Why Are Vertical Angles Equal?

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Objective

Introduce the idea of vertical angles through the lens of supplementary angles

Big Idea

Give students a chance to prove that all vertical angles are equal.

Lesson Beginning

15 minutes

I start class off with a normal class routine and I incorporate a few minutes for students to try to solve various angle problems. The prompt is something like, “measure the angles on our handout.”

I give the students appropriate tools and a chance to get started. When I'm ready, I stop them and ask, “how was I able to find the value of these angles with only one measurement?” Ideally, I will be able to ask this question while talking about an observation as students were working.

Then, I will ask, “how was [insert name] able to find the value of all these angles with only one measurement?”

This is a fun question. Many students will say “they are equal!” When they do, I will respond by discussing which are equal and which are not. I normally ask, "When will they be equal and how do we know they are equal?" Then, I challenge the class by giving them a moment to try and prove that vertical angles will always be equal. If students make progress, then get stuck, I give them a hint, "you only need to use the idea of supplemental angles to show that vertical angles must be equal." 

Usually at least one student can explain or teach why vertical angles are always equal. If so, I have them share their idea with the class at this point. If there isn’t anyone ready to explain, then I move forward and return to the conversation during the discussion.

Lesson Middle

30 minutes

Most of the narrative for this lesson is covered in my video why_are_vertical_angles_equal.

In this section, it is important that students are presented with different levels of challenge as they work with their partners and groups. I use this time to circulate and see what students know. I talk with them about ideas for proving that vertical angles are equal. I collect interesting questions and ideas for the summary at the end.

I find that students ask great questions about vertical angles. One fun question I get often is, “will vertical angles still be equal if they don’t meet at the midpoint?” At first I wasn’t sure what students meant by this question. However, I eventually realized that most vertical angle diagrams look as if they are meeting at the midpoint of at least one of the lines. Upon recognizing this, I decided that this observation leads us into a great conversation around proof and the idea that the rule "All pairs of vertical angles are equal"  will work whenever lines cross, in any way and at any angle. Some students will argue that it should work in any case because lines go on forever and thus could be extended to make the intersection point not the midpoint. The angles at the crossing will never change regardless of how long the lines are. This brings out that angles are a measure of degree (e.g. rotation), not distance. These type of conversations pop up a lot during this lesson, which is fun.

 

Teaching Note: When I teach this lesson, my goal is to allow students to explain how they know that vertical angles must be equal. It is important to encourage students to draw out and model their understanding as they work on these angles. Students should be able to informally explain how supplementary angles and the transitive property help prove that vertical angles are equal. There are also opportunities here to discuss the notion of symmetry. Students can explain how the process of reflecting an angle on a line of symmetry forms vertical angles. This will help all students access the math content in this lesson.

 

Lesson End

20 minutes

To close this lesson, I have students present their solutions to the problems they solved and discuss their emerging ideas around vertical angles. If time allows, I like to take a moment and return to the original proof around why vertical angles should always be equal. I ask students to write their own version of the proof, including illustrations that match their reasoning.