Section 5.4.

Shortest Distance Theorem- the perpendicular segment from a point to a line is the shortest segment from the point to the line.

Triangle Inequality Theorem- the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Corollary 5.1- The perpendicular segment from apoint to a plane is the shortest segment from the point to the plane.

Sec 5.5

Hinge Theorem (SAS Inequality)
-If two sides of one triangle are congruent to two sides of another triangle and the included angle of the 1st is larger than the included angle of the 2nd, then the 3rd side of the 1st is longer than the 3rd side of the 2nd.

Converse of Hinge Theorem (SSS Inequality)
-If two sides of one triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the 2nd, then the included angle of the 1st is larger than the included angle of the 2nd.

Chapter 5-1

Perpindiculars and Bisectors and Altitudes

Perpendicular Bisector Theorem: a point that lies on the perpendicular bisector of a segment is equidistant from endpoints of the segment.

Converse of the Perpendicular Bisector Theorem: if a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.

Concurrent Lines: three or more lines (or rays or segments) that intersect at the same point.

Perpendicular Bisector of a Triangle: a line (or ray or segment) that is perpendicular to the side of a triangle at the midpoint of a side.

Circumcenter Theorem: the circumcenter is equidistant from the vertices.

Angle Bisector of a triangle: bisector of an angle of the triangle

point of concuraccy: incenter

Incenter theorem: the incenter is equidistant from the sides

Median of a Triangle: segment whose endpoints are a vertex and the midpoint of the opposite side

Point of concurracy: centroid

Centroid Theorem: from the centroid to the side is half the distance to the angle.

Altitude of a triangle: the perpendicular segment from a vertex to the opposite side, or the extension of that side

Point of concurracty: orthocenter

Sec. 5.2 Notes

Longer Side Theorem
-If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
Larger Angle Theorem
-If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
Exterior Angle Inequality
-The measure of an exterior angle is greater than the measure of either 2 nonadjacent interior angles.

Section 4.3

In class today we covered Section 4.3. In this section, we learned many things regarding to congruent triangles. To be more specific, we learned about: CPCTC, the Properties of Congruent Triangles, and Congruence Transformations.

CPCTC is a rule that allows us to prove our answers, usually when doing a proof. It stands for Corresponding Parts of Congruent Triangles that are Congruent. This rule tells us that corresponding angles are congruent and corresponding side are congruent. Here’s how that would look.

Properties of Congruent Triangles: are properties that we seen before, but now they can be used to classify steps when doing a proof for a triangle. The properties are:

Reflexive Property of Congruent Triangles: where every triangle is congruent to itself

Symmetric Property of Congruent Triangles: if triangle DOG is congruent to triangle CAT, then triangle CAT is congruent to triangle DOG

Transitive Property of Congruent Triangles: if triangle CAT is congruent to triangle DOG & triangle DOG is congruent to triangle ELK, then triangle CAT is congruent to triangle ELK.

Finally, we learned about the three Congruence Transformations. These show us that you can slide, flip, reflect, or turn a triangle. When doing this, remember that the size and shape will never change.

Triangle Congruence Theorems

THE FOUR:
SSS (side-side-side)
SAS (side-angle-side)
ASA (angle-side-angle)
AAS (angle-angle-side)

CPCTC: Corresponding parts of congruent triangles are congruent
-only to be used after two triangles have been proved congruent
-CANNOT PROVE TWO TRIANGLES CONGRUENT USING CPCTC

November 6 Blog

The Base Angles are the to two adjacent angles to the base,

The Vertex Angle is the angle opposite to the base.

Theorems

Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
(ex. If line BU is congruent to line GU, then angle G is congruent to angle B.)

Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
(ex. Is angle G is congruent to angle B, then line BU is congruent to line GU)



Corollaries

A triangle is equilateral, if and only if it is equiangular.

Angles of equilateral and equiangular triangles = 60 degrees.

Section 4-2

Section 4-2
Vocab: Corollary- An addition to a theorem, that is easily proved using that theorem
Theorems:

• Triangle Sum Theorem(4.1) The sum of the measures of the interior angles of a triangle is 180
  

• Theorem (4.2) -There can be at most one right angle or one obtuse angle in any given triangle. The other 2 angles are always acute
• Exterior Angle Theorem- The measure of an exterior angle of a triangle is equal to the sum of the measures of the opposite 2 nonadjacent angles (remote interior angles)
• Third Angle Theorem- If 2 of the one triangle are congruent to another triangle, the third angles must also be congruent.

Flow Proof:
Examples from class:

4.1 Triangles

This is some of the vocab we learned today.
Notice the plural of vertex is not vertexes, but vertices.
The three types of triangles are listed and the notes I took are on there as well.

