Below are instructions for a detailed investigation, which follows a certain line of thinking to make a conclusion about the interior angles of a triangle, based on the 5th Postulate of Euclid. If you'd prefer a more free-form investigation, you can skip steps #1 - 7, and just click HERE for an open-ended problem to solve.

Constructing an Arbitrary Triangle:

1. Construct a line and a point not on the line, with the point P labeled as shown below.

Hide the two control points on the line, and use the Construct menu to construct two more points on the line. Label them Q and R. Test Q and R; when you drag each, they should move on the line, but you shouldn't move the line itself. Construct the segments from P to Q and from P to R.

2. Use the Construct menu to construct a line through P that is parallel to the line through Q and R. Use the Construct menu again to construct points A and B on this new line and points C and D on the original line. Drag each of the four points so that your diagram looks like Figure 2 below.

Test your dynamic sketch. It should represent an "arbitrary" triangle, one that could take on any appearance just by dragging P, Q, and/or R. Make sure that P can move anywhere in the plane, but the line through P should remain parallel to the original line, now containing C, Q, R, and D.

Using Euclid's 5th Postulate and Lab 4, #5:

3. Referring to Figure 2 above, since lines AB and QR are parallel, what must be true about angle APQ and angle CQP? What must be true about angle PQR and angle CQP? (You could write statements by filling in the blanks below.)

angle APQ _____ angle CQP _______________

angle PQR _____ angle CQP _______________

Putting the above two facts together, what must be true about angles PQR and APQ? _________________________________________________________

Using similar reasoning with angle DRP, what must be true about angles PRQ and BPR? _________________________________________________________

4. What must be true about angles APQ, QPR and BPR?

___________________________________________________________________

Interior Angles of a Triangle:

5. Finally, using the results of #3 and #4 above, what must be true about the interior angles of triangle PQR; i.e., angles PQR, QPR and PRQ?

___________________________________________________________________

6. Write a statement that is true about triangles, based on the above investigation.

___________________________________________________________________

___________________________________________________________________

7. (Bonus) How is the angle CQP in Figure 2 related to angles QPR and PRQ? Write a statement that is true about triangles, based on this observation.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Draw a triangle, and through one of the vertices construct a line parallel to the side of the triangle opposite that vertex. Using Euclid's 5th Postulate, draw a conclusion about the interior angles of any triangle.

___________________________________________________________________