Can you figure out a way to determine the measure of angle BCD in each sketch above without using a measuring device or Sketchpad’s tools? Study your results carefully. Use the Sketchpad file (exterior.gsp) to help you. Next determine angle BCD in the figures above without using any mearement device like a protractor. Explain (1) what you did and (2) why it works.

Goal of Lesson
This problem solving activity asks students to collect some data and then see if they can make a prediction about the measure of the exterior angle based on the measures of two of the interior angles.

Posing the problem
Using a enlargement display show the students a triangle with one of the sides extended as a ray. Ask the question: if you know the measures of angles A and B, could you predict what BCD is? If computers are not available, give the students the hand out below.

What you want students to do is to use their protractors (or Sketchpad measurement tools) to measure the exterior angle.

Make a chart like this:

Notice that <BCD = <CAB + <ABC. Using the Sketchpad model (exterior.gsp) to confirm this conjecture. If it appears true, why should that be?

The Secret Behind the Exterior Angle Challenge

Print out 2 copies of the same screen shot from exterior.gsp.
Cut out the angles CAB and CBA from one sheet and place them over <DCB on the other sheet. Make sure the vertices A, B and C touches and the angles are adjacent to each other. (See below)

Notice that the angles A and B “fit” the space that the red angle BCD makes. This means that

<A + <B = <BCD

But is it really true? Let’s apply some logic.

We know that the three blue angles of triangle ABC together add up to a 180 degrees.

What can we say about the blue angle C (ACB) and Angle BCD (the red angle)? Since AC and CD are part of a ray, they form a straight line. That means that angle ACB and angle BCD must also add up to 180 degrees.

Summarizing these results, we see that

<A (blue) + <B (blue) + <C (blue) = 180
<C (red) + <C (blue) = 180

Therefore it must be true that the sum of angle A and Angle B is the same as red Angle C (BCD).

*Build your own Exterior Angle Model using Sketchpad

If the file exterior.gsp is not available, you can construct a model of a triangle with an exterior angle. Here’s how:

1. Open a new sketch in Geometer’s Sketchpad
2. Use the line segment tool to draw two segments attached at a point and label the end points.

3. Make the third side a ray. Choose the ray tool. (Hold down the line segment tool and slide over and choose the ray tool. )



4. Drag the point on the ray so that it lands on point C.


5. Place a point D on the ray AC past point C.




6. Measure Angles A, B, and BCD*