This picture is of the triangles by their angles.
Equiangular is the same as equilateral except for the fact that angular means by angles and equilateral is by their sides.



When finding the measures of the sides of an equilateral triangle, you can tell they are all equal to each other. So, you can set any 2 of the equations equal to each other, and obviously, the other equations of the sides should come out to be the same because an equilateral triangle's sides are all equal.

For an isosceles triangle, you will take the two sides that are equal to each other and set them equal to each other. From there, you figure out your variable, and you can plug it into each equation, and to the side that is not equal to the other sides.

When given the points of the vertices, it is easiest to put them on a coordinate plane and plot the points. Once you have plotted the points, you may use the distance formula to find the measures of the sides. After you have found the sides, you can classify what type of triangle it is. If all 3 sides are the same, it is an equilateral triangle. If two sides are equal, it is an isosceles triangle and if none of the sides are equal it is a scalene triangle.

Here is the distance formula one more time.

This post is about section chapter 3 section 3 and we learned about finding slopes and finding the perpendicular slope to the original line's slope.


The picture above shows how to find the slope with two given points. Now to find the slope of the line perpendicular to the original one. You simply get the opposite of the slope.
Examples: Original, 3 Perpendicular, -3 Original, 1/3 Perpendicular, -3/1=-3

Section 3-6: Perpendiculars and Distance

In this section we learned how to find the distance between a point and a line. Here is the picture for it:

We also learned a new term, equidistant. 2 things are equidistant if they are the same distance from a given object. An easy way to remember this is the two words that make up the word equidistant. "Equal" and "distant/distance". That way, you can know that it means two things that have an equal distance.
We learned a new theorems too. Here it is:

Today we learned how to find out if two lines are parallel. We do this by using the converse's to angle postulates. (a converse is a statement opposite to a hypothesis)

For example- You could write this statement in two ways, proving two different things;
1. If 2 lines are cut by a transversal so that the lines are parallel, then the corresponding angles are congruent.
OR
2. If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.
The first statement just states that corresponding angles are congruent, corresponding angles postulate.
The second statement is the converse of this, and states that the two lines are parallel. (corrospondign angles converse)

When trying to figure out if lines are parallel you have to look at the converses of normal angle postulates.

Today we learned that:
Slope-Intercept Form: y=mx+b
Point-Slope Form: Y-Y1=M(X-X1)

You use slope intercept form when you are given the slope and the y-intercept
You use point slope form when you are given the slope and a point

If you have two points on a coordinate plane, than it is easy to find the slope.
To find slope you take the two points: (X1,Y1) (X2,Y2) and simply do the slope formula:Here is one of the problems we did today as a class with these formulas and parallel lines. (note- parallel lines will always have the same slope and perpendicular lines will always have opposite slopes.)

Today, on October 9th, 2008, we talked about the many different relationships between alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles when the transversal intersects between two parallel lines. This picture on the left is the picture that I made using geometers sketch pad and it shows the angles measurements which helps to show the relationships amongst the different angles. As well as the following slides are just from the Dyknow presentation from during class and you can use them as notes.

Chapter 3

The main point of the lesson today was adding the vocabulary terms of Parallel Lines, Skew Lines, and Parallel Planes.

Parallel Lines for example are two lines that are on the same plane and do not meet or cross at any giving point.

Skew Lines are two lines that are not what so ever on the same plane and do not meet or cross each other at any given point.

Parallel Planes are planes that run along side one another but do not meet or cross at any given point or area

We also learned what a transversal is. A transversal is a line that intersects two or more co-planar lines.

Corresponding interior/exterior, Alternate interior/exterior, and Consecutive interior/exterior can be made up of these three lines.

For example:

9 and 15 are alternate exterior

6 and 13 are consecutive interior

4 and 6 are alternate interior

Section 2.7 Proving Segments Relationships

Ruler Postulate (2.8)
-Given any 2 points, A & B on a line, A corresponds 0 and B corresponds to a positive real number
Segment Addition Postulate (2.9)
-If point B is between A and C, then AB + BC = AC
Theorem 2.2 : Properties of Segment Congruence
-Segment congruence is:
-Reflexive: For any segment AB, AB=~AB
-Symmetric: If segment AB=~CD, then CD=~AB
-Transitive: If segment AB=~CD & CD=~EF, then AB=~EF

October 1st

In section 2.8, we learned many things dealing with angles. We learned how to prove angle relationships using protractors, we learned angle addition postulates, complements and supplements, properties of angle congrunce, and angle theorems.



