## Learning Math: Geometry

# Polygons Homework

## 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. Scroll down to Solutions to see the filled-in table.

Type of Quadrilateral |
Do diagonals bisect each other? |
Are diagonals congruent? |
Are diagonals perpendicular? |

Trapezoid | |||

Isosceles Trapezoid | |||

Parallelogram | |||

Rhombus | |||

Rectangle | |||

Square | |||

Kite |

### 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.
- Each segment touches exactly two other segments, one at each of its endpoints.

### 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.

### Suggested Reading

**Senechal, Marjorie (1990). Shape. In On the Shoulders of Giants: New Approaches to Numeracy. Edited by Lynn Arthur Steen (pp. 139-148). Washington, D.C.: National Academy Press.**

Reproduced with permission from the publisher. Copyright © 1990 by National Academy Press. All rights reserved.

Download PDF File:

Shape

Continued…

### Notes

**Note 6**

Some people define trapezoid as “at least one pair of parallel sides” and others, as used here, as “exactly one pair of parallel sides.” It’s a matter of taste; there are no strong reasons to do it one way instead of the other, hence the differing opinions.

**Note 7**

Another, more common definition of an isosceles trapezoid is a trapezoid with the nonparallel sides congruent.

### Solutions

**Problem H1**

Answers will vary.

**Problem H2**

Type of Quadrilateral |
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**

**a.**

**b.**

**c.**

**Problem H4**

Our definition requires that polygons divide the plane into an “interior” and an “exterior” region, which is not the case with a “flat” polygon. Also, the segment AB coincides with AC along its entire length; they do not just meet at an endpoint. (Likewise for BC and AC.)

**Problem H5**

Answers may vary, but one reason that “flat” polygons would not be considered “polygons” is that they would be indistinguishable from one another or from a line segment.