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