## SAT SUBJECT TEST MATH LEVEL 1

## PLANE GEOMETRY

## CHAPTER 10

Quadrilaterals and Other Polygons

• The Angles of a Polygon

• Special Quadrilaterals

• Perimeter and Area of Quadrilaterals

• Exercises

• Answers Explained

**A polygon** is a closed geometric figure made up of line segments. The line segments are called

**, and the endpoints of the line segments are called**

*sides***(each one is called a**

*vertices***). Line segments inside the polygon drawn from one vertex to another are called**

*vertex*

*diagonals**.*

Three-sided polygons, called triangles, were discussed in Chapter 9. Although in this section our main focus will be on four-sided polygons, which are called ** quadrilaterals**, we will discuss other polygons as well. There are special names for many polygons with more than four sides. The ones you need to know for the Math 1 test are given in the following chart.

A ** regular polygon** is a polygon in which all the sides have the same length and all the angles have the same measure. A regular three-sided polygon is an equilateral triangle, and, as we shall see, a regular quadrilateral is a square. Pictured below are a regular pentagon, regular hexagon, and regular octagon.

pentagon

hexagon

octagon

**Remember**

In a *regular* polygon, all the angles are congruent and all the sides are congruent.

### THE ANGLES OF A POLYGON

A diagonal of a quadrilateral divides it into two triangles. Since the sum of the measures of the three angles in each of the triangles is 180°, the sum of the measures of the angles in the quadrilateral is 360°.

*a* + *b* + *c* = 180 and *d* + *e* + *f* = 180*a* + (*b* + *e* ) + (*c* + *d* ) + *f* = 360

**Key Fact I1**

**In any quadrilateral, the sum of the measures of the four angles is 360°.**

Similarly, any polygon can be divided into triangles by drawing in all of the diagonals emanating from one vertex.

pentagon

hexagon

octagon

Notice that a five-sided polygon can be divided into three triangles, and a six-sided polygon can be divided into four triangles. In general, an *n*-sided polygon can be divided into (*n* – 2) triangles, which leads to KEY FACT I2.

**Key Fact I2**

**The sum of the measures of the n**

**angles in a polygon with**

*n***sides is (**–

*n***2)**

**180°.**

**EXAMPLE 1:** To find the measure of each angle of a regular octagon, first use KEY FACT I2 to get that the sum of all eight angles is (8 – 2) 180° = 6 180° = 1,080°. Then since in a regular octagon all eight angles have the same measure, the measure of each one is 1,080° 8 = 135°.

An ** exterior angle** of a polygon is formed by extending a side. Surprisingly, in all polygons, the sum of the measures of the exterior angles is the same.

**Key Fact I3**

**In any polygon, the sum of the measures of the exterior angles, taking one at each vertex, is 360°.**

*x* + *y* + *z* = 360

*a* + *b* + *c* + *d* + *e* = 360

**EXAMPLE 2:** KEY FACT I3 gives us an alternative method of calculating the measure of each angle in a regular polygon. In Example 1 we used KEY FACT I2 to find the measure of each angle in a regular octagon. By KEY FACT I3, the sum of the measures of the eight exterior angles of any octagon is 360°. As a result, in a regular octagon, the measure of each exterior angle is 360° 8 = 45°. Therefore, the measure of each interior angle is 180° – 45° = 135°.