# Making Geometry fun *tolerable*

# Chris is a Stanford-educated tutor with over 10 years experience tutoring Geometry to students of all abilities, from students struggling to get from a C to a B, to go-getters trying to move an A- up to an A, to struggling students just hoping to pass. In that time he got a lot of experience learning how to explain this stuff in a way it actually makes sense to non-math people. Through his videos he has helped countless students, and he can do the same for you.

# Geometry

##### Whether you're taking a class called "Geometry" or you're just reviewing for an entrance or exit exam, below are all the topics you may need.

# Geometry Basics:

#### Videos covering the definition of midpoint, as well as how to find the coordinates of the midpoint of a line segment in the X-Y plane.

#### This video introduces the number line and explains how to use it to add and subtract negative numbers. It also explains how to determine if segments on the number line are congruent.

#### This video covers the Ruler Postulate, which is basically "geometry speak" for finding the distance between two points on the number line by subtracting their coordinates. Good stuff!

#### This video covers the Segment Addition Postulate, which basically says that if you break a segment into two parts, the lengths of the two parts are going to add up to the length of the original segment. Shocking!

# X-Y Stuff:

#### The videos in this topic introduce you to the X-Y plane (aka "rectangular coordinate system"), explain what the "origin" is, explain how to graph points, explain "the four quadrants", and tell you how to plot points.

#### If you're new to X-Y graphs, this is the place for you. We'll start with segments and lines without equations (for equations of lines from algebra, click here). Then we'll look at graphs of circles and rectangles, and x- and y-intercepts.

#### This chapter teaches you the formulas for finding the distance and midpoint between two points on the X-Y plane, as well as word problems using these formulas.

#### Slope-Intercept Form, Point-Slope Form, Standard Form, Vertical Lines, Horizontal Lines, Perpendicular Lines: in this chapter, we experience the splendor of all the different types of linear functions, and master the equations and graphing of each.

#### In this chapter we'll emphasize the similarities and differences between the equations of these four shapes, and we'll discuss why conic sections are called that.

#### These videos tell you what translations, rotations, and dilations are, as well as showing you the steps to do reflect a shape or function across a line using graphical techniques.

# Triangles:

#### This chapter covers the basic formula and how to use it on right triangles, as well as the most common mistake students make with it. Also included is a video about simplifying radicals, since that comes up a lot with Pythagorean.

#### This video covers all your basic triangle vocabulary, as well as that particular type of SAT question where they ask you if a triangle is "possible" with the given side lengths.

#### The 180 rule is one of the most important things you'll learn in geometry, and it will haunt you forever if you don't get it down! The exterior angle theorem, on the other hand, is almost useless and you'll never see it or it's "inequality" version ever again. Yet both may be on the test.

#### Once you get into the chapters of your book that cover congruent triangles, you're going to get so sick of congruent shapes that you'll be able to shoot your own video on them. But if you're just getting to congruence for the first time, these quick videos introduce the basics of congruent shapes and corresponding parts that you'll need so that you can understand what your teacher is talking about.

#### The chapter with every possible combination and permutation of S's and A's except for one: SSA. What are these good for, how do we use them in proofs, and why exactly is SSA not a thing? Prepare to enter the fray of congruent triangles!

#### Corresponding parts of corresponding triangles really are congruent. This isn't the first chapter that mentions corresponding parts, but this is where we'll get into how to use them in proofs.

#### Many geometry teachers don't cover all of these because they're redundant, right-triangle-only versions of all the congruence theorems you already know. The only one that every teacher covers is H-L.

#### Between the main theorem and the corollaries, you wouldn't believe how often you're going to come across this theorem throughout your academic career, from trig to physics to calculus to architecture and design. And the SAT!

#### The videos in this section cover a few different random topics in triangles that didn't really fit in anywhere else.

#### These videos cover a couple of theorems about the bisectors of angles and segments. I'll save you the suspense: both involve things being equidistant, and result in congruent triangles.

#### These are pretty confusing for everyone when you first see them, but then at some point you hopefully are given step-by-step instructions to make these easy. This is that moment for you!

#### These videos cover a few theorems that all involve "inequalities". In previous chapters, you were always trying to prove two triangles or segments congruent. In this one, you're trying to prove things unequal, whether it's saying that one side in a particular triangle is longer than another side, or that a side in one triangle is longer than a side in another.

#### These videos introduce the aspects of ratios and proportions that you will need whether you're in Geometry or you're about to take the SAT.

#### The videos in this chapter explain what similar triangles and similar polygons are, and how to solve various types of problems including: scale factors, missing sides, similar trapezoids. Also explained are which types of polygons are always similar (i.e. squares, equilateral triangles, special right triangles, etc).

#### This chapter covers the very specific type of triangle problem where you are supposed to solve for "x" in a right triangle where an altitude to the hypotenuse divides a right triangle into two smaller right triangles, both of which are similar to the original. It's very meta.

#### If you need to find missing sides of similar triangles, that is mostly covered in the similar polygons chapter videos. Instead, this chapter of videos on similar triangles focuses on similarity theorems and proofs for triangles. Also covered are right triangle problems involving the "geometric mean" of various sides and the "altitude to the hypotenuse" of right triangles.

# Parallel Lines & Quadrilaterals:

#### This section covers all those problems where a couple of parallel lines are crossed by another line (the transversal) and you're then supposed to figure out all the angles, and show which ones are congruent.

