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Geometric Analysis Grants

thindes57 — Fri, 27 Nov 2009 14:05:24 +0000

Funding Source: National Science Foundation
Funding Type: Discretionary, Grant
Total Available: $8.8 Million
Award Ceiling: N/A
Deadline: 11.02.10
Eligibility: Unrestricted

Description:
The program in Geometric Analysis supports research on differential geometry and its relation to partial differential equations and variational principles; aspects of global analysis, including the differential geometry of complex manifolds and geometric Lie group theory; geometric methods in modern mathematical physics; and geometry of convex sets, integral geometry, and related geometric topics.

The Footprint--Determinism

bwinwnbwi — Thu, 26 Nov 2009 13:02:56 +0000

Yes, God has a physical footprint and it’s grounded in the Logos of existence as it is described in the “the new model of the observer/observed relationship.” Accordingly, we live in a universe that, on one level, is deterministic, while, on another level, is less deterministic. However, the entire universe is comprehensible by people who can comprehend—you, me, and the scientist. Also, according to this Logos, death is not “the end;” rather, death is like the off ramp of one highway merging on to another highway—all energy far from equilibrium, eventually, must take this “off ramp.” However, information generated on the highway of life moves full speed ahead (by reproduction and natural selection, on the one hand, and by culture—language, books, libraries, etc., on the other hand). And, finally, we live in a universe where comprehensibility begins and ends in duality. Initially, this duality begins with the wave/particle duality of conjugate variables, and later, this duality is defined by human intelligence embedded in the physical events. The boundaries that shape God’s footprint then are defined by the duality that constitutes the comprehensibility of the universe, e.g., ~~b (wave/particle duality), ~bb (accommodation/assimilation of living creatures duality), and, b~b~bb (the duality of physical event/human intelligence).

One of my favorite scenes in the movie Godzilla was when Matthew Broderick found himself in a huge hole searching for Godzilla’s footprint. The craterlike hole and the footprint were one, but Godzilla’s footprint was off the scale of any footprint Matthew had ever encountered so it remained hidden from him until a military officer pointed out that he was standing in the middle it. In a like manner, we are standing in the middle of God’s footprint, the breath of which begins in the quasi-material world described by quantum physics, extends up into Einstein’s space-time continuum and is as deep as what is humanly possible to imagine. Since we know the parameters of the footprint, we can extrapolate a shape that is much more manageable. The footprint is shaped like a piece of pie! The space-time continuum exists in the mind’s eye of the physicist, but the rest of us know this continuum only through its effect on (some) physical events, so let the physical event represent one end of the piecrust and at the other end of the crust sits the observer. Both the physical event edge of the pie and the observer edge of the pie comes together at the narrow slivered end of the pie piece. Let the slivered part of the pie represent the quasi-material world of quantum phenomena.

The physical event, or that which we see, smell, taste, touch, and hear, occurs along the physical event edge of the pie piece while the comprehensibility aspect of the universe occurs along the pie piece’s observer’s edge. In other words, the physical event side represents what I (and Northrop) call the aesthetic continuum while the observer’s edge of the pie— or that which, in one form or another, senses an environment, — represents “liberation from the aesthetic continuum.” As always, from the human observer’s point of view, the aesthetic continuum is subject to an analytical account, or the hypothetical deductive method which postulates the public side of the continuum, and of course, there is the more personal, relative, experiential aspect of that continuum, one’s own individual, relative experience of it. The public side of the continuum, though, thanks to the advances of Relativity and quantum physics has changed the meaning and significance of the physical event, and that change woke me from my drunken slumber (my drunken slumber comment is a very loose paraphrase of Kant’s comment on Hume’s critique of Locke’s theory of knowledge). Of course, the implications of Relativity theory and quantum mechanics are still being debated (after ninety years and counting) and I, like so many more, am eagerly waiting to see how it turns out. Fortunately, I’m not holding my breath,—which brings me to a brief description of my upcoming posts.

