Tour 5.2: Exterior Angle Theorem and Parallel Lines

Tour 5.2: Exterior Angle Theorem and Parallel Lines

(there is no writing on this paper—please return it when you are finished)

DEF: An exterior angle of a triangle is formed whenever a side of the triangle is extended to form an angle supplementary to the adjacent interior angle (see ÐABC below).

Constructing a Triangle with an exterior angle

1. Start Sketchpad if it isn’t already running. If it is running, choose New Sketch from the File menu.

2. Use the Segment tool to draw a segment.

3. Construct a second segment that shares one endpoint with the first.

4. Construct another segment to complete the triangle with an exterior angle.

5. Control-Click on each of the points and choose Label Point to name each of your points as shown above.

Measuring the Angles

Now that you have a triangle, we need to measure the angles. To do this, you must select three points successively with the vertex as the second point chosen.

6. Hit Esc twice to deselect any objects. Then select points A, B, D in that order.

7. Choose Angle from the Measure menu. (mÐABD=### should show up.)

8. Use steps 6 & 7 to find mÐBAC and mÐACB.

9. Move the angle measures to place them close to each angle.

Dragging and Measuring to Confirm Your Conjecture

Now simply by looking at what you have, you cannot prove anything except in the specific case on your screen. In fact, as long as we are using numbers, you cannot prove anything unless you look at every possible case (and that is impossible!). We can, however, look at many cases to come upon an unproven conclusion.

10. Hit Esc twice to deselect any objects. Then move the four different points around to create a variety of different types of triangles.

QUESTION: What do you notice about the exterior angle measure compared to either of the remote interior angles?

We can look at several cases at once in a table.

11. Hit Esc twice to deselect any objects. Then select the three angle measures and choose Tabulate from the Graph menu. (A small table should show up.)

12. Hit Esc twice to deselect any objects. Then select the four points and choose EditàAction ButtonsàAnimation. Select OK from the menu. (A button labeled “Animate Points” should appear.)

13. Practice turning the Animate Points button on and off to watch your triangle “dance”.

14. Control-Click on your table and choose Add Table Data.

15. Select the second option and click OK.

16. Animate the points and watch the table grow.

QUESTION: As you analyze your table, what do you notice about the exterior angle measure compared to either of the remote interior angles at any given time?

Creating a Caption and Using the Text Palette

To finish things off, let’s create a caption for this sketch so that others who see it will know what’s going on.

17. Choose the Text tool from the Toolbox.

18. Double-click in a blank part of the sketch plane to start a new caption.

19. Use your keyboard to type the caption, stating your conjecture about the exterior angle compared to the remote interior angles.

Click outside the caption to close it. You may have noticed a new set of tools when working with captions (or other text objects). This is called the Text Palette. You can use the Text Palette to apply formatting options—such as italics, font size, font color, and more advanced mathematical formatting—to text. (You can manually show or hide the Text Palette using Show Text Palette or Hide Text Palette from the Display menu.)

20. Use the Text Palette to change the font, text color, and other available settings of the caption. Changes will affect the entire caption because it’s currently highlighted. Using the Text tool, click and drag within the caption to highlight a range of text and apply changes just to that text.

Looking at a proof

As stated before, what you have shown is not a proof, even if you are able to find a conjecture that appears to be true. Look in your textbook, p216, to see a proof of this theorem (Theorem 30).

Making Pages

This document will have a total of seven pages.

21. Choose Document Options from the File menu. In the dialog box that appears, choose Blank Page from the Add Page popup menu. Do this six times so you have a total of 7 pages. Use the Page Name field to name the first page THM30, the second page THM31, and so on to match up with Theorems 30-36 in section 5.2 of your book.

22. You can now click on the tabs in the bottom left corner to move from page to page.

Theorems 31 to 33

We will now build a pair of parallel lines by looking at alternate interior angles. Before you begin, look back over your notes to remind yourself of what alternate interior angles look like (they are also defined on p193).

23. Draw a short segment on your screen.

24. Hold down the mouse button on the segment tool to select the extended line tool.

25. Click on the first segment, and then finish drawing a line that crosses the first

line. (This new line will be the transversal.)

26. Add two additional points (one to each of the lines as shown in the picture to the right).

27. Hit Esc twice to deselect any objects. Then select the new point on the transversal.

With this point selected, choose the Mark Center command from the Transform menu.

(The point should flash!)

28. Select the original line, the point of intersection, and the new point on the original segment. Select the Rotate tool from the Transform menu. Change the angle to 180 and select Rotate. This will flip a copy of your angle from the bottom to the top.

29. Hit Esc twice to deselect any objects. Then select the five points and choose Label Points from the Display menu. Rename each of the points so they are the same as the picture to the right.

29. Move the points around to see the two angles move together.

30. To confirm that the angles are indeed the same, measure both of these alternate interior angles (ÐABC & ÐEDC. Use steps #6-8 if you need a reminder of how to do this.).

QUESTION: You created your picture by setting alternate interior angles congruent. What do you notice to be true about the lines ED and BA?

31. Move your points around to see if your conjecture appears to hold true. Add a textbox that states your hypothesis in “If…then…” form. Check on p 217 to see if your hypothesis is correct. Read the proof of this theorem (Theorem 31). Does this make sense?

32. Theorem 32 is very similar. Look at the picture on p 217. Can you create this image on the THM32 page of your work? The steps are very similar to Theorem 31, but you are working with Ð’s 1 and 8. Add a textbox to this page explaining your hypothesis as well.

Theorem 33 also has a similar picture. However, to create this one, we will need to use a new tool.

33. On the THM33 page, complete steps 23-26. Label points A, B, & C as shown.

34. Hit Esc twice to deselect any objects. Select B, then C in that order. Then select Mark Vector from the Transform menu. (A dotted line should flash from B to C.)

35. Select points C, A, and the BA segment (but not point B). Choose the Translate tool from the Transform menu. Label points D and E as shown.

36. Measure your two corresponding angles.

37. Use a textbox to write your hypothesis.

Theorems 34 & 35

These two theorems are simple variations of 31 to 33. For these we will worry less about an accurate construction of the picture, but write out simple paragraph proofs.

38. Copy and paste the picture from Theorem 31 onto page THM34.

39. Measure the appropriate angles (you may need to add a point. Use your book p218 for guidance).

40. Use a textbox to write a simple paragraph proof explaining how this theorem is basically the same as the Alternate Interior Angle Theorem. (It should only take 3-5 sentences).

41. Repeat this process for page THM35.

Perpendicular Lines Theorem

For this theorem we are given a line with two lines that are perpendicular. We can use another new tool to demonstrate this.

42. On the THM36 page create an extended line across the screen.

43. Add two additional points on your line. Label these points A & B.

44. Select points A & B and the line. Select the Perpendicular Lines tool from the Construct menu.

This theorem may be so obvious that it may seem trivial to even discuss it, much less prove it. However, it can easily be proven using the Corresponding Angles Theorem.

45. Add a textbox with a short paragraph proof of Theorem 36.

Congratulations! You have now worked your way through demonstrating and proving all of the theorems from section 5.2.

Saving

Once you have finished, save your file in the dropbox. (Group Shared/Student/Nussbaum/Drop Box)

Find Mr. N’s dropbox. Make sure you save it with the following filename:

Class Number_Last Name_First Name_5.2 ExtAng&Parallel.gsp