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# Geometric Theorem Help!

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foumy6
This is a guide of all the theorems that I can find or remember 1-5, so this should be a helpful resorce for you. (Credit to my Geometrey teach for handingus the lists he made :)and to myself for taking additional notes along with the for your notebook section of my text book
NOTE:You may need to shift through comments but I always post my info in red so I ask unless you are posting new information please dont post in red
)

If I have any of these wrong or more you would like to add please feel free to comment.

Theorems: POINTS, LINES, PLANES, AND ANGLES
Theorem 1-1: If two lines intersect, then they intersect in exactly one point.
Theorem 1-2: Through a line and a point not in the line, there is exactly one plane.
Theorem 1-3: If two lines intersect, then exactly one plane contains the lines.

DEDUCTIVE REASONING
Theorem 2-1: (Midpoint Theorem)- If M is the midpoint of AB, then AM =½ AB and MB =½ AB.
Theorem 2-2: (Angle Bisector Theorem)- If BX is the bisector of angle ABC, then ABX =½ ABC
and XBC =½ ABC.
Theorem 2-3: Vertical angles are congruent.
Theorem 2-4: If two lines are perpendicular, then they form congruent adjacent angles.
Theorem 2-5: If two lines form congruent adjacent angles, then the lines are perpendicular.
Theorem 2-6: If the exterior sides of two adjacent acute angles are perpendicular, then the
angles are complementary.
Theorem 2-7: If two angles are supplements of congruent angles (or of the same angle), then
the two angles are congruent.
Theorem 2-8: If two angles are complements of congruent angles (or of the same angle), then
the two angles are congruent.

PARALLEL LINES AND PLANES
Theorem 3-1: If two parallel planes are cut by a third plane, then the lines of intersection are
parallel.
Theorem 3-2: If two parallel lines are cut by a transversal, then AIA are congruent.
Theorem 3-3: If two parallel lines are cut by a transversal, then SSI angles are supplementary.
Theorem 3-4: If a transversal is perpendicular to one of two parallel lines, then it is
perpendicular to the other line as well.
Theorem 3-5: Two lines are cut by a transversal and the AIA are congruent, then the lines are
parallel.
Theorem 3-6: If two lines are cut by a transversal and SSI angles are supplementary then the
lines are parallel.
Theorem 3-7: In a plane two lines perpendicular to the same line are parallel.
Theorem 3-8: Through a point outside a line, there is exactly one parallel to the given line.
Theorem 3-9: Through a point outside a line, there is exactly one perpendicular to the given
line.
Theorem 3-10: Two lines parallel to a third line are parallel to each other.
Theorem 3-11: The sum of the measures of the angles of a triangle is 180 degrees.
C 1: If two angles of one triangle are congruent to two angles of another triangle, then
the third angles are congruent.
C 2: Each angle of an equiangular triangle has a measure of 60 degrees.
C 3: In a triangle, there can be at most one right angle or obtuse angle.
C 4: The acute angles of a right triangle are complementary.
Theorem 3-12: The measure of an exterior angle of a triangle equals the sum of the measures
of the two remote interior angles.
Theorem 3-13: The sum of the measures of the angles of a convex polygon withn sides is (n-
2)180.
Theorem 3-14: The sum of the measures of the exterior angles of any convex polygon, one
angle at each vertex, is 360 degrees.

CONGRUENT TRIANGLES
Theorem 4-1: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
C 1: An equilateral triangle is also equiangular.
C 2: An equilateral triangle has three 60 degree angles.
C 3: The bisector of the vertex angle of an isosceles triangle is perpendicular to the
base at its midpoint.
Theorem 4-2: Converse of theorem 4-1.
Theorem 4-3: AAS; if two angles and a non-included side of one triangle are congruent to the
corresponding parts of another right triangle, then they are congruent.
Theorem 4-4: HL; if the hypotenuse and a leg on a right triangle, are congruent to that of
another, then the two triangles are congruent.
Theorem 4-5: If a point lies on the perpendicular bisector of a segment, then the point is
equidistant from both ends of the segment.
Theorem 4-6: Converse of theorem 4-5.
Theorem 4-7: If a point is equidistant from the endpoints of a segment, then the point is
equidistant from the sides of the angle.
Theorem 4-8: If a point is equidistant from the sides of an angle, then the point lies on the
bisector of the angle.

