# Geometry

We are surrounded by space. And that space contains lots of things. And these things have shapes. In geometry we are concerned with the nature of these shapes, how we define them, and what they teach us about the world at large--from math to architecture to biology to astronomy (and everything in between). Learning geometry is about more than just taking your medicine ("It's good for you!"), it's at the core of everything that exists--including you. Having said all that, some of the specific topics we'll cover include angles, intersecting lines, right triangles, perimeter, area, volume, circles, triangles, quadrilaterals, analytic geometry, and geometric constructions. Wow. That's a lot. To summarize: it's difficult to imagine any area of math that is more widely used than geometry.

## Tools of geometry

- Introduction to Euclidean geometry
- Lines, line segments, and rays
- Points, lines, and planes
- Geometric definitions
- The golden ratio

## Angles and intersecting lines

This topic continues our journey through the world of Euclid by helping us understand angles and how they can relate to each other.

- Angle basics and measurement
- Angle types
- Vertical angles
- Complementary and supplementary angles
- Angles between intersecting and parallel lines
- Angles with triangles and polygons

## Special properties and parts of triangles

You probably like triangles. You think they are useful. They show up a lot. What you'll see in this topic is that they are far more magical and mystical than you ever imagined!

- Triangle inequality theorem
- Perpendicular bisectors
- Angle bisectors
- Medians and centroids
- Altitudes
- Bringing it all together

## Quadrilaterals

## Transformations

In this topic you will learn about the most useful math concept for creating video game graphics: geometric transformations, specifically translations, rotations, reflections, and dilations.
You will learn how to perform the transformations, and how to map one figure into another using these transformations.

- Introduction to rigid transformations
- Translations
- Rotations
- Reflections
- Dilations or scaling around a point
- Sequences of transformations

## Congruence

Learn what it means for two figures to be congruent, and how to determine whether two figures are congruent or not. Use this immensely important concept to prove various geometric theorems about triangles and parallelograms.

- Transformations and congruence
- Triangle congruence
- Theorems concerning triangle properties
- Working with triangles
- Theorems concerning quadrilateral properties
- Proofs of general theorems that use triangle congruence

## Similarity

Learn what it means for two figures to be similar, and how to determine whether two figures are similar or not. Use this concept to prove geometric theorems and solve some problems with polygons.

- Definitions of similarity
- Introduction to triangle similarity
- Solving similar triangles
- Angle bisector theorem
- Solving problems with similar and congruent triangles
- Solving modeling problems with similar and congruent triangles

## Right triangles and trigonometry

Triangles are not always right (although they are never wrong), but when they are it opens up an exciting world of possibilities. Not only are right triangles cool in their own right (pun intended), they are the basis of very important ideas in analytic geometry (the distance between two points in space) and trigonometry.

- Pythagorean theorem
- Pythagorean theorem proofs
- Pursuit of a Pythagorean proof
- Special right triangles
- Introduction to the trigonometric ratios
- Solving for a side in a right triangle using the trigonometric ratios

## Circles

Explore, prove, and apply important properties of circles that have to do with things like arc length, radians, inscribed angles, and tangents.

## Perimeter, area, and volume

A broad set of tutorials covering perimeter area and volume with and without algebra.

- Basic area
- Advanced perimeter and area
- Area formula proofs
- Advanced area with triangles
- Area and circumference of circles
- Surface area

## Analytic geometry

In analytic geometry, also known as coordinate geometry, we think about geometric objects on the coordinate plane. For example, we can see that opposite sides of a parallelogram are parallel by by writing a linear equation for each side and seeing that the slopes are the same.

- Distance and midpoints
- Problem solving with distance on the coordinate plane
- Dividing line segments
- Parallel and perpendicular lines on the coordinate plane
- Equations of parallel and perpendicular lines
- Coordinate plane proofs

## Geometric constructions

We now have fancy computers to help us perfectly draw things, but have you ever wondered how people drew perfect circles or angle bisectors or perpendicular bisectors back in the day. Well this tutorial will have you doing just as your grandparents did (actually, a little different since you'll still be using a computer to draw circles and lines with a virtual compass and straightedge).

- Constructing bisectors of lines and angles
- Constructing regular polygons inscribed in circles
- Constructing circumcircles and incircles
- Constructing a line tangent to a circle