Polygon Exercises

1. Carefully draw several distinct examples of each of the following quadrilaterals: a kite, a parallelogram, a rhombus, a rectangle and a trapezoid. Try to make the shapes as varied as possible. Think about how varied you can make the shapes and still satisfy the definition.
2. The square is a subclass of which other groups of quadrilaterals given in problem 1?
3. Show that if the opposite angles in a quadrilateral have equal measure, then that quadrilateral has to be a parallelogram.
4. True or False:
   1. Every square is a rectangle.
   2. Every equilateral triangle is an isosceles triangle.
   3. Every rhombus is a parallelogram.
   4. The sum of the angles in a triangle is 180°.
   5. Every diagonal bisects a quadrilateral into two triangles.
   6. There exists a diagonal that bisects a quadrilateral into two triangles.
5. Draw an equilateral triangle, and draw the lines of symmetry. How many lines of symmetry are there?
6. Draw a right triangle, an acute triangle and an obtuse triangle. Find the midpoint of each of the sides, and connect the midpoints. (This should create 4 smaller triangles.) Now repeat this subdivision for the three smaller triangles that contain one of the original vertices. (i.e. subdivide the three "corner triangles"). What other triangles are the small triangles congruent to? What other triangles are the small triangles similar to? (Remember that congruent means that they should have the same size and shape. Similar means that they should have the same shape, but can be of different size.)
7. Draw a right triangle, an acute triangle and an obtuse triangle. Pick one of the sides, and rotate the triangle 180° about that midpoint. What shapes do you get? Show that this rotation always results in a parallelogram.
8. Using paper, make copies (pattern blocks) of each of the polygons indicated. In each case, see if there is a tessellation. If there is, sketch it.
   1. Isosceles triangles that are not equilateral.
   2. Scalene triangles.
9. Which triangles can tessellate a plane? Give a conjecture based on what you have seen in experiments. Are there some which will obviously tessellate? Which ones? Are there some which are harder to see but you think may work. Which ones will be the most difficult to determine if they tessellate or not? We will discover the full answer later.
10. In each of the following cases, see if there is a tessellation. If there is, sketch it
    1. Rectangles that are not squares.
    2. Parallelograms that are not rectangles or rhombuses.
    3. Kites.
    4. Trapezoids that are not isosceles.
    5. Other quadrilaterals (make one up).
11. Which quadrilaterals can tessellate the plane? Give a conjecture based on what you have seen in experiments. Are there some which will obviously tessellate? Which ones? Are there some which are harder to see but you think may work. Which ones will be the most difficult to determine if they tessellate or not? We will discover the full answer later.
12. Create a geometric tessellation by triangles, then make a drawing inside the triangles to make it look more Escher-like.
13. Which of these shapes are convex?

    a.

    b.

    c.

    d.

    e.

14. Below are three early cubist paintings (beginning of the twentieth century). When answering the following questions think about how these paintings are similar and how they are different. The intent is to discuss the paintings using the terminology we have developed for geometry.
    1. What geometric shapes do you recognize in Braque's Woman and Guitar?
    2. What geometric shapes do you recognize in Picasso's Ambrose Vollard?
    3. What geometric shapes do you recognize in Metzinger’s Cubist Landscape?
    • Early Cubist Paintings
    • 
      Braque (1913)Woman and Guitar
    • 
      Picasso (1910) Ambrose Vollard
    • 
      Metzinger (1911) Cubist Landscape

Instructor:Polygon Exercises Solutions (restricted access)