Theorem 5-1: Opposite sides of a parallelogram are congruent
Theorem 5-2: Opposite angles of a parallelogram are congruent.
Theorem 5-3: Diagonals of a parallelogram bisect each other.
Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent, then it is a
parallelogram.
Theorem 5-5: If one pair of opposite sides is both congruent and parallel, then the quadrilateral
is a parallelogram.
Theorem 5-6: If both pairs of opposite angles of a quadrilateral are congruent, then it is a
parallelogram.
Theorem 5-7: If the diagonals of a parallelogram bisect each other, then the quadrilateral is a
parallelogram.
Theorem 5-8: If two lines are parallel, then all points on one line are equidistant from the other
line.
Theorem 5-9: If three parallel lines are cut off into congruent segments on one transversal,
then they cut off congruent segments in every transversal.
Theorem 5-10: A line that contains the midpoint of one side of a triangle and is parallel to
another side passes thought the midpoint of the third side.
Theorem 5-11: The segment that joins the midpoints of two sides of a triangle (1) is parallel to
the third side (2) is half as long as the third side.
Theorem 5-12: The diagonals of a rectangle are congruent.
Theorem 5-13: The diagonals of a rhombus are perpendicular.
Theorem 5-14: Each diagonal of a rhombus bisects two angles of the rhombus.
Theorem 5-15: The midpoint of the hypotenuse of aright triangle is equidistant from the three
vertices.
Theorem 5-16: If an angle of a parallelogram is a right angle, the parallelogram is a rectangle.
Theorem 5-17: If two consecutive sides of a parallelogram are congruent, then the
parallelogram is a rhombus.
Theorem 5-18: Base angles of an isosceles trapezoid are congruent.
Theorem 5-19: The median of a trapezoid (1) Is parallel to the bases (2) has a length equal to
the average of the base lengths.

sorry if this is hard to read I typed it up as fast as I could .
To find more look through comments and if you have helpful things to add feel free to post tham and also please answer the poll on the page
_AVG_
Wow, that is quite a comprehensive list although you did leave out Circle Theorems. And, of course, you have only provided 2-dimensional theorems (but that's alright, I find 3-dimensional geometry very taxing).

Perhaps, we could build this topic into a sort of Geometrical Encyclopaedia of sorts (with a list of theorems and their proofs - of course, we'll need to start off from Euclid's postulates for that).

Anyway, thanks for sharing the list with us.
foumy6

I have some new info to add to this and if you have more to add feel free.
For this one I will be giving you some 2D geometric formulas enjoy:)

 MEANS ANGLE
@ MEANS PI OR 3.14
^MEANS TO THE POWER OF THE FOLLOWING NUMBER, SO 6^3 WOULD BE 6 TO THE THIRD POWER
** MEANS SQUARE ROOT
* MEANS THE END OF THE RADICLE
EXAMPLE: SQUARE ROOT OF THREE OVER TWO WOULD LOOK LIKE **3*/2
NOW THE SQUARE ROOT OF THREE HALVES WOULD LOOK LIKE THIS **3/2
------------------------------------------------2D----------------------------------------------------
SQUARE
s = side
Area: A = s2
Perimeter: P = 4s
-------------------------------------------------------------------------------------------------------
RECTANGLE
l = length, w = width
Area: A = lw
Perimeter: P = 2l + 2w
-------------------------------------------------------------------------------------------------------
TRIANGLE
b = base, h = height
Area: A = 1/2bh
Perimeter: P = a + b + c
-------------------------------------------------------------------------------------------------------
EQUILATERAL TRIANGLE
s = side
Height: h = p**3*/2 s
Area: A = p**3*/4 s^2
--------------------------------------------------------------------------------------------------------
PARALLELOGRAM
b = base, h = height, a = side
Area: A = bh
Perimeter: P = 2a + 2b
---------------------------------------------------------------------------------------------------------
TRAPEZOID
a, b = bases; h = height; c, d = sides
Area: A = 1/2 (a + b)h
Perimeter:
P = a + b + c + d
--------------------------------------------------------------------------------------------------------
CIRCLE
r = radius, d = diameter
Diameter: d = 2r
Area: A = @r2
Circumference: C = 2@r = @d
----------------------------------------------------------------------------------------------------------
SECTOR OF CIRCLE
Area: A = 1/2r^2
Arc Length: s = r
----------------------------------------------------------------------------------------------------------
ELLIPSE
a = semimajor axis
b = semiminor axis
Area: A = @ab
Circumference:
C ~ @(3(a + b)) −**(a + 3b)(b + 3a)*))
----------------------------------------------------------------------------------------------------------
ANNULUS
Average Radius = 1/2(r + R)
Width: w = R − r
Area: A = @(R2− r2)
----------------------------------------------------------------------------------------------------------
REGULAR POLYGON
s = side length,
n = number of sides
Circumradius: R = 1/2 s csc(@/n)
Area: A = 1/4ns^2 cot(@/n )
or A = 1
-----------------------------------------------------------------------------------------------------------