#### Theorems, problems, and proofs involving parallelograms.

#### These quadrilaterals are kind of like The Beatles: most people have only heard of three of them: squares, rectangles, and rhombuses (rhombusi). Kites, on the other hand, are only covered in some geometry classes. This chapter has theorems, problems and proofs for all of them, but if your class doesn't cover kites, you probably shouldn't use those theorems on your next test.

#### Trapezoids are technically quadrilaterals with a single pair of parallel sides, but the reality is that most of the problems you'll ever see are isosceles trapezoids. So in this chapter most of the problems and proofs are about isosceles trapezoids, but we also discuss the little trick questions where teachers and test makers throw in the weird non-standard trapezoids to try and throw you off.

# Circles:

#### The first video in this chapter covers the basis of the following vocab items: radius, diameter, circumference, chord, secant, tangent, arcs (major & minor), inscribed angles, Central angles, concentric. The second video covers some very basic proofs based on the fact that all the radii of a given circle are the same length.

#### The first video in this chapter covers basic circumference problems, where you need to get circumference when you know diameter or radius, or vice versa. Then there are two videos about pulleys and gears, car wheels and bike tires, how far and how fast a car is going when its tires rotate a certain speed. Finally we'll discuss what gear ratios are even for, and why your bike has 20 speeds in the first place.

#### These videos introduce arcs, arc length, central angles, and inscribed angles, then they go through lots of different examples involving solving for missing angles and arcs between.

#### These videos cover everything you could want to know about chords of circles, from solving for missing angles to doing proofs about equidistant chords.

#### Not to be confused with trig tangents, circle tangents are segments which sidle up alongside a circle and intersect it at exactly one point. This chapter covers the definition, problems and theorems, including how inscribed angle theorems apply to tangents.

#### Videos covering problems where two secants meet outside a circle, and the problem asks you to calculate the length of the segments, or the angle between them.

#### This chapter is super-important if you're ever planning on taking the SAT or another tough standardized test. You won't see sectors too much after Geometry class, but circles and circle-ish regions, especially "shaded regions" involving circles and triangles, are favorite topics for test-makers everywhere!

#### This video covers equations of circles in X-Y coordinates. This topic isn't covered by most geometry teachers, but if it's in your class, here you go. For equations of other conics, like ellipses and parabolas, check out the Algebra 2 page.

# Area & Volume:

#### Besides rectangles and squares, this chapter covers other basics of area such as how to convert units for area problems, such as from square cm to square meters (cm^2 to m^2) or square inches to square feet (in^2 to ft^2).

#### This topic includes videos covering just about every type of triangle area problem, all listed by type. Plus, a couple of videos that are going to SAVE YOUR BACON on every triangle problem you'll ever see, including on the SAT!

#### Examples and word problems using parallelograms (parallelagrams sp) and trapezoids.

#### Rhombuses unfortunately use the same area formula as kites -- A = ½ d

_{1}d_{2}. "Unfortunate" because you won't remember that formula a year or even 3 months from now when you need it on a big test! So in this chapter, whenever I work an area problem I'll also show you how to work it using parallelograms or triangles, since that's how you'll remember to do it long after the diagonals formula is lone gone from your mind.#### The first video in this chapter covers what an apothem is and how to find it. The second shows you how to use the apothem formula to find the area of regular polygons such as pentagons, hexagons, octagons, etc., and it also explains how to find the area of regular polygons a better way.

#### This chapter is super-important if you're ever planning on taking the SAT or another tough standardized test. You won't see sectors too much after Geometry class, but circles and circle-ish regions, especially "shaded regions" involving circles and triangles, are favorite topics for test-makers everywhere!

#### This chapter covers a particular type of problem, where they give you two similar 2-D or 3-D objects -- triangles or polygons or circles or squares or cylinders -- and expect you to figure out the scale factor, area, surface area and volume of one based on the other.

#### This chapter covers regular problems and word problems for how to find the volume of every type of solid: pyramids and cones, oblique and right prisms, spheres, and cylinders. Lots of word problems too, including unit conversions for units of volume: cubic inches, cubic meters, liters, etc.

#### These videos cover how to find the surface area of every type of solid: pyramids and cones, oblique and right prisms, spheres, and cylinders. Lots of word problems too.

# Trig, SohCahToa, & "Special" Triangles:

#### With this chapter we'll start trig off on the right foot -- triangles -- which has worked great for my tutoring students over the years. Also covered: what SohCahToa is (other than a weird abbreviation for the sinusoidal functions); what opposite, adjacent, and hypotenuse mean; how to find sine, cosine and tangent; and how to work a bunch of "solving triangles" problems.

#### In this chapter we'll get into the 30-60-90 and 45-45-90 triangles, with special emphasis on how to find their sides, do SohCahToa on them, and not get confused between which is which. Also, some super-tricky examples for students with tough teachers.

# Random Topics:

#### This topic covers those problems where they give you a couple shapes, or a shaded region, or a shape inside another shape, or a shape next to a shape with a shaded region in between, and then you're supposed to figure out the probability that a random event (like impact by a bug or lawn dart) will strike one region or the other.

#### This chapter first defines everything you'd need to know about polygons: definition of polygon, names of polygons, concave vs convex, equilateral vs equiangular, regular vs irregular. Then it gets into the various types of problems you'll see, like finding interior and exterior angles, apothems of polygons, and area of polygons.