### Exterior Angles

Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees.

### Similar Polygons

If two polygons are similar, their corresponding angles are equal and the ratios of the corresponding sides are in proportion.

ExampleThese two polygons are similar because their angles are equal and the ratios of the corresponding sides are in proportion.

### Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides.

In this figure, line *AB* || *CD* and *BC* || *AD*.

A parallelogram has:

- opposite sides that are equal (
*AB*=*CD*and*BC*=**AD**) - opposite angles that are equal (m
*a*= m*c*and m*b*= m*d*) - and consecutive angles that are supplementary (m
*a*+ m*b*= 180°, m*b*+ m*c*= 180°, m*c*+ m*d*= 180°,m*d*+ m*a*= 180°)

### Special Types of Parallelograms

- A
**rectangle**is a parallelogram that has four right angles. - A
**rhombus**is a parallelogram that has four equal sides. - A
**square**is a parallelogram in which all angles are equal to 90 degrees and all sides are equal to each other.

### Diagonals

In all parallelograms, **diagonals** cut each other into two equal halves.

- In a rectangle, diagonals are the same length.
- In a rhombus, diagonals intersect to form 90-degree angles.
- In a square, diagonals have both the same length and intersect at 90-degree angles.

### Solid Figures, Perimeter, and Area

The GED provides you with several geometrical formulas. These formulas will be listed and explained in this section. It is important that you be able to recognize the figures by their names and to understand when to use which formulas. Don't worry. You do not have to memorize these formulas. They will be provided for you on the exam.

To begin, it is necessary to explain five kinds of measurement:

**Perimeter****Area****Volume****Surface Area****Circumference**

The perimeter of an object is simply the sum of all of its sides.

Area is the space inside of the lines defining the shape.

Volume is a measurement of a three-dimensional object such as a cube or a rectangular solid. An easy way to envision volume is to think about filling an object with water. The volume measures how much water can fit inside.

The surface area of an object measures the area of each of its faces. The total surface area of a rectangular solid is double the sum of the areas of the three faces. For a cube, simply multiply the surface area of one of its sides by six.

Circumference is the measure of the distance around the outside of a circle.

### Coordinate Geometry

Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A **coordinate plane** is a grid of square boxes divided into four **quadrants** by both a **horizontal** (*x*) axis and a **vertical** (*y*) axis. These two axes intersect at one coordinate point, (0,0), the **origin**. A **coordinate point**, also called an **ordered pair**, is a specific point on the coordinate plane with the first number, or coordinate, representing the horizontal placement and the second number, or coordinate, representing the vertical. Coordinate points are given in the form of (*x, y*).

### Graphing Ordered Pairs

- The
*x*-coordinate is listed first in the ordered pair and tells you how many units to move to either the left or to the right. If the*x*-coordinate is positive, move to the right. If the*x*-coordinate is negative, move to the left. - The
*y*-coordinate is listed second and tells you how many units to move up or down. If the*y*-coordinate is positive, move up. If the*y*-coordinate is negative, move down.

The *x*-coordinate:

The *y*-coordinate:

ExampleGraph the following points: (2,3), (3,–2), (–2,3), and (–3,–2).

Notice that the graph is broken up into four quadrants with one point plotted in each one.

Here is a chart to indicate which quadrants contain which ordered pairs based on their signs:

### Lengths of Horizontal and Vertical Segments

Two points with the same *y*-coordinate lie on the same horizontal line and two points with the same *x*-coordinate lie on the same vertical line. The length of a horizontal or vertical segment can be found by taking the absolute value of the difference of the two points, or by counting the spaces on the graph between them.

ExampleFind the length of line

ABand lineBC.

### Midpoint

To find the midpoint of a segment, use the following formula:

Midpoint *x* =

Midpoint *y* =

ExampleFind the midpoint of line segment

AB.