Introduction

In Chapter 5 we considered the geometric properties of triangles, whose sides are composed of three straight line segments. In this chapter we expand the topic to more general geometric figures whose sides are straight line segments. These are called polygons.

Definitions

DEFINITION 1 A polygon is a geometric figure in the plane whose sides consist of straight line segments, and no two consecutive sides lie on the same straight line.

DEFINITION 2 An n-sided polygon is called an n-gon. The most common n-gons have special names:

1. • A 3-gon is a triangle;

2. • A 4-gon is a quadrilateral;

3. • A 5-gon is a pentagon;

4. • A 6-gon is a hexagon;

5. • A 7-gon is a heptagon;

6. • An 8-gon is an octagon;

and so on.

When n > 3 we classify n-gons as either concave or convex. We will be primarily interested in convex polygons.

DEFINITION 3 A polygon is convex if every line segment between two points in the interior of the polygon is contained entirely within the polygon. A polygon that is not convex is said to be concave. When a polygon is concave the extension of some side of the polygon intersects some other side.

The most commonly seen n-gons are the convex polygons whose sides all have the same lengths and meet their adjacent sides at the same angle.

DEFINITION 4 A polygon that is equilateral and equiangular is said to be a regular polygon.

Results about Quadrilaterals

Some of the results we consider in this section will be special cases of the results in the general polygons section, but quadrilaterals are seen so frequently that it is good to have them for ready access.

DEFINITION 1 Quadrilaterals have some special definitions:

1. • A regular quadrilateral is a square;

2. • An equiangular quadrilateral is a rectangle;

3. • An equilateral quadrilateral is a rhombus;

4. • A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

5. • A quadrilateral with two sides parallel is a trapezoid.

– If the two nonparallel sides of a trapezoid are have equal length it is isosceles.

From the definitions, we have the hierarchy of quadrilaterals shown as follows. Every statement about a specific quadrilateral is also true about those quadrilaterals that point to it.