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