Intermath | Workshop Support
Write-up
Title
Angles in a Circle
Problem Statement
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?
Problem setup
What is the relationship between an angle created by two points on the circle and the center and the arc it
creates on the circle? What is the relationship between an angle created by three points on the circle and the
across from that angle? What is the relationship between the angle created by two points on the circle and an
arbitrary point inside the circle and the two arcs across from that angle? What is the relationship between the
angle created by the intersection of two cotangent lines of a circle and the arcs they create on the circle?
Plans to Solve/Investigate the Problem
The plan for solving this problem was to construct each situation in GSP. I would then calculate the measure
of the angles and the arcs then compare their measures.
Investigation/Exploration of the Problem
Using GSP, I created each circle and proceeded to investigate each situation.
I created the circle for the first scenario. I then created one more point on the circle. I constructed the segme
connecting the center of the circle to each point. I measured the angle created by the segments. I selected
points A, C, and the circle (in that order). I then constructed the arc on the circle. I measured the arc. I
compared the measurement of the arc and the central angle and found that they were equal.
m ABC = 114.60
m AC = 114.60
B
A
C
I created the circle for the second scenario. I constructed two more points on the circle. I then constructed
segments to connect the one point to the other two points on the circle. I then measured the angle created by
segments. I selected points D, F, and the circle (in that order) and constructed the arc on circle. I measured t
angle of the arc. I then compared the measurement of the angle created by the segments and the measuremen
of the arc angle and determined that the arc angle was twice the measurement of the angle created by the
segments.
m DEF = 54.03
m DF = 108.05
m DF
m DEF
= 2.00
E
F
D
I created the circle for the third scenario. I put three more points on the circle. I then created a segment to
connect two points and a second segment to connect the other two points. I constructed a point at the
intersection of the two segments. I measured the angle created by the segments. I then measured the arcs on
either side of the angle. I compared the measurements of the angle with the measurement of the arcs and fou
that half the sum of the arcs is equal to the measurement of the angle.
m JK = 62.21
K
m GI = 137.63
J
m GHI = 99.92
H
m JK+m GI
2
= 99.92
G
I
I created the circle for the fourth scenario. I created a point outside the circle. I then created a point on the
circle and constructed a line connecting the point outside the circle to the point on the circle. I then created a
point at the intersection of the line and the circle. I then created another random point on the circle and create
a line from the point outside the circle to that point on the line. I constructed a point at the intersection of tha
line and the circle. I then constructed segments from the point outside the circle to the points on the circle an
hide the lines. I created the arc by selecting points L, N, and the circle (in that order) and constructing the arc
created the other arc by selecting points O, P and the circle (in that order) and constructing that arc. I measur
the angle created by the line segments and the arc measurements. I then compared the measurements and fou
that the angle created by the segments is half the difference between the two arc measurements.
m LN = 111.57
m LN-m OP
m LMN = 34.26
m OP = 43.05
2
P
L
O
N
Author & Contact
Kelly Elder
kelder@rockdale.k12.ga.us
= 34.26
M