Summary and Review

Warm-Up

• Try this quiz

DO

• SET I: #1-8
• SET II: #20-25, 26-33

Polygons

Read pages 74-75

Things to Note:

• definition of a diagonal
• definition of a perimeter (you probably already know this)
• definition of convex and concave.
• names of polygons by number of sides (on the sidebar of the book)

DO

• this quiz on identifying polygons by number of sides
• SET I: #17-19
• SET II: #30 - 31 (these are both proofs -- if you need help, ask me).

Line Segments

Read pages 69-70.

Particularly study Theorem 2 and its proof. This course does lots of proofs. If you want more information on the steps of making a proof, look at Mastering the Formal Geometry Proof.

DO:

Set I: #1-6, 12-19
Set II: #29-30, 34 -35

Example of an Algebra Proof

5(x + 5) = 16x - 8
5x + 25 = 16x - 8
5x + 25 - 16x = - 8
-11x + 25 = -8
-11x = -33
x = 3



Given: 5(x + 5) = 16x - 8
Prove: x = 3

Statements | Reasons
5(x + 5) = 16x - 8 | Given
5x + 25 = 16x - 8 | Distributive Property
5x + 25 - 16x = - 8 | Subtraction Property of Equality
-11x + 25 = -8 | Distributive Property
-11x = -33 | Subtraction Property of Equality
x = 3 | Division Property of Equality

Algebra Review #2

Go to page 82

I thought we would move to an algebra review now.

Do the even numbered problems on page 82. Try starting with #30 and going backwards, then the first ones will be easier. Try to go fast -- if you take more than 30 minutes, stop and show me where you are.

OR
Do this ThatQuiz on distributive properties

Example of using distributive property to solve a problem:

4 (x - 11) = 3x + 16

simplify the number in parentheses

4x - 44 = 3x + 16

transfer numbers and expressions by doing the opposite operation

4x - 3x = 16 - 44

solve

x = -28 (don't forget the negative!)

Also, take a few minutes to review the Postulates and Theorems on page 79.

For review: Algebra Properties

Properties of Equality

Read pages 60 - 61

The properties of real numbers should be familiar to you. If the letters confuse you, try to plug in some actual numbers. Then it should be clear.

Example:

Reflexive Property

a = a

2 = 2

(a here means any real number, and what is true for 2 is true for all numbers)


DO:

• Set 1: #1-6, 20-23
• Set II: #24-26, 35-40

The Ruler Postulate

Read pages 56-57.

The Ruler Postulate is not as complicated as it sounds. Take a ruler:

You can see pretty easily that:

1. A given number corresponds to a certain point. This is called its coordinate.
2. There is never more than one point to a number.
3. If you take any two points on a given ruler, the distance between the two points is given in terms of a real number (say, 20 is the distance between 10 and 30 on the purple ruler).
4. You can express this as a formula |x-y| given that x is 30 and y is 20. This formula can be used whenever you need to know the distance between two points (this is important! HJ will be asking several of this kind of question in the exercises!

EXAMPLE:

If 400 in the first ruler was A and 200 was D, then the distance between the two in terms of letters would be |A-D|. Does that make sense?

DO;

• Set I -- # 10 - 19
• Set II -- #24-31

Points, Lines and Planes #2

Review pages 51-52 if necessary.

Do SET I, problems 9-23, and SET II, #28-37 (you can do these orally to me, but get them ready first)

Points, Lines and Planes

Read pages 50-52 in Jacob's Geometry.

Things to Know:

Jacob's definition of a postulate:

A postulate is a statement that is assumed to be true without proof (page 40).

Jacob calls some terms "undefined" meaning that if you attempt to define them you will go in circles. Euclid tries to start with as little as possible and build from there, which is the classical way of doing things .... look at his definitions

Look at these first postulates of Euclid's -- many of them are constructions, which means that Euclid asks that we draw them or at least agree that they can be drawn. Postulates are also called axioms.... see here. Wikipedia says:

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

In this course we are going to start at the end of each chapter, since Jacob sums up what he has covered and this can be useful.

Turn to page 79

There is a list of postulates. Take a pencil and try to draw the postulates. This will probably help you understand them better since they won't be just abstract sentences. If you need help, ask.

Jacobs defines a theorem this way:

A theorem is a statement that is proved by reasoning deductively from already accepted statements.

Look at the theorems on page 79 and see if they seem to make sense given the postulates.

To DO:

Try this quiz. Hint: the proportions should match.