Geometer’s Sketchpad Activities

Exploring Geometry

The Exterior Angles of a Triangle

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In this activity, you’ll investigate relationships between the interior angles of a triangle and the exterior angles formed by the rays that extend beyond the triangle’s vertices.


To measure an angle, select three points making the vertex the middle selection. Then from the menu MEASURE, choose Angle.
Double-Click on the measured angle to activate the calculator. Then click on the measure you want to use. / Sketch and Investigate
Use Geometer’s Sketch Pad (GSP) to do the following:
·  Construct a Triangleby drawing 3 rays, so that each ray has an end point that becomes the vertex of the triangle. Label each of the vertices ABC.
·  Select the ray, and Construct a point on the ray, then move the point so that it is outside of the triangle.Do this for each ray, labeling the points D, E, & F.
·  Use GSP to measure each interior and exterior angle and record their measures in a data table.

·  To help you identify patterns and relationships, calculate the sum of any two of the interior angles. You may do this with a calculator or you may use GSP to add the measures of angles, by choosing the CALCULATE option in the menu.
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Things to Do . . .

Begin your investigation
Record the initial measured angles on your data sheet, then “Grab” one of the vertex points of the triangle and relocate it. Record on your data with the new information in the second row of your table. Continue doing this till you can establish a pattern of then measures of the angles. Be sure to compare the sum of two interior angles with an exterior angle.
mÐABC / mÐBCA / mÐCAB / mÐCBD / mÐECA / mÐFAD
Trial 1
Trial 2
Trial 3
Do as many trials as necessary to discover a pattern.
From your investigation, you will complete:
1. A Sketch of your initial triangle labeling the points.
2. Your Data Table
3. Your conjecture.
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Points to Ponder

In your investigation, you recorded the measures of the interior angles in a triangle, and you measured the exterior angles ÐFAB, ÐECA, ÐCBD in a data table. You also studied the sum of the measures of two of the interior angles in a triangle.
·  What do you notice about the relationship between the exterior angle and the two remote interior angles?
Write a conjecture, about them, and then write a mathematical equation to express this relationship: ______
·  After writing your conjecture, write a mathematical equation using the symbols and letter of the angles to express the relationship you described in your conjecture.
(an example of a mathematical equation that would describe a linear pair relationship would be something like this: )
·  Once you have written both a conjecture and an equation, test them by moving a point on the triangle again and see if your results confirm your thoughts. Have Mr. M check your conjecture & Equation before moving on to the next step.

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Going Further & Deeper . . .

Once Mr. M confirms your conjecture, confirm that your conjecture is true for all of the angles. Do this by writing three mathematical equations. One for each of the exterior angles and the relationship they have with their remote interior angles. Then, test these three equations by moving each vertex and doing the calculation to confirm your conjecture.

GSP Triangle Ext Angles.doc, Created by S. Middleton 2 revised: 09/24/07