# IM2: 5.1 How do you know that?

You probably know that the sum of the interior angles of any triangle is 180°. But an important question to ask yourself is, “How do you know that?” We know a lot of things because we accept it on authority—we believe what other people tell us; things such as the distance from the earth to the sun is 93,020,000 miles or that the population of the United States is growing about 1% each year. Other things are just defined to be so, such as the fact that there are 5,280 feet in a mile. Some things we accept as true based on experience or repeated experiments, such as the sun always rises in the east, or “I get grounded every time I stay out after midnight.” In mathematics we have more formal ways of deciding if something is true. Below, slide the "Slide Me!" button to see how the angles of the triangles relate. Feel free to change the size of the triangle by dragging the white dots around. Do this for several sizes of the triangle. Then answer the questions below.

## Triangle Angle Experiment

1. What does this experiment reveal about the sum of the interior angles of the triangles you cut out, and how does it do so?

2. Since you and your classmates have performed this experiment with several different triangles, does it guarantee that we will observe this same result for all triangles? Why or why not?

Perhaps a different experiment will be more convincing. See the triangle given below. Slide the two sliders in any order which rotates the triangle on one of the two midpoints. Again, feel free to change the size and shape of the triangle. Answer the questions below.

## Experiment #2

3. What does this experiment reveal about the sum of the interior angles of the triangles you rotated, and how does it do so?

4. Do you think you can rotate all triangles in the same way about the midpoints of its sides, and get the same results? Why or why not?

5. Will the second figure in the sequence always be a parallelogram? Why or why not?

6. Will the last figure in the sequence always be a trapezoid? Why or why not?