While trying to comprehend the meaning of the “new physics” awhile back, I wrote some dialogue. The dialogue below deals mostly with Relativity theory. Next week’s post wanders in and out of Relativity theory and quantum mechanics. After that, well, I’m only sure of a post on the observer, or the connecting link that shapes God’s footprint. After that maybe a post on temporality etc. etc., time will tell.

Our old Professor friends, — the philosopher, Noel, the physicist, Tony, and the English Professor, Stan, — have been discussing this situation (the significance of the physical event), so perhaps they can make this idea more clear?

“Maybe Noel,” interrupted Tony, “you’re referring to a different Einstein. The one that I thought we were talking about is the one who eliminated the confusion concerning space and time. We have known for a long time that people in other cultures experience space and time differently. But that’s the beauty of Einstein’s work; now we can all agree that space-time intervals are the same for everybody, even for space aliens traveling at close to the speed of light. We now know that the length of a space-time interval between any two events is the same for everybody.”
“Okay, Tony, if you want to jump into the thick of it, than lets do it,” replied Noel. “The space-time interval, what’s it based on?”
“The speed of light, or rather the constancy of the velocity of light,” Tony responded. “You and I share the same space-time, but my space and your space, and my time and your time, are the same only when we are at rest relative to each other. We live in our own private worlds of space and time, but in the new public domain of space-time, space and time are the same for everybody. In fact, the intrinsic structure of space-time accounts for the constancy of the velocity of light for all observers.”

“Do you know why?” said Noel.
“Sure,” responded Tony, “it has to do with the implications of relativity theory. In the mathematics of space-time, Minkowski, Einstein’s mathematics professor, showed that even though the Pythagorean theorem does not work in space-time, something like the Pythagorean theorem is still at work. In Euclid’s geometry the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two sides. In the geometry of space-time, the distance between two events, like in the Pythagorean theorem, is equal to the time interval squared minus the space interval squared, however, that minus is the reverse of what takes place in the geometry of Euclid. Subtracting, instead of adding the two intervals, produces four-dimensional space-time. In space-time the distance between two events connected by a light ray becomes zero. Light rays coming at us from outer space take time to reach us, but in space-time no distance is traveled. That’s one of the incredible results that follow from Einstein’s theory. And that is also why the speed of light is constant for all observers. In space-time light is just there, everywhere.”

“I’m just a little confused,” said Noel, “If light doesn’t go anywhere, how can we know that the length of a space-time interval between any two events is the same for everybody?”
“Because of the constancy of light’s velocity,” Tony replied.
“So what you’re saying is that time doesn’t change, just space?” said Noel. “Is that the answer? Don’t answer that. There’s ‘no’ time to answer, right? Anyway, Einstein’s field equations dictate the space of space-time, and, as you have all ready pointed out Tony, we can agree upon the measured value of space-time. Is that about right?”
“Well, a stab in time will get you nine,” Tony muttered. “You know damn well what I’m talking about Noel. It’s just that you don’t like it. You won’t accept that in the cosmic scheme of things, you and I, and everybody else, are just world lines. That past, present, and future may, or may not, possess meaning scares the hell out of you. You hate the idea that your private frame of reference might be limited and meaningful only to you. Einstein’s universe attacks your sense of freedom, your dignity. Well I’ve got news for you. Nobody was more concerned about dignity than the old man. He didn’t bemoan the fact that he wasn’t God. It was enough for him to peer into the heart of nature, or the mind of God if you prefer to call it that, and understand what was really going on. It was enough for him to know that all human beings had this gift, but how it was used was a person’s own business. Denying it, however, was not dignified. It was just plain stupid; and anyway, what about the effects, the predictable consequences of Einstein’s theory? If they don’t occur in reality then where do they occur?”