NEXT UPDATE: My next update which I hope to have up soon will be 3D formulas!
Bikerman
Please do not simply cut-paste from other sites without:
b) Using quote tags and citing the source.
Bikerman (moderator)
foumy6
 Bikerman wrote: Please do not simply cut-paste from other sites without: a) Adding something of your own b) Using quote tags and citing the source. Bikerman (moderator)

:O u deleted the thing i typed up yesterday!!!!!!!!!!!!! I didnt copy and paste that I got it from my text book and notes!!!!!!!!!!!!!!!!!!!!!!!!!!!!
honestley I cant bileave you did that.............. I spent an hour typeing that up and if i copied and pasted it then how would it take me an hour todo!!!!!!!!!!!! I am very upset because I came on today just to update, and now i find out im going to have to re-type up everything couldn;t you have asked and not just gone ahead and deleted it?????
seriously not cool!
Bondings

I think the confusion is due to these mathematical theorems and facts being quite similar. The information of your notes and your books is probably very similar to what is available on the internet, which caused it to look like copy-paste. (And please note that a lot of people copy-paste whole articles from the internet like they wrote it themselves, which is not only against the rules and annoying but can get us into trouble.)

Anyway, back to geometric theorems now.
Bikerman
Thanks Bondings.

foumy6 - I use some software to check if postings are plagiarised (it does a pattern match using google and other sources and returns a 'hit probability'). In the case of your posting it returned a 95% probability, which is normally a sure-fire sign that it has been copied.
I'm prepared to accept that you didn't simply copy the formulae out of the text book, but the next question is - what are you trying to achieve? These are all very standard formulae which most people here know. If you are trying to construct a resource which people can use to look-up a particular formula then this is a very bad way to do it, because it is not easily searchable.
You also have the problem that the editor is not optimised for displaying formulae and it is very messy.
Ideally you need a LaTeX system to display formulae correctly.

It is up to you - I have no problem with you spending time typing this stuff in, but I think you would be much better advised to construct a simple database with the equations/formulae in - then it could be a useful resource which could be searched.
foumy6
I assure you bikerman that this is straight from my notes, and notes my teacher prints off and hands out. It may be possible (These notes are from him) that he printed these off i don't really know. Ill ask him next chance I get, and if they were i'll ask him for the site and make sure to include that in the thread as a source anyway I hope to have my new update up tomorrow or the next day
IceCreamTruck
Honestly I've been wanting to brush up on my math. It was my favorite subject in school, and just about the only homework that I ever enjoyed doing because math has logical answers as opposed to sociology questions like "Who was the most influential person of the year 1454". I hate speculating, and then being graded on the answer I "made up" to satisfy the question.

I am curious if some of these characters might help a little bit here: http://www.zytrax.com/tech/web/entities.html

 Code: © ® ² ³ " & < >

Well, I guess I got my answer because they are not translated in the forum
IceCreamTruck
I suppose I could talk a little bit about AstroTwister (planet widget) that I created for yahoo widget engine here, because it is some pretty heavy math, and posed some interesting questions about robust calculations being kicked off 10 times a second or so, and how I improved it dramatically.

When I first approached the solar system widget idea my mind wanted to take it all the way up to elliptical orbits for the planets, mainly so they could go behind the sun and come back out on the other side and swing around out front, and go behind again (most apparent in mercury because it's closest to the sun).

Calculating the 9 planets positions on ellipses ten times a second was crushing my computer, but the previous flat model where I constructed circular orbits for each of the planets while I was originally building the widget did not command all of the computers processing power, but the ellipses and the planets going behind each other and coming back out looked amazingly cool, so I brain stormed on the subject.

My solution was to keep the circular orbits, but to shrink their height (Y values) by a percentage. This produces simulated ellipses that are just as convincing as the real thing, and keeps my widget running below 1 percent processor usage on modern computers even when it's set to full speed!

Any math people out there want to explain what you call a circle that is shrunk by a percentage on the Y axis? Is that truly an ellipse, or something else completely?

Here is ellipse formula: "(((x-h)*(x-h))/(a*a)) + (((x-h)*(y-k))/(a*a)) = 1"

What I basically did in my widget is for every point on a circle: "(x*x) + (y*y) = (r*r)"

Except I was solving for "y", so my radius was a known value: "y = SquareRoot[(r*r) - (x*x)]"

I would then shrink all "y" values creating my circle by a percentage... say shrinking to 80%

(y/100)*80 = 80% of "y" Yes, this can also be displayed as "y*0.8" but what fun are percentages if you don't solve for 1 percent first? Maybe that's just me.

It's pretty easy on a graph if you are already graphing points of a circle... just evaluate "(y/100)*80" in each x/y coordinate for the circle.

