# Read e-book Geometry Quick Review: Triangles - Key Theorems and Proofs (Quick Review Notes)

Tangent Lines and Rates of Change — In this section we will introduce two problems that we will see time and again in this course: Rate of Change of a function and Tangent Lines to functions. Both of these problems will be used to introduce the concept of limits, although we won't formally give the definition or notation until the next section. The Limit — In this section we will introduce the notation of the limit. We will also take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.

We will be estimating the value of limits in this section to help us understand what they tell us. We will actually start computing limits in a couple of sections. One-Sided Limits — In this section we will introduce the concept of one-sided limits. We will discuss the differences between one-sided limits and limits as well as how they are related to each other. We will also compute a couple of basic limits in this section. Computing Limits — In this section we will looks at several types of limits that require some work before we can use the limit properties to compute them. We will also look at computing limits of piecewise functions and use of the Squeeze Theorem to compute some limits.

Infinite Limits — In this section we will look at limits that have a value of infinity or negative infinity. We will concentrate on polynomials and rational expressions in this section. Continuity — In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval. The Definition of the Limit — In this section we will give a precise definition of several of the limits covered in this section.

We will work several basic examples illustrating how to use this precise definition to compute a limit. The Definition of the Derivative — In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function. Interpretation of the Derivative — In this section we give several of the more important interpretations of the derivative.

We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function. Differentiation Formulas — In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers. Product and Quotient Rule — In this section we will give two of the more important formulas for differentiating functions.

We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate. Derivatives of Trig Functions — In this section we will discuss differentiating trig functions. Derivatives of Exponential and Logarithm Functions — In this section we derive the formulas for the derivatives of the exponential and logarithm functions.

Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine.

Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions.

### What Are Triangles?

As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation — In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e. Implicit differentiation will allow us to find the derivative in these cases.

Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates the next section. Related Rates — In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one or more quantities in the problem.

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## Circle Geometry

This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work. Higher Order Derivatives — In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives. Logarithmic Differentiation — In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients sometimes easier than using product and quotient rule.

More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i. Critical Points — In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions. Minimum and Maximum Values — In this section we define absolute or global minimum and maximum values of a function and relative or local minimum and maximum values of a function.

We also give the Extreme Value Theorem and Fermat's Theorem, both of which are very important in the many of the applications we'll see in this chapter. Finding Absolute Extrema — In this section we discuss how to find the absolute or global minimum and maximum values of a function. In other words, we will be finding the largest and smallest values that a function will have. The Shape of a Graph, Part I — In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative or local minimum and maximum values of a function and where a function will be increasing and decreasing.

We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum. The Shape of a Graph, Part II — In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points i. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points but not all as relative minimums or relative maximums.

With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter. We will discuss several methods for determining the absolute minimum or maximum of the function. Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc. More Optimization Problems — In this section we will continue working optimization problems. The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the 'simple' geometric objects we looked at in the previous section.

Linear Approximations — In this section we discuss using the derivative to compute a linear approximation to a function. We can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this. We give two ways this can be useful in the examples. There are three different types of triangle: A scalene triangle has no congruent identical sides and no congruent angles.

An isosceles triangle has, at least, two congruent sides and two congruent angles. An equilateral triangle has three identical sides and three identical angles. Knowing these types of triangles helps you identify properties and postulates associated with them. All equilateral triangles are isosceles, but not all isosceles triangles are equilateral.

Triangles can also be classified by their angles: Know the difference between similar and congruent shapes. Similar shapes are those that have identical corresponding angles and corresponding sides that are proportionally smaller or larger than each other. In other words, the polygon will have the same angles, but different side lengths.

Congruent shapes are identical; they are the same shape and size.

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In a right triangle, the degree angles in both triangles are corresponding. The shapes do not have to be the same size for their angles to be corresponding. Learn about complementary and supplementary angles. Complementary angles are those angles which add together to make 90 degrees, supplementary angles add to degrees.

Remember that vertical angles are always congruent; similarly, alternate interior and alternate exterior angles are also always congruent. Right angles are 90 degrees, while straight angles are Vertical angles are the two angles formed by two intersecting lines that are directly opposite each other. They are on opposite sides of the line they both intersect, but on the inside of each individual line.

### Point, Line, Plane and Solid

They are on opposite sides of the line they both intersect, but on the outside of each individual line. When you want to find the sine, cosine, or tangent of an angle, you use the following formulas: Draw a diagram after reading the problem. Sometimes the problem will be provided without an image and you will have to diagram it yourself to visualize the proof. Once you have a rough sketch that matches the givens in a problem, you might need to re-draw the diagram so that you can read everything clearly and the angles are approximately correct.

## Calculus I

Make sure to label everything very clearly based on the information provided. The clearer your diagram, the easier it will be to think through the proof. Make some observations about your diagram. Label right angles and equal lengths. If lines are parallel to each other, mark that down as well. If the problem does not explicitly state two lines are equal, can you prove that they are?

Make sure you can prove all of your assumptions. Write down the relationships between various lines and angles that you can conclude based on your diagram and assumptions. Write down the givens in the problem. In any geometric proof, there is some information that is given by the problem.

Writing them down first can help you think through the process needed for the proof. Work the proof backwards. When you are proving something in geometry, you are given some statements about the shapes and angles, then asked to prove why these statements are true. Sometimes the easiest way to do this is to start with the end of the problem. How does the problem come to that conclusion?

Are there a few obvious steps that must be proved to make this work? Make a 2-column grid labeled with statements and reasons.

## Pythagoras' Theorem

In order to make a solid proof, you have to make a statement and then give the geometric reason that proves the truth of that statement. Under the reason, you will write the proof for this. If it is given, simply write given, otherwise, write the theorem that proves it. Determine which theorems apply to your proof. There are many individual theorems in geometry that can be used for your proof.

There are many properties of triangles, intersecting and parallel lines, and circles that are the basis for these theorems. Determine what geometric shapes you are working with and find the ones that apply to your proof. Reference previous proofs to see if there are similarities. There are too many theorems to list, but here are a few of the most important ones for triangles: Make sure your steps flow in a logical fashion. Write down a quick sketch of your proof outline.

Write down the reasons for each step. Add the given statements where they belong, not just all at once in the beginning. Re-order the steps if necessary. The more proofs you do, the easier it will be to order the steps properly. Write down the conclusion as the last line.

The final step should complete your proof, but it still needs a reason to justify it. When you have finished the proof, look it over and make sure there are no gaps in your reasoning. Once you have determined that the proof is sound, write QED at the bottom right corner to signify it is complete.

What is a good way to learn the concept of angles, tangents and transverses? It is a math so it requires practice. Try at least one hour a day to practice it. First, try to learn all concepts and then directly jump on solving equations. You would not immediately be perfect as it takes time to learn math.

It is necessary to know all the very basic concepts as well. Not Helpful 6 Helpful Some people think mathematically; others are more comfortable in some other pursuit. If math is not your area, you'll find something else that is. Meanwhile, keep looking for math help until you find someone whose explanations make sense to you. Not Helpful 2 Helpful 2.

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