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# Unit 1: Two-Dimensional Geometry

Here at Shmoop, we like our geometry the way we like our soda: flat and full of sharp angles.

Wait a second, that can't be right. We like our geometry to be cherry flavored.

Uh, no, not that either. Maybe comparing geometry to soda was a bad idea. Anyway, 65% of the Geometry EOC will cover 2-D shapes: y'know, the flat ones that you can cut out of construction paper.

We'll start by talking about the basic building blocks of two-dimensional geometric shapes: points, lines, angles, and planes. We then use those to make polygons, including regular polygons such as quadrilaterals (those of the four sides), triangles (right, wrong, and otherwise), and circles (not actually a polygon, but roll with us here). And we'll do it all while keeping the subject from being too obtuse.

Except, of course, when we talk about angles.

Although we'll be covering a lot of different two-dimensional shapes, each with their own quirks, some issues are common to pretty much all of them.

**Angles and sides**, for one. If we know these, we understand (or can find) everything there is to know about a polygon. Just knowing what figure we're working with can tell us a lot about the sides and angles; a square will always have 90-degree angles, for instance. Sometimes we'll use one set of angles to find what another set are equal to, as with parallel lines. Other times, there will be some connection between the angles and the sides of a figure, as with certain right triangles, and we can find one from the other. Knowing the relationships between different angles and sides will be the difference between breaking down in tears on the first page of the EOC and never breaking a sweat.

The **area** of a shape can be thought of as the number of 1 times 1 squares needed to fill up that shape. Normally filling something up with a lot of squares gets us a pretty sad party, but in geometry it gets us the area.

The **perimeter** of a shape is another way of thinking about its size, and it's the length of all the sides. The **circumference** of a circle is pretty much the same thing, except a circle has no sides, just a big curve. Which honestly makes them sound pretty different, but they're not.

The EOC draws its figures consistently, so it pays to pay attention to how they do it. Points on a line are labeled with CAPITAL LETTERS. For polygons, each **vertex** (i.e. pointy bit) and accompanying angle on a figure is labeled with a CAPITAL LETTER, while a lowercase letter is used to label the sides.

Sometimes, symbols are put on figures to show certain special relationships.

**Parallel lines** will never meet each other. An arrow, set into each line, symbolizes their relentless, unidirectional drive to always move forward.

The universal symbol for "these things are the same," also called **congruency**, is sticking a line through it. It works for both the sides of a figure and its angles.

A **right angle** is exactly 90 degrees, no more, no less. If an angle is right, it will have a little box in the corner. This next part is very important: if there is no box, then you can't assume that angle is a right angle, no matter how much it may look like one. You can't eyeball the rightness of an angle on the EOC.

A **proof** in geometry is a way to show that some statement is always true, if you start off already knowing some basic facts about the situation. It is set up like as formal argument, which is like a normal argument but with less swearing and thrown furniture.

We start off with the **given**, which is one or more statements that we assume to be true, because they are given (to us). We are also told what needs to be **proved** (by us). We then build a logical sequence of statements, using the information we already have to move closer and closer to what we want to prove, while showing that every new statement must be true given what we already know.

At the end of it all, we have an unassailable, rock-solid proof for our conclusion, which isn't too shabby.