My question remains... does this satisfy the definition of an ellipse? I have never reverse engineered my equation to see if they truly form ellipses, but I can't imagine what else you would call these orbits
foumy6
in this one I will show you how to add, subtract and multiple matrices. For those of you who done know what a matrices looks like i will sort of have a totorial in this post to show you what one looks like and how to read them in my guide.
---------------------------------------------------------------------------------------------------------
this matrix would be called a 1 by 1 matrix = [ x ] (x representing any number + or -)
then you can have any soze of matrix but I'll keep it simple for now, so this would be called a 1 by 2 matrix = [ x1 x2 ] (not x to the first and x squared it is saying the first x and the secind x.
------------------------------------------------------------------------------------------------------------
Now we get into marices having "y"s as well as x's I will start with a simple 2 by 2 matrix =
x's[ x1 x2 ] (now sence there is now way for me to show you what a full matrix like this would look y's[ y1 y2 ] like I am going to just put "[ ]" again but I will label them x and y to help simplify it.)
--------------------------------------------------------------------------------------------------------------
You can get very large with matrices. Now most commonly the numbers in the martix stand for poins in a cordinate grid, but not always so if it is they will most likley have lables over each row of elements (elements are the x's and y's)like this:
A B C D
x's[ 3 2 4 6 ]
y's[ 6 5 8 3 ]

So this matrix would represent the cordinateds:
A(3,6)
B(2,5)
C(4,8 )
D(6,3)
So this whould make a qudrilateral of some sort.
In my next post I will be showing you haw to add subtract, and multiply matrices.
foumy6
[ 2 5 ] + [ 1 8] = [ 3 13 ]
x1 in the first matrix=2
x1 in the second matrix=1
2+1=3 then you do the same with x2's - 5+8=13
-------------------------------------------------------------------------------
Now lets get to a bit harder addition the hardest you will need to know is a 2 by 2 the rest will be easy then after:
x[ 3 6 ] + [ 4 2 ] = x[ 7 8 ]
y[ 2 4 ] + [ 1 6 ] = y[ 3 10]
So all you do is add the x1's and x2's and the y1's and y2's
-------------------------------------------------------------------------------
Im not going to write up a whole guide for subracting because its the same as addition just subtracting so just subtract the x1's and x2's and y1's and y2's and so on.
------------------------------------------------------------------------------
I 've decided to move multiplying to a different post because im out of time to work on posts todays srry
IceCreamTruck
 foumy6 wrote: I 've decided to move multiplying to a different post because im out of time to work on posts todays srry

OK, you can't stop there because you have my attention... just FYI. Call it a refresher course for me because I used to know this, and it's coming back as I read your post, but I have forgotten a lot of this.

x[ 3 6 ] + [ 4 2 ] = x[ 7 8 ] this equation could be for sales.... say you have 6 sales in the first three hours, and two more by the fourth hour, and you want to know how many sales per hour you have... your equation let's us keep this in grid form without breaking down and separating ideas. Just add hours to hours, and sales to sales, and you get total hours, total sales. More importantly this equation let's you do this math on the board in front of your boss instead of getting back at him when you've had time to count up all the hours and sales. Now you can just put that point on the grid and move on with your presentation. Hopefully you've had more sales than 8 in 7 hours... unless you are selling Lamborghini's.

I follow, but I'm ready for the good stuff now! Your set up is good, maybe too good, but we require more meat! errr... um, math! lol
foumy6
IceCreamTruck wrote:
 foumy6 wrote: I 've decided to move multiplying to a different post because im out of time to work on posts todays srry

OK, you can't stop there because you have my attention... just FYI. Call it a refresher course for me because I used to know this, and it's coming back as I read your post, but I have forgotten a lot of this.

x[ 3 6 ] + [ 4 2 ] = x[ 7 8 ] this equation could be for sales.... say you have 6 sales in the first three hours, and two more by the fourth hour, and you want to know how many sales per hour you have... your equation let's us keep this in grid form without breaking down and separating ideas. Just add hours to hours, and sales to sales, and you get total hours, total sales. More importantly this equation let's you do this math on the board in front of your boss instead of getting back at him when you've had time to count up all the hours and sales. Now you can just put that point on the grid and move on with your presentation. Hopefully you've had more sales than 8 in 7 hours... unless you are selling Lamborghini's.

I follow, but I'm ready for the good stuff now! Your set up is good, maybe too good, but we require more meat! errr... um, math! lol

Lol I will try to get more up as soon as I can lol but for now IM Playing on the new iPad and am loving it!!!!!!!
IceCreamTruck
 foumy6 wrote: Lol I will try to get more up as soon as I can lol but for now IM Playing on the new iPad and am loving it!!!!!!!