# Engaging students: Measures of the angles in a triangle add to 180 degrees

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Claire McMahon. Her topic, from Geometry: the proof that the measures of the angles in a triangle add to $180^o$.

One of the hardest concepts in math is learning how to prove something that is already considered to be correct.  One of the more difficult concepts to teach could also be said on how to prove things that you had already believed and accepted in the first place.  One of these concepts happens to be that a triangle’s angles are always going to add up to 180 degrees.  Here is one of the proofs that I found that is absolutely simplistic and most kids will agree with you on it:

This particular proof is from the website http://www.mathisfun.com.  This is a great website to simply explain most math concepts and give exercises to practice those math facts.  For the more skeptical student, you can use a form of Euclidean and modern fact base to prove this more in depth.  I found this proof on http://www.apronus.com/geometry/triangle.htm.  Here you will see that there is no question as to why the proof above works and how it doesn’t work when you do a proof by contradiction.

I stumbled across this awesome website that very simply put into context how easy it would be to prove that a triangle’s angles will always add up to 180 degrees.  In this activity you take the same triangle 3 times and then have them place all three of the angles on a straight line.  This proves that the angles in a triangle will always equal 180 degrees, which is a concept that should have already been taught as a straight line having an “angle” measure of 180. The website for this can be found here: http://www.regentsprep.org/Regents/math/geometry/GP5/TRTri.htm.

The triangle is the basis for a lot of math.  There is one very important person that really started playing with the idea of a triangle and how 3 straight lines that close to form a figure has a certain amount of properties and similarities to parallels and other figures like it.  We base a whole unit on special right triangles in geometry in high school and never know exactly where the term right angle is derived.  This man that made the right angle so important in math is none other than Euclid himself.  While Euclid never introduced angle measures, he made it very apparent that 2 right angles are always going to be equal to the interior angles of a triangle.  Not only did Euclid prove this but he did so in a way that relates to all types of triangles and their similar counterparts using only a straight edge and a compass, pretty impressive!!

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