Proving angle relationships:
Protractor Postulate- Given segment AB and a number "r" between zero and one eighty is exactly one ray, with endpoint "A" such that the measure of angle is "r"




Angle Addition Postulate:
If point "S" is in the interior of angle PAT then.. measure of angle PAS + measure of angle TAS = measure of angle PAT

measure of angle PAS and measure of angle TAS are adjacent angles... they share a vertex and one side and no other points.. this is just like segment addition postulate.

Complemens and Supllements:
Supplement Theorem- if two angles form a linear pair, then they are supplementary
Complement Theorem- if the noncommon sides of 2 adjacent angles form a right angle, then the 2 angles are complementary

Properties of Angle Congruence:
Angle congruence-
reflexive= for any angle ABC, angle ABC is congruence to angle to ABC
Symmetric= if angle ABC is congruent to angle DE, then angle is DEF is
congruent to angle ABC.
Transitive= if angle ABC is congruent to angle DEF and angle DEF is
congruent to angle GHI then angle ABC is congruent to angle GHI

Angle Theorems:
Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to the congruent angles) then they are congruent
Example: if the measure of angle 1 + measure of angle 2 and measure of angle 3 + measure of angle two = 180 degrees, then angle 1 is congruent to angle 3
COngruent Complements Theorem
if two angles are complementary to the same angle (or to congruent angles) then they are congruent.
Example: if measure of angle 4 + measure of angle 5 = 90 degrees and measure of angle 5 + measure of angle 6 = 90 degrees.. then measure of angle 4 is congruent to measure of angle 6.

More Angle Theorems
perpendicular lines intersect to form right angles
all right angles are congruent
perpendicular lines form congruent adjacent angles
if two angles are both congruent and supplementary then each is a right angle
if two congruent angles form a linear pair, then they are right angles.

Last Angle Theorem:
Verticle angles Theorem- verticle angles are congruent

Section 2.6 Reasoning With Algebra

The properties:
*The transitive property is very similar to the law of syllogism that we just used.
*The first step in two column proofs is almost always going to be given.

Proofs and Postulates

In this seccion we learned about Paragraph Proofing and Postulates. Above are the postulates we learned in class and need to be memorized. Also above is the definition that we learned, axiom or postulate.

Theorem- a statement which has been proven true.
Postulates, definitions, undefined terms and algebraic properties are used to prove theorems true.

Chapter 2 Section 4

Conjecture: an educated guess based on known imformation.

Inductive reasoning: Examining several specific situations to arrive at a conjection is called inductive reasoning.

Truth value: The truth or falsity of a statement

Law of detachment: If p=q and p is true, then q is also true

Law of syllogism: If p=q and q=r is true then p=r is true

In Chapter 2, Section 3, we learned about Conditional Statements.

A conditional statement is a statement that can be written in if-then form. When a statement is in if-then form, it is written in the form if p, then q. The part of the sentence after the word "if" is the hypothesis, and the part of the sentence after the word "then" is the conclusion.

Example 1: If you buy the television, then you will get $100 cash back.
hypothesis- You buy the television
conclusion- You get $100 cash back

Example 2: She will get a promotion if she works hard.
hypothesis- She will get a promotion
conclusion- She works hard.
hypothesis- She will get a promotion.

*There is no if-then statement in example 2. However, you can identify the hypothesis and conclusion by first identifying the hypothesis and then finding the conclusion.

Writing a Conditional Statement

First, identify the hypothesis and conclusion. Then, put them into a statement using if-then form.

Examples:

The boy is Tom's son, and he is four feet tall.
hypothesis- The boy is Tom's son
conclusion- He is four feet tall.
Conditional Statement: If he the boy is Tom's son, then he is four feet tall.

An angle with a measure of 180 degrees is a straight angle.
hypothesis- An angle has a measure of 180 degrees.
conclusion- it is a straight angle.
Conditional Statement: If an angle has a measure of 180 degrees, then is it a straight angle.

Finding the Truth Value of a Statement

The truth value of a statement is either true or false. Both parts of the statement, the hypothesis and the conclusion, must be true in order for the statement to be true.

Example:

If there is a sale, then you will buy her a pair of shoes.

a. There was a sale at the mall; you bought her a pair of shoes.
The hypothesis is true, because there was in fact a sale. The conclusion is true, because you bought her a pair of shoes, which is what was promised in the statement. Both parts are true, and therefore, the conditional statement is TRUE.

b. There was a sale at the mall; you bought her two pairs of shoes.
The hypothesis is true, because there was a sale. The conclusion is false, because you bought her two pairs of shoes, which is not what was promised. Both parts are NOT true, and therefore, the conditional statement is FALSE.

