# Two-Dimensional (plane)

## Conditional Statements

**Logic** is a big part of geometry and although one does not need to know formal logic in order to do geometry, it helps to start geometry by starting to think about logic and how it works. Geometry logic starts with conditional statements.

## Using Coordinates in Geometry

Although we often like to think of algebra and geometry as different subjects, they are very much related and being able to use algebra in geometry (and vice versa) can be very helpful when it comes to solving more advanced problems!

As we know from algebra, coordinates (x,y) are points on a plane. Two points (two sets of coordinates) make up a line (and have a slope, remember: $m=\dfrac{y_2-y_1}{x_2-x_1}$?).

## Similar Triangles

Triangles that are **congruent** are exactly the same. If $\triangle ABC \cong \triangle DEF$ (this is the notation that shows that triangles are congruent), then all of the sides and angles in $\triangle ABC$ are equal to all of the corresponding sides and angles in $\triangle DEF$.

## Properties of Right Triangles

Right triangles are special. They are also useful. So much of geometry is taking a tiny bit of information and using it to generate all kinds of other information (e.g., the measure of one angle telling you the measure of all of the other angles in the figure!). Right triangles are super helpful in this process for two reasons:

## Using Circles to Find Angle Measures

Another common theme in geometry problems is circles. Questions will often ask you to figure out the degrees of angles that are embedded in circles. The main rule you want to remember is: **Whenever a set of angles forms a circle, they add up to 360° (every circle is 360° around).**

**Other important rules for Circle problems:**

## Finding Angle Measures with Parallel Lines

The final set of angle rules that commonly come up in tests are parallel line rules.

It is important to remember that whenever two lines are crossed by a transversal, specific angles (corresponding, alternate interior, alternate exterior) are created. However, those angles are EQUAL when the lines cut by the transversal are parallel. First, some vocab:

## Finding Angle Measures

Geometry involves scores of rules about shapes, angles, and figures. Once you've memorized those rules, the trickiest part of geometry is figuring out how to put different rules together to solve a problem.

Some of the most common types of questions that tests use to test students geometry knowledge ask about angles. There are many rules about angles in geometry and good geometry students learn to use lots of those rules in a single problem.

## Circles: Area

The area of a shape is the size of the area inside the boundaries of that two-dimensional shape (once you start working with three dimensional shapes, area will refer to the two-dimensional area of one of the faces of that shape).

## Circle: Circumference

The distance around the outside of a circle is called the circumference (sometimes we think of it as the perimeter of a circle).

The formula for the circumference of a circle can be written in two ways:** $d\pi$** or **$2r\pi$**

## Circle (Diameter v. Radius)

We usually measure the size of circles by either their diameter or their radius. If you buy a circular rug, for example, and check its measurements, they will almost certainly report either the diameter or radius of the rug. Everyone once in a while you'll see a circular object defined by its **circumference, which is the distance around the outside of the circle**.