Inside Arcs and Angles

In the middle section of this lesson, teachers will use the same format described in the video narrative.  Students will work in pairs on Exploration #2, and hopefully try to discover the rule for finding internal angles in circles.  I’ve included my thought on how students can answer the questions for Exploration #2 below:

2)   What names can you give for <GLJ and <HLK?   Try to think of 2!

- We’d love for students to recognize that these are vertical angles and hence congruent.  We’d also like for students to realize that these angles are both interior angles or angles inside the circles.  Some students also will recognize these as acute angles since they are less than 90 – this is also a great connection!

3)   How is the measure of <GLJ related to intercepted arcs GJ and HK?  Write a rule!

- Students may recognize that when we add the two intercepted arcs and divide by 2, we will be able to find the measure of the interior angle.  

4)   What mathematical concept in Algebra connects with the rule you wrote in #3?

-       Teachers can introduce the idea of finding the average for students in this part of the exploration.  The rule asks students to find the average of the intercepted arcs to find the measure of the interior angles.

For exploration #3, we have similar questions but are asking students to look at an exterior angle.  Here are possible responses for this exploration: 

2)   What kind of angle is <D?  Try to think of 2 names!

Students will hopefully realize this is an acute angle that is exterior or outside of the circle.

3)   How is the m<D related to arc BGF and arc BH?  Write a rule!

-       This rule is the trickiest for students, and requires them to work backwards in finding the average between the three angles (the intercepted and exterior angle).  We’d love for students to realize the rule on their own, however, I often ask students to consider our formula for interior angles and to consider how this may vary for an exterior.  To lead students to subtraction, I often ask them to consider taking the small arc out of the larger arc and then see how they can get their exterior angle.   

4)   How is the rule for this problem different than the rule for Exploration #2?

At the most straightforward level, this problem is different because we are subtracting not adding arcs.  However, it’s important for students to note that exterior are formed by secants, and we are not just using the “average” to calculate – we are literally subtracting arcs and dividing by two. 

After reviewing all examples in class notes, students can work in small groups on the practice questions attached.   I would suggest that teachers either put answers on the board or review at least one example from the following examples, #1, 6, 8 or 9. 

To wrap up class, students will complete an exit ticket, which asks students to find the measure of an angle inside a circle (like exploration #2).