4. Angles in a Circle: Investigate how the location of an angle's vertex affects its
relationship to the measure(s) of the arc(s) it intercepts. (10 pts)
The vertex of an angle can appear on, inside, or outside a circle. How does the location
and measure of the vertex angle compare with the measure(s) of the arc(s) it intercepts?
Initial Analysis or Plan: Briefly analyze the problem and your plan for solving it.
The goal of this problem is to compare and find a relationship between the
highlighted blue vertex angle and the highlighted arc measures. For the first two no
investigation was required as I have taught about inscribed angles on the circle and
on the center for the past few years. The last two I however had no idea of the
relationship so I began by making sketches in GSP. I tested quite a few cases in each.
Attempted Solution: Carry out your plan and write up a solution.
Well considering I teach high school mathematics I have the rules in the first two
cases memorized. I’m not sure if that is cheating or not. In the first example
above you have an angle that’s vertex intercepts the center of the circle. In this
instance the arc is always the same measure as the central angle. In the second
example the vertex is on the circle and the angle is ½ of the arc. If the arc were
60 the angle would be 30.
In the last two I had to do a little investigating. In the bottom left you have an
angle similar to the first except the vertex is not on the center of the circle. So
after doing some investigation I came up with the following conclusion. The
vertex angle is ½ the sum of the two arcs. So if the vertex angle is 70 then the
two arcs the smaller and the larger must add up to be 140. Or the sum of the two
vertex angles equals the sum of the two arcs is another way to look at it.
The bottom right that would have been easy to solve if the two rays were tangent
to the circle and the vertex was some external point but that is not the case. The
rays are secants touching the circle in exactly two places. The closest solution I
could come up with and there is some rounding inaccuracy is the major arc minus
the minor arc cut in half equals the vertex angle. I’m still not sure that is accurate
however.
Closing Analysis:
 Analyze the activities from the learning window; usefulness for learning
the content?
Students need to know the relationships between arc measure and
vertex angle measure for most standardized tests. It is also very
useful when dealing with Circle Geometry.

Analyze the activities from the teaching window, usefulness for teaching
the content?
The first two the patterns are very easy to find. The second two
are not so easy. This is good to do in an honors class to see who
will go beyond and really delves into finding an answer providing
sufficient proof.
Final Personal Reflections:
In general I learned that patterns can and at times are difficult to
find. Sometimes when you stumble on one that you think is
accurate this one case will defeat it. Many students would give up
half way through that is why I suggest doing the second two shapes
in an honors class. Students would have to provide several
examples testing their claim. I would also extend it to go beyond
and actually demonstrating using tangible objects the proof.