5.4

There were 2 theorems in 5.4, triangle inequality thm. and short distance thm. The triangle inequality thm. states that the sum of the lengths of any 2 sides of a triangle is greater that the length of the third side.

The short distance thm. states that the perpendicular segment from a point to a line is the shortest segment from the point to the line.


Also in 5.4 you have the corollary which states that the perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

Longer Side Theorems

Longer Side Theorem- If one side of a triangle is longer than the other, than the angle opposite the longer side is larger than the angle opposite of the shorter side.

Larger Angle Theorem- If one angle 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 the 2 nonadjacent interior angles

4.1 Review

Section 3.6

:)
Perpendiculars and Distance
Key Concepts:
The distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point.

The distance between 2 parallel lines is the distance between one of the lines and any point on the other line.

2 lines in a plane are parallel if they are everywhere equidistant. Equidistant means that the distance between two lines measured along a perpendicular line to the lines is always the same. The distance between parallel lines is the length of the perpendicular segment with endpoints that lie on each of the 2 lines.

Theorem 3.9- In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other.

Chapter 5 section 1

Today in class we learned about perpendiculars, medians, attitudes and bisectors

Perpendicular Bisectors Theorem-
An example of this theorem is shown in the picture above; it means that if for example segment CM is Perpendicular bisector of segment AB then that means that segments AC and BC are congruent to each other

Perpendicular Bisectors Theorem Converse-
Exactly the same as the perpendicular bisectors theorem except this is when you would know that segment AC and BC are congruent first. This converse would then prove that C is the perpendicular Bisector of segment AB.


Concurrent lines:
Three or more lines that intersect at the same point.

Point of Concurrency:
The point where the lines

INCENTER:
To find the incenter you must do as the picture shows below:

1.Bisect all the angles of the triangle
2.The point where the lines meet is where the incenter is
In this triangle with the incenter, since the angle was bisected meaning both sides are equal, this allows us to mark them congruent as shown in the picture above.

Incenter theorem- the incenter is equidistance from all sides



CIRCUMCENTER:
To find the circumcenter you must do as follows:

1.Find the midpoint of all three sides of the triangle
2.Draw a perpendicular line from the midpoint
3.The point where all three lines intercept is called the circumcenter
4.The sides where the midpoint is splitting the line are congruent as shown in the above picture

Circumcenter theorem- the circumcenter is equidistance from all three vertices




CENTROID:
To find the centroid you should:

1.Find the midpoint of all three sides of the triangle
2.Draw a line from the midpoint to the midpoint on the opposite side
3.The point where they intersect is called the centroid
4.The two sides of the midpoint are congruent to each other as shown below
Centroid Theorem: From the centroid to the side is half of the angle








ORTHOCENTER:
To find the orthocenter of a triangle you should do:

1. Make a perpendicular line from a side of the angle to the vertex across form it
2.Do this for all three sides
3.The point where the lines intercept is known as the orthocenter as shown in the picture below

Congruent Triangles Section 4.3

Practice--------------------------------->


<--The Properties
CPCTC-Corresponding Parts of Congruent Triangles are Congruent

4.4- 4.5

In 4.4 we learned the theorems we use to prove triangles congruent.
SSS- Side, Side, Side
SAS- Side, Angle, Side
AAS- Angle, Angle, Side
ASA- Angle, Side, Angle

The ones that don't work are
ASS
AAA

We learned about CPCTC (congruent parts of congruent triangles are congruent)
If you have two congruent triangles all their parts are equal.

4.4- 4.5

In

Section 4.7 Blog

Section 4.6

4.2

According to Theorem 4.1 (Angle Sum)
The measures of the angles in a triangle are always equal to 180
m

If we know the measures of two angles, we can find the third measure.

if two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are not congruent.

Exterior Angle Theorem
The measure of one exterior is equal to the two interior angle measures added together.

A corollary is a statement that is easily proven with a theorem.
Corollary 4.1: acute angles of right triangles are always complimentary.
Corollary 4.2: there can only be one right angle or one obtuse angle in a triangle.

Section 3.5

Section 3.5 was about how you can prove that lines are parallel. You can prove that lines are parallel in many ways. One way is to use all of the converses that are in the dyno that we went over. They are corresponding, alternate exterior, and alternate interior converse. Also if the consecutive interior angles are supplementary then the line would be parallel. These are the ways to prove that lines are parallel.

Section 3-3 Review

This section covered slopes, slope formulas, slope relationships, slope-intercept form, and some review on parallel and perpendicular lines.

slope the ratio of a line's vertical rise to it's horizontal run

rate of change how much a quantity changes over time

The slope of parallel lines are always equal
The slope of perpendicular lines always are opposite reciprocals.
Skew Lines that are not coplanar and do not intersect at any point.

Slope-Intercept Form y=mx=b
point-slope form y-y1=m(x-x1)

Slopes Section 3.4

On October 13, we reviewed slopes.

Slope Intercept Form: y = m(x) + b
You must have the slope and the y-intercept.

Point Slope Form

y - y1 = m(x-x1)
You must have the slope and a point.

Section 3-2

In section 2-3 we learned about five different theorems and postulates for transversals



Alternative Interiorior and Exterior Theorums:
This states that so long as there are two parallel lines that are split by a transversal then the Alternate interior angles of those two parallel lines are congruent to each other. The same is true for Alternate exterior angles


Corresponding angles Postulate:
This one states that corresponding angles are congruent, so long as they are on two parrallel lines that are split by a transversal. At right is a picture showing two corresponding angles that are congruent.









Consecutive Interior Angles Theorem:
Consecutive Interior angles are supplementary so long as they are on two parellel lines that are split by a transversal. This means that the sum of both he consecutive interior angles is 180 degrees. To the right is an example.




Perpendicular transversal theorem:
States that if a transversal is perpendicular to one of two parallel lines then it is perpendicular to the other one. example shown below

October 8, 2008

In class today we learned about lines and transversal. There are three kinds of lines, parallel lines, skew lines and parallel planes. Parallel lines are 2 lines that are coplanar but don't intersect. Skew Lines are 2 lines that are not coplanar and do not intersect. Parallel planes are 2 planes that do not intersect. We also talked about about transversal. Transversal is a line that intersect two or more non coplanar lines and creates angles. There are 4 types, corresponding angles, alternate exterior, alternate interior, consecutive interior. Corresponding angles are angles that correspond from each other, nonadjacent, same side of the trensversal, and one exterior while one is interior. Alternate exterior angles are exterior angles that lie on either side of the rensversal making a linear pair. Alternate Interior angles lie inbetween the non coplanar lines, lie on either side of the transversal, and are non adjacent. Consecutive Interior angles are angles that lie on the same side of the transversal on the interior.

Examples

Parallel Lines

Skew Lines

Parallel Planes

Corresponding angles

Alternate Exterior

Alternate Interior

Consecutive Interior

Section 2.6 Reasoning With Algebra

This is what we learned in the 2.6 chapter. These are the Dyknow slides. We learned about the properties, and it was the beginning of learning about the proofs.