Exploring the angles of a triangle with Geogebra®.

We are going to explore three questions in this lesson.

  1. What is the sum of the three interior angles of any triangle?
  2. What is the relationship between the two remote interior angles of a triangle and the related exterior angle?
  3. Can the collinear remote interior angle ever be greater than the related exterior angle?

We will explore the answers to these questions with dynamic geometry software.

Before we begin, we need to know the commands of the software.

The menu bar at the top has the following icons

We will reference these icons from left to right. The first icon is the cursor control and looks like your typical cursor. The second icon is the point menu and has a point with the label A on it. To ensure you are reading the menu bar correctly, the 5th icon is the triangle menu.

Notice that each icon has an arrowhead in the bottom right corner. Clicking on this arrow will bring up a drop down submenu.

Step 1. / Click on the arrowhead of the third icon of the menu bar and select Line through Two Points from the submenu.
Step 2. / Move your cursor to the point (-3, 1) and click you cursor to establish a point A. Then move your cursor to (3,1) and click your cursor to establish a point B. You should have a line on your screen similar to that in the figure below.
Step 3. / Click the second icon in the menu bar and move your cursor to the point
(0, 4) and click your mouse one time to create a point C at 4 on the y-axis.
Step 4. / Move your cursor to a point on line AB to the right of point B and click your cursor once to create a point D on line AB. Your screen should look similar to the figure below. Note: in the figure below, point D is at (6,1).
Step 5. / Click the arrowhead on the third icon in the menu bar and select Segment between Two Points from the submenu.
Step 6. / Move your cursor to point A and click your cursor once. Then move your cursor to point C and click your cursor a second time. You should notice a line segment following your cursor as it moves from A to C. The line segment stops moving when you click on point C. Now click your cursor on point B once and move your cursor to point C and click the cursor on point C. You should have a triangle as seen in the figure below.
Step 7. / Click the 8th icon on the menu bar (looks like an angle). Move your cursor to point B and click it then move your cursor to point A and click it and finally move your cursor to point C and click it. You should now have the measure of angle A (45˚). (Be sure to click the points in order as listed)
Step 8. / Repeat Step 7 to find the measures of the other angles by placing clicking on points A, C and then B to get a 90˚ measure for angle C; clicking points C, B, and then A to get a measure of 45˚ for interior angle B; and finally clicking points D, B and then C to get a measure of 135˚ for exterior angle B. Your screen should look like the figure below.
(Be sure to click the points in order as listed)
Step 9. / In this step, we are going to clean up our diagram so we can see all parts. Click your cursor on the first icon in the menu bar and then move your cursor to the letter A on your figure. Click and drag your cursor and the letter A will follow your cursor. Move the letter A out away from the triangle a little way to the right so it can be seen easily. Now move your cursor to the 45˚ label near point A and click and drag this up and to the right so it can be easily seen but close enough you know it belongs to angle A.
Step 10. / Repeat Step 9 for all labels on your diagram. It should look similar to the figure below.

This is the tool we will use to answer the three questions of this lesson.

Answering our first question.

What is the sum of the three interior angles of any triangle?

Step 1. / Move your cursor to the input command line at the bottom of the Geogebra screen. You should get a new icon to the right of the command line.
Step 2. / Click on the icon with the Greek letter alpha to get a table menu seen in the figure below. Notice that the angle measures in your diagram on your screen have the Greek letters that are found in this table. We have named the first row of Greek letters in blue for your reference.
Step 3. / Select the Greek letter alpha from the table then press the + key on your keyboard. Press the icon again and select beta from the table and then press the + key on your keyboard. Press the icon again and select gamma from the table and press ENTER on your keyboard. You should have a new value in the table at the top left of your screen stating epsilon equals 180˚ as seen in the figure below.
Step 4. / Place your cursor on point C and click and drag the point around the screen changing the original triangle as C moves. Notice that the values for the angle measures changes as the position of point C changes. What do you notice about the value of the sum of the three interior angles as you move point C around making different triangles?

Answering our second question.

What is the relationship between the two remote interior angles of a triangle and the related exterior angle?

Step 1. / Place your cursor on point C and click and drag the point around the screen changing the original triangle as C moves. Notice that the values for the angle measures changes as the position of point C changes. What do you notice about the value of the sum of the two remote interior angles and the measure of the exterior angle (delta) as you move point C around making different triangles?
Step 2. / Click on the icon with the Greek letter alpha to get a table menu.
Step 3. / Select the Greek letter alpha from the table then press the + key on your keyboard. Press the icon again and select beta from the table and then press the ENTER key on your keyboard. You should have a new value in the table at the top left of your screen stating epsilon equals 180˚ as seen in the figure below.
Step 4. / Compare the measure of the sum of the two remote interior angles with the measure of the exterior angle (compare zeta to delta). What do you notice?
Step 5. / Place your cursor on point C and click and drag the point slowly around the screen changing the original triangle as C moves. Notice that the values for the angle measures changes as the position of point C changes. What do you notice about the value of the sum of the two remote interior angles (zeta) and the measure of the exterior angle (delta) as you move point C around making different triangles?

Answering our third question.

Can the collinear remote interior angle ever be greater than the related exterior angle?

(We are asking if alpha can ever be greater than delta in our diagram.)

Step 1. / Place your cursor on point C and click and drag the point around the screen changing the original triangle as C moves. Compare the values of alpha and delta as point C changes locations.
Step 2. / Now lets move point C back to the location (0,4).
Step 3. / Select the last icon in the menu bar that looks like four arrows pointing up, down, left and right. This icon will allow us to move the screen to a different view so we can see more of the x-axis to the left.
Step 4. / Move your cursor to the figure and notice the cursor has changed from an arrow to a hand. Click and drag the figure until you can see more of the x-axis to the left of point A and you can still see point D. Release the point and drag to stop moving the figure.
Step 5. / Select the first icon in the menu bar to get your arrow cursor back.
Step 6. / Place your cursor on point C and click and drag the point in an arc towards the line containing points A, B and D. As you slowly move point C, notice the values for alpha and delta. What is the biggest value you can get for alpha without going past the line containing points A, B and D? What is the value of delta when alpha is at its greatest value?