GSP LAB: DISCOVERING, DEFINING, REVISING AND PROVING A THEOREM ABOUT ARCS AND ANGLES
1.
Open a new Sketchpad file and create a circle. Label the center point as O. Label the control
point (the point on the circle) as P.
2. We will now follow the steps below to create an arc on the circle (but we do not want to use the
control point P).
First plot 3 new points on the circle and label them with A, B and C (making sure B is between A
and C). See below:
B
C
A
O
P
Click on the arrow tool and open space to make sure nothing is highlighted. Now, highlight, in
order, points A, B and C. Then from the construct menu, choose “Arc through 3 Points”. You
will see a darkened arc that now connects all points on the circle from A to C and through B.
Click in the open space so that nothing is highlighted.
3. Click on the arc TWICE to highlight the arc. (On the first click, the circle is highlighted. You need
the second click in order to highlight just the arc). Then, from Measure menu choose Arc Angle.
The arc measure will be shown (in degrees). Notice that you can drag points A and C to create
large and small arcs. We always want to keep B in between A and C however. Drag A and C so
that your arc angle is something less than 180 degrees. Then, drag the arc angle measure near
to your arc.
4. Draw segments from O to C and from O to A to create two radii. As a check, your image should
look something like the one below:
B
C
m ABC = 114.90
A
O
P
5.
Next, measure central angle COA. Move the measurement near to your central angle. You will
see that the central angle measure matches the arc angle measure. See below:
B
C
mCOA = 114.90
O
m ABC = 114.90
A
P
6. There is a limitation to Geometer’s Sketchpad that we need to discuss here. Sketchpad does not
allow central angle measures to be larger than 180 degrees. You will see this happen when you
make your arc more than 180 degrees. Let’s do so now – your image should look something like
the following:
B
m ABC = 200.00
A
O
mAOC = 160.00
P
C
Notice the arc angle is 200 degrees. So, the central angle that goes with that arc should also be
200 degrees. However, Sketchpad will not give us the angle we want. Instead, GSP is
programmed to only give central angle measures between 0 and 180 degrees. This is a problem
. What we really wanted was for Sketchpad to give us the 200 degree central angle that
corresponds with the arc. See below:
7.
To get around this problem, we will hide the central angle measure. However, we know the
central angle measure will always match the arc angle. So, from this point forward we will use
the knowledge that the central angle measure will always match the arc measure, even when
the angle is more than 180 degrees. Now we are ready to investigate a new relationship .
8.
Drag your arc back to where the measure is between 0 and 180 degrees. Make sure point B is
still between points A and C. Plot a new point on the circle but NOT on arc ABC. Label this as
point D. See below:
m ABC = 141.10
B
C
A
O
P
D
9.
Now, create segments CD and AD. Notice that we have also created angle CDA. Highlight
points C, D and A in that order. Then, use the Measure menu to find the measure of angle CDA.
See below:
m ABC = 141.10
B
C
O
A
P
D
mCDA = 70.55
GSP LAB, DISCOVERING AND PROVING A NEW ANGLE THEOREM:
RECORDING SHEET
10. Now, drag point C (but keep point C between points D and B). What conclusion can we draw?
11. Now drag point A (but keep point A between points D and B). Does this change your conclusion
in #10?
12. Finally drag point D (but keep D between A and C). Does this change your conclusion in #10.
13. Try to summarize your findings in the form of a theorem. (This is challenging.)
GSP LAB (continued): REFINING YOUR THEOREM
You have just stated a conclusion in the form of a theorem based on our investigation of arcs
and angles in a circle. However, if we have knowledge of the related vocabulary, we may be
able to more easily communicate our theorem.
Below, you will find vocabulary that may be useful in defining your theorem:
Inscribed Angle – An angle whose vertex is on a circle and whose sides are determined by two
chords.

Draw two examples of an inscribed angle below. In each sketch, label the inscribed angle as
. :
C
C
Intercepted Arc – An arc that lies in the interior of an angle with endpoints on the sides of the
angle.

Draw two examples of an intercepted arc below. In each sketch, label the center point of
your circle as C. Then, “bold or highlight” each intercepted arc.
C
C
Finally, we will revise our theorem!

Restate your conclusion in #13 above, making use of the new vocabulary!