# Engaging students: Defining the terms complementary angles, supplementary angles, and vertical angles

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 Brittany Tripp. Her topic, from Geometry: defining the terms complementary angles, supplementary angles, and vertical angles

How could you as a teacher create an activity or project that involves your topic?

I really enjoy getting the students up, out of their seats, moving around, and engaging with one another. One way that you could do this, that would fit for this topic, would be to create different matching cards. You could makes cards that have the names: complementary angles, supplementary angles, and vertical angles. Then to go along with those you could have different cards that have the definitions of the different types of angles and other characteristics about them. You would give each student a card and then turn on some music and have them dance around the room looking for the people that pair with them. I love the idea of music because it gets the students more engaged than just lazily walking around the room. In my opinion, music and dancing your awakens senses and increases student engagement in a way that just walking around doesn’t. After everyone finds the people they pair with you could have each group read off their cards so that everyone else has to opportunity to gain all the information on all of the cards. You could take this one step further and get different angles and measurements and cards ask the students to first find their complementary angle, then supplementary. This would give them the opportunity to actually practice the different types of angles and put definitions to actually problems.

What interesting things can you say about the people who contributed to the discovery and/or development of this topic?

A variety of people have contributed to the discovery and development of this topic, for instance: Proclus, Eudemus, Euclid, and that is just to name a few. They all have varying definitions of what an angle itself is and while none of them use the terms complementary, supplementary, or vertical angles, they state things that we now know to be those things. Proclus, or Proclus of Athens, is known to have been “the most authoritative philosopher of late antiquity.” “[He] was eager to demonstrate the harmony of the ancient religious revelations and to integrate them in the philosophical tradition of Pythagoras and Plato.” He also wrote commentaries on a variety of other philosophers and mathematicians works including, but not limited to, Euclid, Aristotle, and Plato. In a commentary of Euclid’s first book of elements, Proclus’s idea, of what an angle is, is presented.

There are also two corollaries given by Proclus in association with Book I Proposition 32 of Euclid’s Elements which discusses the three angles of a triangle. Eudemus, or Eudemus of Rhodes, was a Greek philosopher very present before 300 B.C. He worked closely with Aristotle and Theophrastus. “[Eudemus’s] history of geometry, arithmetic, and astronomy completed the Doctrines of the Natural Scientists of Theophrastus.” There are three known works on the history of Mathematics that were contributed by Eudemus, those, as stated in the previous quote, being: History of Arithmetic, History of Geometry, and History of Astronomy. Eudemus’s idea of what an angle is, is also presented in the commentary of Euclid’s first book of elements.

Of course, considering Eudemus is known for one of his works History of Geometry, I think it is safe to say he contributed much more to Geometry than this simply idea of an angle. Now on to maybe one of the most well known mathematicians, who contributed to the understanding and development of angles, Euclid. There is not very much that anyone knows about Euclid besides when and where he was born. He is mostly known for this contribution to geometry in the Elements. In Euclid’s book, Euclid’s Elements, there are propositions outlining a variety of different types of angles such as supplementary angles and vertical angles. For instance, Book I Proposition 13 of Euclid’s Elements is about supplementary angles and Book I Proposition 15 is about vertical angles.

How can technology be used to effectively engage students with this topic?

There is a website that I have used before with students that I have tutored and I have really enjoyed using it and see how much the students enjoy using it! The website is IXL Learning (https://www.ixl.com/). I love this website because it has a huge variety of different mathematics topics ranging from grades Pre-K, counting objects, to Twelfth grade, Pre-Calculus and Calculus. The website alone is super appealing because it is very colorful which instantly helps to catch the attention of whomever is using it. All of the grades levels are presented on the homepage so it makes finding the grade you are looking for extremely easy. When you click on the grade level it takes you to a screen that is broken up into categories and within each category there are subcategories. These make it even easier to find/access the specific topic you are looking for. For instance when looking for complementary, supplementary, and vertical angles you can click on tenth grade geometry. The page it takes you to contains a category called Angles. Under the angles category there are subcategories that include angle vocabulary, angle measures, identify different types of angles, as well as other things.