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

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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