Draw any quadrilateral. Then draw a point anywhere inside the quadrilateral, and connect that point to each of the vertices, as shown below:
Now answer the following questions:
How many triangles have been formed?
What is the total sum of the angle measures of all the triangles?
How much of the total sum from part (b) is represented by the angles around the center point (i.e., what is their sum)?
How much of the total sum from part (b) is represented by the interior angles of the quadrilateral?
Repeat the activity with a five-sided polygon and an eight-sided polygon, and then attempt to generalize your result to an n-sided polygon. Note 8
Estimate the number of degrees between two adjacent legs of the starfish below. Then, using a protractor, measure one of the angles. (You may want to print the PDF image in order to do this.) How close was your estimate?
Print the figures below from the PDF document. For each figure, cut out a and e. When are a and e congruent? What other angles have the same measure as a? As e?
How many angles can be formed with the rays below?
Look at all the possible combinations. For example, three rays can form three distinct angles. Do you see them? Close Tip
Predict the number of angles formed with seven rays and with 10 rays. Can you generalize your prediction to n rays? Note 9
Write a series of commands in the style used in the Interactive Activity (or Geo-Logo commands) to draw a regular decagon (a 10-sided figure). You can test your commands using the Interactive Activity. How many total degrees did you turn to make the decagon?
A sled got lost in the darkness of a polar night. Mayday emergency calls were received all night, but the darkness prohibited a search. The next morning, planes searched the area, and the pilots saw these tracks made by the sled:
Use turns to describe the route of the sled as if you had been in it.
If the sled continued in the same way, it might have returned to its starting point. How many turns would the sled have had to make to return to its starting point?
The pilot described the track as follows: "It looks like the sled made three equal turns to the right. The four parts of the track seem to be equally long, and the resulting angle between each part measures about 150 degrees."
What do you think the pilot means by "the resulting angle"?
How does the pilot's description differ from your own?
If you were to make a 40-degree turn on the sled, what would the resulting angle be? If the sled track forms a 130-degree resulting angle, what is the size of the turn?
You are interested in making a quilt like the one shown below. In the center, a star is made from six pieces of material: Note 10
Is it possible to make the star with the piece below? Why or why not?
Print and cut out several copies of this image from the PDF document.
What about the highlighted piece below -- could it be used to make the star in the quilt? Without measuring, determine the measures of angles A, B, and C, and explain how you arrived at your solution.