Session 3:
Homework

A quadrilateral is a polygon that has exactly four sides. Here are some definitions of special types of quadrilaterals:

 • A trapezoid has one pair of opposite sides parallel. Note 6 • An isosceles trapezoid is a trapezoid that has congruent base angles. (The base angles are the two angles at either end of one of the parallel sides.) Note 7 • A parallelogram has two pairs of opposite sides parallel. • A rhombus has all four sides congruent (the same length). • A rectangle has three right angles. • A square is a rhombus with one right angle. • A kite has two pairs of adjacent sides congruent (the same length).

Problem H1

Go through the process of understanding a definition for each of the quadrilaterals named above. Does the given definition define the figures as you know them? For each type of quadrilateral, try to list alternative definitions, or at least several properties of the figures. (Think about sides, angles, diagonals, and so on.)

Problem H2

Fill in the chart below with yes, no, or maybe in each cell. When you click "Show Answers," the filled-in table will appear below the problem. Scroll down the page to see it.

Do diagonals bisect each other?

Are diagonals congruent?

Are diagonals perpendicular?

 Trapezoid Isosceles Trapezoid Parallelogram Rhombus Rectangle Square Kite

Do diagonals bisect each other?

Are diagonals congruent?

Are diagonals perpendicular?

 Trapezoid No Maybe Maybe Isosceles Trapezoid No Yes Maybe Parallelogram Yes Maybe Maybe Rhombus Yes Maybe Yes Rectangle Yes Yes Maybe Square Yes Yes Yes Kite Maybe Maybe Yes

Problem H3

Put the following quadrilaterals in the appropriate spaces in each Venn diagram to indicate properties these types of quadrilaterals must have: trapezoid, isosceles trapezoid, parallelogram, rhombus, rectangle, square, and kite.

 a. b. c.

Mathematicians often define things in a way that makes other work more convenient. For example, the definition for a prime number specifically excludes the number 1 as a prime. (A prime number is an integer whose only factors are itself and 1. A prime number has exactly two factors.) Why is the number 1 not a prime, when it fits the rest of the definition very well? Because a lot of proofs work based on the "fundamental theorem of arithmetic": Every number can be uniquely factored into primes (where a different order of the primes does not count as factoring differently). Suppose the number 1 were a prime. Then you could have all of these factorizations for the number 6:

6 = 3 • 2
6 = 3 • 2 • 1
6 = 3 • 2 • 1 • 1 • 1 • 1 • 1 • 1

Recall our definition of a polygon: Polygons are two-dimensional geometric figures with these characteristics:

 • They are made of straight line segments. • Each segment touches exactly two other segments, one at each of its endpoints. • Polygons are closed -- they divide the plane into two distinct regions, one "inside" and the other "outside" the polygon.

 Problem H4 Explain why our definition does not allow for "flat" polygons like the one shown here. What part(s) of the definition fails? "Flat Triangle" ABC, with one 180° angle and two 0° angles.

 Problem H5 Write down some reasons why we would not want to consider figures like the one above polygons.