Section 2.5

Postulates and Paragraph Proofs

Postulate or axiom- a statement that is accepted as true

Postulates

2.1 Through any two points, there is exactly one line

2.2 Through any three points not on the same line, there is exactly one plane

2.3 A line contains at least two points

2.4 A plane contains at least three points not on the same line

2.5 If two points lie in a plane, then the entire line containing those points lies in that plane

2.6 If two lines intersect, then their intersection is exactly one point

2.6 If two planes intersect, then their intersection is a line

Theorem- once a statement or conjecture has been shown to be true

Proof-is a logical argument in which each statement you make is supported by a statement that is accepted as true

Paragraph proof- write a paragraph to explain why a conjecture for a given situation is true

Key Concept

1. State the theorem or conjecture to be proven
2. List the given information
3. If possible, draw a diagram to illustrate the given information
4. State what is the be proved
5. Develop a system of deductive reasoning

[__]=Vocabulary


[__]=Postulates

Section 2.1

Section 2.1 is about Conjectures and inductive reasoning. A conjecture is an educated guess based on known information. It is like a hypothesis. Inductive reasoning is reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction. An example of a conjecture is if you were given the numbers 2,4,6,8,10. Then you were asked what number would come next. You could guess that 12 would come next. This is also an example of inductive reasoning. A counterexample is an example that you can use to prove that a conjecture is wrong. This is what section 2.1 was about.

2.3

The main points in 2.3 are the if, then way of writing a statement. The part after the if is the hypothesis and the part after the then is the conclusion. Not all conditional statements are in if, then form. The other way is very similar to the if, then form but the difference is that there is no if, then just the hypothesis and conclusion by themselves. The truth value of conditional statements is if the hypothesis is true than the conclusion will be true. There are three other kinds of if, then statements. They are converse, inverse and contrapositive. Conditional statements and contrapositive statements are always true and converse and inverse statements can be true or false, it depends on what the statement is saying.

ch.1 Section 6 Polygons

Polygons

Today in class we learned all about polygons!! Polygons are:

• shapes that have AT LEAST three sides
• each side MUST intersect with exactly two other sides

We also learned that there are many different types of polygons. To the left is a chart that lists all the different types of polygons.



We also learned about five different types of polygons:

concave polygon- this is a polygon that if you were to extend its sides then the lines would go through he interior of the shape

convex polygon- this is a polygon that if you were to extend the sides then the lines would NOT go through the interior of the polygon.

equilateral polygon- all the sides are congruent, meaning they are all the same length

equiangular polygon- All the interior angles are congruent

regular polygon- a polygon that is both equiangular and equilateral

Another thing we learned during class was about how to find the area and perimeter of polygons and also a circle

AREA:
square - a side of the square to the second power or squared (written out on right)
rectangle- base times height

triangle: 1/2bh this means the base of the triangle times the height. To recognize where the height is make sure that it is both of these two things:

• The height is always perpendicular to the base
  
• always forms a right angle

PERIMETER:
All polygons- the sum of all the sides equals the perimeter

CIRCLE:
area- area is pi times radius squared.

• radius is half of the diameter
• pi is approx. 3.14

circumference (perimeter)- two times pi times radius

Chapter 1 Section 3 Blog Entry

Chapter 1 Section 3

Distance and Midpoints

In this section, the main concepts are finding distance on a number line, coordinate plane, midpoints of segments, and the coordinates of these. For the first part the problems were using the number line and finding midpoints. This was just finding averages pretty much. We also had to find midpoint coordinates on a plane by finding the averages of the two points, which is similar to finding the midpoint on a number line except with 2 axes. Then we used algebraic equations to solve harder types of these problems. Also compass’ were used to bisect segments on page 25. After that Pythagorean theorem to find the distance between points, which was review from the last section.

What we learned today

Today we learned about

Adjacent Angles - Two angles in a plane which share a common vertex and a common side but do not overlap

Vertical Angles - Angles opposite one another at the intersection of lines. Vertical angles are congruent.

Linear Angles - Two adjacent angles at the intersection of two straight lines. Adds up to 180 degrees.

Supplementary Angles - Two angles that add up to 180 degrees.

Complementary Angles - Two acute angles that add up to 90 degrees.

Chapter 1 summary

Chapter 1 summary
Key Concepts

Point- Drawn as a dot, labeled as a capital letter.

Line- The letters representing 2 points on the line or a lowercase script. Does not end; ever.

Midpoint- The in-between of two points on a line. Basically the point exactly in between the two end points of a line segment.

Example of using algebra to find measures
AB-BC Definition of midpoint
4x – 5 – 11+2x AB-4x-5, BC-11+2x
4x-16+2 Add 5 to both side
2x- 16 Subtract 2x from each side
x-8 Divide each side by 2
Now substitute 8 for x in the expression for BC
BC-11+2x Original measure
-11+2(8) x-8
-11+16 or 27 The measure of BC is 27

September 4th...What we did today!

Today we learned about... ANGLES!!!
Angles are two rays that intersect or meet at the same point. This point is called a VERTEX. There are four types of angles;
Acute- 0-89 degrees
Right- 90 degrees
Obtuse- 91-179 degrees
Straight- 180 degrees
When titling angles based on points, the letter corresponding with the vertex point should ALWAYS BE IN THE MIDDLE!

PRECISION AND MIDPOINTS.

PRECISION is based off of the smallest unit available when measuring something. The measurement should be within .5 of the measure. For example, if you measure something that is 5 centimeters, the actual length cannot be under 4.5 centimeters and more than 5.5 centimeters.

NOTE: measurements of like, 50 centimeters and 50.0 centimeters show differences in precision.

how??? Well a measure of 50 cm means that the ruler is divided that way. 50.0 shows that the ruler was divided into millimeters.

The precision of CUSTOMARY UNITS is determined before reducing the fraction.

to find the betweenness of points, jus substitute the measures of the points around the one you want to find the measure of, and then add them.




To find the measure of y and qp if p is between q and R
you substitute the values into an equation, and work it like an algebra problem as seen here. >>>