Course:Math 124 F07 Bart Homework 1

Homework 1 MT-A124 Math and the Art of Escher

Due Wednesday September 12

Instructions: On this assignment collaborative work is permitted. You may discuss the problems with other students, but the work you turn in must be your own (i.e. written up by you). You may of course use hints and suggestions given during class or office hours. The solutions you turn in must be

• Neatly written up (using complete sentences), on one side of each page.
• Labeled appropriately with the problem number.
• Stapled (if there is more than one page).

Problems:

1. Draw good examples of the following quadrilaterals: a kite, a parallelogram, a rhombus, a rectangle and a trapezoid.
2. The rectangle 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. Hint: Use the Corresponding Angles Converse Postulate: If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.
4. True/False:
   • Every square is a rectangle.
   • Every equilateral triangle is an isosceles triangle.
   • Every rhombus is a parallelogram.
   • The sum of the angles in a triangle is 180 degrees.
   • Every diagonal bisects a quadrilateral into two triangles.
   • 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 about that midpoint. What shapes do you get? Show that this rotation always results in a parallelogram.