yes! As a follow up, a new diagram was given. Below I will outline my steps in the process of solving for each of the 4 angles presented.

The Process

The first thing I noticed when looking over the above diagram was the 44 deg. angle at the bottom. Since this time the lines are parallel and recalling that opposite side exterior angles are congruent, I knew the value of A is 44 deg. as well. At this point, recalling the vertical angle theorem, I was able to fill in 2 other 44 deg. angles(in blue).
After writing in the values, I noticed the triangles at the top and bottom of the diagram. They now had 2 of there 3 angles filled in. Knowing that the angles in a triangle always add to 180 deg., I was able to find the third angle in each:
180 - (12 + 44) = 124
180 - (44 + 30) = 106

Once those angles were filled, my focus became on B. By the vertical angle theorem would also we 124 deg. And by the same theorem I was able to find another angle of 106.
If you can't tell, the tiny orange dot in the small triangle represents the other 106.
Next, I looked to solving C, because I felt like if I'm going in order thus far I should keep with it. But as I looked at the diagram, D felt like the clearer next step. All I would have to do is find all the angles of the little triangle and I'd be a step away from finding the measure of D. I recalled that angles on the same line add up to 180 deg. This allowed me to find the angle beneath 112 deg, because they are on the same line - so the angle must be 68. By the same reasoning I was able to find another angle of 68 deg.
Now there's a tiny pink dot in the tiny triangle. 
Again, I know all the angles of a triangle add up to 180 degrees. So 180 - (106 + 68) equals the measure of the third angle, which is 6 deg. Then I noticed that the measure of angle D with the angle I just found makes a circle, which has an angle of 360 deg. So using this fact and the found angle, D is 354 deg.
So now I finally had to face C. This one took me a minute. As I looked at the diagram, I kept thinking "how will I find C?" and wishing "if only I could find the measure of the angle that with C results in 360 deg". So how could I find this angle? The shape was irregular and I didn't know anything extra about the lines around C (no parallel lines to be had). I kept thinking, if only I could somehow break the shape in triangles and find other angles that could lead to C. I wished the shape was regular, then maybe I could find the sum of the angles and then find C. As I thought about this I realized that the regularity of the shape was irrelevant. This shape was five sides, so it's a pentagon, and pentagons (all and any) have 540 deg as the sum of their interior angles! 
Since I now have 4 of the 5, finding the fifth became extremely simple: 540 - (124 + 22 + 68 + 52) = 274. To find C, all I had to do was subtract this angle from 360. So C ends up being 86 deg.


For not having done high school geometry in a while, I felt good about working through this problem. I especially liked the way in which it seems a little daunting (all the lines and angles and shapes) but as you dive in, you realize that each piece of the puzzle isn't too bad. It was also helpful in using several different theorems that I was taught in high school geometry to figure everything out. While the pentagon through me off a little in my search for C, I enjoyed that aspect. It was nice to struggle and face a minor challenge along the way - once reaching the solution, how obvious it now seemed! I think it would be fun to present the problem to high school students, since a lot of the knowledge and understanding required for the puzzle they are in the process of learning. And because it's a process, that one thing leads to the next and has more purpose than simply filling in the blank for one question, students could get a lot out of it - applying the skills they are gaining and diving into critical thinking and problem solving!

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