Related Conditionals

Statements based on a conditional statement are called related conditionals.

Here is a chart of the related conditonals:


Example:
Conditional: If you eat healthy food and exercise daily, then you are a healthy person.
Converse: If you are a healthy person, then you eat healthy food and exercise daily.
Inverse: If you do not eat healthy food and do not exercise daily, then you are not a healthy person.
Contrapositive: If you are not a healthy person, then do do not eat healthy food and exercise daily.

Statements with the same value are considered logically equivalent.

September 16 2009

Today in class we started logic puzzles.
We did one called five boys and five dogs. There was a situation given and then 6 clues given to help narrow down the answers.
Clues:
1) No dog's name begins with the same letter as that of his master.
2) Rover is not Bart's or Sidney's dog.
3) Spot's master and the owner of the spaniel both have names beginning with the same letter.
4) Neither Eric's dog nor Bernard's dogs is the basset, nor is Snoopy.
5) Bart's dog and the collie are not called Spot or Snoopy.
6) Ralph's dog is not a terrior

Going through these steps, some are obvious and other you have to logically think to cross other options out. Obvious ones are such as 1, 2, and 6. The others you have to think if Spots master and the owner of the spaniel..., that means that Spots master cannot own a spaniel. First go through the list and cross out the obvious and then go through a couple more times gainning more infomation as you cross more out.
Answers: Eric had a collie named Rover, Bernard had a terrior named Spot, Bart's dog was a spaniel named Fido, Sideny's dog was a basset named Bowser, Ralph's dog was a poodle named Snoopy.

In section six, we learned about polygons and the different kinds of polygons. A polygon has to have three or more segments(sides) and each side intersects to other sides and each endpoint of a side is a vertex of the polygon.
TYPES OF POLYGONS!
3 sides triangle
4 sides quadrilateral
5 sides pentagon
6 sides hexagon
7 sides heptagon
8 sides octagon
9 sides nonagon
10 sides decagon
12 sides dodecagon
If there are 11 sides or more than 12 sides, than call it an n-gon or the number and then -gon.
a shape with 11 sides= an eleven-gon
a shape with 16 sides= a sixteen-gon

We also learned about Special Polygons
Concave Polygons: A side extended goes thru the interior of the polygon.
Convex Polygons: No line that contains a side goes thru the interior of the polygon.
Equilateral Polygons: All sides are congruent
Equiangular Polygons: All interior angles are congruent
Regular Polygon: Equilateral and equiangular

Area Formulas

Area for a square: length of a side squared
Area for a rectangle: product of base and height
Area for a triangle: one-half the product of the base and height.
Perimeter of any polygon: the sum of the lengths of all the sides
Area of a circle: the product of pi and the radius squared
Circumference of a circle: 2 multiplied by pi multiplied by the radius

In Section 1-5, we learned about the different types of angle pairs and relationships. The different types of angle pairs are: adjacent angles, vertical angles, and linear pairs.


Adjacent angles are two angles that lie on the same plane, have a common vertex and common side, but no common interior points. Note that, both of the angles will always equal the total measure when added together.

Vertical angles are two nonadjacent (no common side)angles formed by two intersecting lines. The angles opposite each other will always be the same.

A Linear pair is a pair of adjacent angles with noncommon sides that are opposite rays. Both of the angles will always add up to 180 degrees.

The different types of angle relationships are: complimentary angles and supplementary angles.



Complimentary angles are two angles that have a sum of 90 degrees.

While, supplementary angles are two angles that have a sum of 180 degrees.

Types of Angles

Hey guys, I'm posting the Types of Angles picture again, because it kind of got blocked out. Sorry about that. Here it is

Section 4 (Aditya Nellore)

In section 4, we learned all about angles. An angle is made up of 2 rays, originating from the same point, which is called the vertex. Also, we learned about naming an angle. The letter that represents the vertex always has to be in the middle.(This is on the slide directly on the left). When you want to find the measure of angles, you pretty much just do it like this: m'angle'ABC=m'angle'DEF. This is also on the slide on the top left, since it's probably not very clear here. Notice you have to put an 'm' in front of the names of the angles. There was also some information about congruent angles. Congruent angles are angles that have the same measure. In other words, if you put one on top of the other, then they will fit perfectly on top of each other. We also got a list of all the types of angles and their degrees. That is in the slide on the top right. I hope this helps, guys!

In section 1.2, we covered how to be precise when measuring different types of lines. Some key terms to remember in this section were;

The reason that we have names for different types of lines in the term list is because we need to know how to measure those line precisely. We basically learned how to measure the length of lines that we were given.