Reading: EG Chapter 6

The goal of this problem is to perform some experiments to come up with some data. Then you should be able to come up with a conjecture and possibly and explanation.

• Make a paper cone. Measure and record the cone angle.
• Then carefully with a straightedge, draw a triangle on the cone. Label the vertices. Measure with a protractor the 3 angles of the triangle and record the 3 angles and the angle sum in a table.
• On the same cone (or an identical one), repeat this experiment at least 5 times and compare your results. You should include some examples where the cone point is inside the triangle and some where the cone point is outside. Note in your table whether the triangle has the cone point inside or outside.
• What relationship in the angle sums do you observe among the triangles where the cone point is outside? What relationship when the cone point is inside?

Do the EXPERIMENT ON MODELS above for at least 3 cones. Do a nice, careful job. If the cones are too small, the angles will be hard to measure.

Then, write a CONJECTURE that tells for a cone with cone angle A, what is the angle sum for a triangle with the cone point inside the triangle and what is the angle sum when the cone point is outside.

Finally, write your best EXPLANATION for why your conjecture is true.

Bring to class your models and the full written set of tables, your conjecture and your explanation. Write clearly or type.

This problem is similar to Problem 2.1, except that the triangles will be on cubes or rectangular solid.

• First, make a model of a cube or find a cardboard box (which is a rectangular solid) that you can draw on. Then proceed much as in problem 1. Draw triangles on the box and label the vertices.
• Measure the angles of the triangle and record the angle sum in a table. Also in the table record the number of corners of the box inside the triangle.
• For your experiments, draw at least two triangles that contain no corners in the interior, two that contain 1 corner in the interior, two that contain 2 corners in the interior. Also draw one triangle with 3 corners in the interior and one with 4 corners in the interior. So you will have eight triangles recorded.

Do the EXPERIMENT ON ONE MODEL for the 8 triangles as described above. Do a nice, careful job. Choose a box big enough so that the angles are not too hard to measure.

Write an analysis of your results. Are the angle sums the same if the number of corners inside is the same? Write a CONJECTURE describing how the angle sum depends on the number of corners.

For the case of 1 corner, explain the connection between the one-corner case and the cone experiments of Problem 2.1.

Finally, write your best EXPLANATION for why your conjecture is true.

Bring to class your models and the full written set of tables, your conjecture and your explanation. Write clearly or type.

Consider a triangle ABC on a cube. Notice that the triangle divides the surface of the cube into 2 pieces T1 and T2. If you think about it, you can consider either piece the interior of a triangle with vertices ABC. If you make T1 the interior, the angle at A will be t and for the other choice T2, the angle is 360 - t.

• If the angle sum of triangle ABC is x with one choice of interior, tell what is the angle sum for the other choice. Show your work.
• Assuming that none of A, B, C are corner points, if the number of corners in interior T1is n, tell what is the number in interior T2.
• Reexamine your data from Problem 2.2. Make another table in which you use the same triangles but record the angle sum for the other choice of interior and also the number of corners inside for that choice of interior. Notice that you DO NOT have to make more measurements. You can compute it all from the table data.
• Re-examine your conjecture to show it fits with this new data.

Bring to class and turn in this written work. Write clearly or type.

Draw 3 "random" great circles on a sphere. The sphere surface is divided into disjoint triangles. How many triangles are there? Are some of the triangles congruent? How many different triangle shapes are there in general? Suppose one of the triangles is ABC with angles measuring a, b, c. Then there are two triangles with edge BC in common. What kind of shape do the two triangles form when they are fit together? What is the sum of the areas of these two triangles (suppose the sphere has area S, so the answer should be in terms of a, b, c and/or S).