1.2 Measurement of Segments and Angles

An acute angle is an angle whose measure is greater than 0 and less than 90 degrees.

A right angle is an angle whose measure is 90 degrees.

An obtuse angle is an angle whose measure is greater than 90 and less than 180 degrees.

A straight angle is an angle whose measure is 180 degrees.

Types of angles

Parts of a degree

A degree of an angle is divided into 60 minutes (‘) and each minute is divided into 60 seconds (“).

Convert the following into minutes and seconds.

87 ½ 60.4 90 2/3 180

Congruent Angles and Segments

Congruent angles are angles that have the same measure.

Congruent segments are segments that have the same length.

Time to work

Look at page 11 for examples of different types of angles, then the top of page 13 to show different ways we denote congruency. Look through the sample problems and then begin working problems 1-4, 6-9, 11, 13-18, 19, 21.

1.3 Collinearity, betweenness, and

assumptions

Collinearity

Points that lie on the same line are called collinear. Points that do not lie on the same line are called noncollinear.

If three points are collinear, then we can say that one point is between the other two.

Triangle Inequality

Any three points that are noncollinear form a triangle.

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

Assumptions from Diagrams

Assume Do not assume

Straight lines and angles Right angles

Collinearity of points Congruent segments

Betweenness of points Congruent angles

Relative positions of points Relative sizes of segments and angles

Time to work

Look at page 19 and read through the assumptions from diagrams. Then check the sample problems for understanding. After that, begin working on problems 1, 3-4, 6-15.

1.4 Beginning proofs

Two-column proofs and theorems

We will use theorems often when performing proofs (and you will need to memorize them).

A theorem is a mathematical statement that can be proved.

Theorem 1 – If two angles are right angles, then they are congruent.

Theorem 2 – If two angles are straight angles, then they are congruent.

Two-column proofs

Here is what a two-column proof should look like.

Statements Reasons1. 1.2. 2.3. 3.4. 4.5. 5.

Proof Example Given: Angle RST is 50 degrees. Angle TSV is 40 degrees. Angle X is

a right angle.

Prove: Angle RSV is congruent to angle X.

Statements Reasons1. 1.2. 2.3. 3.4. 4.5. 5.6. 6.

Time to work

Look through the sample problems and then begin working problems 1-14.

1.5 Division of segments and angles

Midpoints and Bisectors of Segments

A point (or segment, ray, or line) that divides a segment into two congruent segments bisects the segment. The bisection point is called the midpoint of the segment.

Only segments can have midpoints.

Trisection Points and Trisecting a Segment

Two points (or segments, rays, or lines) that divide a segment into three congruent segments trisect the segment. The two points at which the segment is divided are called the trisection points of the segment.

Angle bisectors and trisectors

A ray that divides an angle into two congruent angles bisects the angle. The dividing ray is called the bisector of the angle.

Two rays that divide an angle into three congruent angles trisect the angle. The two dividing rays are called trisectors of the angle.

Two-column proofsLook at problems 4-6. Notice the first statement and reason for each of these is given. Then they take that information and use a theorem we just learned.

Statements Reasons1. 1.2. 2.3. 3.4. 4.5. 5.

Time to work

Look through the sample problems. If you notice problems 4-6 what is the first reason in each of these? They then take that and use that information to prove something using the theorems we learned earlier in this chapter.

Now you may begin working problems 1-21.

1.6 Paragraph Proofs

Paragraph Proofs

Paragraph proofs serve the same purpose as two-column proofs. Each has advantages and disadvantages. We will be using two-column proofs most of the year. The only difference is that we write paragraph proofs in paragraph format instead of listed.

Time to work

Let’s look at each of the sample problems together and talk through them to show the differences between two-column proofs and paragraph proofs.

After that you may begin working on your homework which is problems 1-10.

1.7 Deductive Structure

Deductive Structure

Geometry is based on the idea of deductive reasoning. This means that we deduce or find conclusions based on information we can justify or prove. We use undefined terms, assumptions known as postulates, definitions, and theorems to do this.

An example of an undefined term is a point, which we earlier gave examples of but have not defined.

A postulate is an unproved assumption.

Definitions and Theorems

Definitions state the meaning of a term and are REVERSIBLE.

Theorems and postulates are not always reversible.

Read page 40 and 41 in your textbook. After reading, discuss with your partner what we mean by reversible.

Time to work

After reading and understanding the sample problems, work problems 1-5, 8-12, and 14.

1.8 Statements of Logic

Writing Statments of Logic Video

Reading

Start by reading this section of the text. We will then discuss the terms and ideas represented here.

Conditional Statements

Can be written in either declarative (declares something to be true) or conditional (if… then…) format.

The negation of it is raining would be it is not raining. To write the negation of something we use this symbol ~

Converse, Inverse, and Contrapositive

Let’s look at the statement “If you live in Lexington, then you live in Kentucky.”

Converse – If you live in Kentucky, then you live in Lexington.

Inverse – If you don’t live in Lexington, then you don’t live in Kentucky.

Contrapositive – If you don’t live in Kentucky, then you don’t live in Lexington.

Theorem 3 and Chains of Reasoning

Theorem 3 states that if a conditional statement is true, then the contrapositive of the statement is also true. Using our previous example, if you live in Lexington then you live in Kentucky, the contrapositive was if you don’t live in Kentucky then you don’t live in Lexington. Both of these statements are true.

Chains of reasoning allow us to use reasoning to connect things. So if p then q, and if q then r, means that we can connect p and r and say if p then r. An example might be if it’s hot then I sweat, if I sweat then I stink, so if it’s hot then I stink.

Time to work

To gain a better understanding of this topic try problems 1-5, 8, and 9.

1.9 ProbabilityLAST SECTION OF THIS CHAPTER!!!

Steps for Probability Problems

Determine all possibilities in a logical manner. Count them.

Determine the number of these possibilities that are “favorable.” We will call these winners.

Probability = number of winners / number of possibilities

Time to work

Read through the sample problems. Focus on how they make an organized list of all the possibilities (look at problem 3 for a good example). Then you may begin working problems 1-11 and 15.