Part
1:
- Discuss the characteristics of a triangle.
- Draw a triangle in your sketch window. There are several ways to
draw a triangle using The Geometer's Sketchpad. One way is to use segments to connect
three non-collinear points.
| To create a new sketch window:
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To create three points:
-
Choose the Point Tool 
by clicking and
letting go. You will see the Point tool highlighted, meaning
you've selected it as your active tool.
- Move the pointer into the sketch window (where
it becomes a cross-hair) and click to create a point.
- Move the cross-hair to where you want to create
a second point and click. Do the same for a third. Note that
the last point you created is still selected (i.e., it has a
bold outline). Choose the Selection Arrow Tool
from the
toolbox, and click anywhere in the sketch window. This should
de-select the third point.
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To construct a segment:
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Select two points. Position the tip of the Selection Arrow
over one of the points, and click to select it. To select the
second point hold down the shift key as you point and click on
it. For future use, remember to select more than one object at
a time you must hold down the shift key.
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Under the Construct menu, select Segment. The
two little boxes on the constructed segment indicate that it is
selected. Repeat this procedure to construct the remaining two
sides of the triangle.
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- Once a triangle is drawn we can manipulate it. Click on one of the triangle's vertices (or sides), hold down your mouse button and
drag it. What happens to your triangle?
Part 2:
-
Verify that your triangle is a right triangle by measuring its interior
angles. Among other things, Sketchpad is capable of measuring angles
and lengths of segments. Prior to measuring any objects, you might want
to select a standard unit of measure for both distance and angle measurement.
To select a standard unit of
measure:
- Under the Display menu, select Preferences.
- Choose a unit for distance, a unit for angle measurement and
the displayed precision for each.
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To measure an angle:
- Select three points which define the angle. Here, selection
order is important! You must select: point, vertex, point.
- Choose the Angle command under the Measure menu.
The measure of the angle will appear in your sketch window.
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To measure the length of a
segment:
- Select the segment.
- Choose from the Measure menu, the Length command.
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Draw an isosceles triangle. Confirm that you have an isosceles triangle
by measuring the lengths of the sides of the triangle. If your triangle is
not isosceles, drag one of its vertices until it becomes isosceles.
-
As you drag the sides or vertices of your triangle,
you might notice that your displayed measurements might be slightly off from what
you expect. How can you explain this?
-
Label your triangle as "isosceles" in a text box in your sketch window.
To create a text box:
- Select the Text Tool in the tool box.
- Drag a text box in your sketch window.
- When a flashing cursor appears in the text box, you may type
in the box.
- To edit your text box once you've clicked outside of it, select
the Text Tool and click inside the text box. A flashing
cursor should appear, allowing you to edit.
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Part 3:
- Draw a sketch and record the three angle measurements of a scalene,
obtuse triangle in your sketch window. What is the sum of the three interior
angles of your triangle? Compare the sum with your neighbor's sum. Comment on your findings.
- Use Sketchpad to calculate the sum of the three interior angle
measures.
To use the Sketchpad's built-in
calculator:
- With the Selection Arrow, select all three angle measures.
- Choose Calculate under the Measure menu.
- The angle measures have been stored under the Values
pull down menu and can be moved into the calculator display by
highlighting.
- Use those measures and the keypad to form an expression for
the sum of the three angle measures in the calculator display.
- To evaluate, click on the OK box.
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What is the sum of the measures of the three angles? Drag any vertex
to manipulate your triangle. What do you observe about the angle measurements
and the sum while manipulating your triangle? What conclusions can
you draw from your observations? (Instructor Note: A
sketchpad
file illustrating the sum of the interior angles has been
saved as
intangles.gsp)
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Do these manipulations and observations constitute a valid proof?
Explain your reasoning.
-
Formally prove that the sum of the interior angles of a triangle is
180°.
Extensions:
-
We have formally proven that the sum of the interior angles of a triangle
is 180°. We would like to further investigate the sum of the interior
angles of other polygons. In a new sketch, draw a quadrilateral and
find the sum of its interior angles. What do you observe about the angle
measurements and the sum while manipulating your quadrilateral? Is
the sum of the interior angles of any quadrilateral equal? Explain your
answer. What mathematical conclusions can you draw from your observations?
Investigate a concave quadrilateral. What conclusions can you draw about The Geometer's Sketchpad from your
observations?
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Continue the investigation using both pentagons and hexagons. On
paper, organize the sum of the interior angles of each of the four figures
in a table. Predict the sum of the interior angles of an octagon. Draw
an octagon in a new sketch window and assess your prediction. Generalize
the pattern to produce a formula for the sum of the interior angles
of an n-gon. Verify your formula by an informal proof.
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In each of your drawn polygons, investigate the sum of the
exterior
angles. What do you observe about the sum while manipulating your polygon?
Discuss and write a conjecture about your findings. Why is the sum of
the exterior angles of a polygon not dependent on the number of sides
of that polygon? Explain your reasoning by a formal or informal proof.
To extend segments to rays:
- Point your selection arrow on the Point tool in the Toolbox.
- Hold down your mouse button and select the Ray option.
- Select the initial point of your ray and another point
for the ray to go through. Remember to hold down the shift key
while you select the second point.
- Under the Construct menu, select Ray.
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