“Right where they are predicted to occur,” Noel replied, “in the surrounding manifold of our sensual experience. Nature, or the name that we give to that manifold, takes in everything we can see, hear, feel, taste, smell, and explain. Space, as an ontological entity, in the theory of general relativity, doesn’t exist. The being of space has been replaced with purely methodological considerations. What space ‘is,’ or whether any definite character can be attributed to it, is no longer a concern. Rather, we must be concerned with the geometrical presuppositions, the ‘ideal meanings’ that get used in the interpretation of the phenomena that we ascribe to nature according to law.”
“I’m getting tired of this,” said Tony. “Science gets done and benefits follow, which, really, is all we have to worry about, right Stan? How come you’re so quiet, anyway? That’s not like you. Are you sick or something?”
“I’m fine. You know me, quiet as a mouse, but sharp as a tack,” said Stan. There’s a time for talking and time for listening. I’ve been enjoying the latter. Let me try to simplify this conversation, eh fellows; that is, after I throw another log on the fire.”
“Always the educator, eh Stan,” said Tony, “but that’s why we love ya.”

“Take nature for instance,” responded Stan, “for you Tony, its independent of the observer. It’s a bit complicated, but knowable, and it exists before one begins to experiment on it. That’s not the case for Noel. For him, nature does not exist independent from the observer. In fact, questions asked concerning nature, for Noel at least, actually brings nature into existence. And, he looks to quantum mechanics to substantiate that claim. On that level, the physical world seems to emerge from observations made on it. Any argument there fellows?”

“You’ve got the stage,” replied Noel, “go for it.”
“Now for the hard part,” said Stan, “On the one hand we have Einstein’s theory of general relativity, and on the other hand we have quantum theory. Both theories are proven successes, but when taken together they are out of joint. The equations that describe the gravitational field are completely different from the one’s that describe subatomic interactions. Moreover, space and time are intimately related in relativity theory. They are dependent on the state of motion of the observer. In quantum theory space and time are not tied to existence at all. As far as a person’s limited reason is concerned, there is no quantum world, just an abstract quantum physical description. Given this confusing state of affairs, it would be doctrinaire and dogmatic to say that one theory is better than the other, or that one is talking sense and the other is lacking in it. Right fellows?”
“Who’s patronizing now,” said Tony.
“Guilty as charged,” responded Stan, “I guess nobody’s perfect. For you Tony, the mind’s ability to discover reality’s true nature is a religious belief, just like it was for Einstein. If Einstein had a religious belief, it was that the world is comprehensible and objective.”
“I’d probably go to church, if I could sit next to Einstein,” Tony replied.

“As I was saying,” said Stan, “under the rule of cause and effect everything has its place and time, but that is not what works for you Noel. Knowledge, for Noel, constitutes what we take to be the physical world, and new knowledge may substantially alter that world. In other words, over time, both knowledge and the perceived field that we find ourselves in changes. Both Cassirer and Kant agreed on this. The function of the mind’s capacity to connect meaning to sensual contents goes beyond sensual contents and establishes an order among the connections between them. The necessary elements of every assertion—being and non-being, similarity and dissimilarity, unity and plurality, identity and opposition—cannot be represented by any content of perception, but through them ‘ideal meanings’ get created, and when applied to the perceptual field those elements fill our perceptions with meaning. That process, over time, alters both the meaning and the content of our perceptual field. But, what it comes down to in the end is testing the deductive consequences of those ‘ideal meanings’ against the sensual contents in the field of our perceptions. That was the way it worked for Einstein and, in any universe that will not change.”

Based on the above dialogue, for me at least, the physical event seems a little less obvious! But it’s still there; the foundational attribute of our knowledge of the objective world is still there. It’s just that it seems a little more open to interpretation at this point. Anyway, the physical event is only one aspect of God’s footprint. To get a better perspective on the footprint, (and I’m sure Matthew Broderick would agree here), we need to climb out of the hole in order to see the whole pie piece—errrr footprint!

Interesting Facts About Angles

asv3 — Thu, 26 Nov 2009 11:18:20 +0000

Angles formed between two interesting straight lines, there are 3 types of angles

1. Acute angles – lesser than 90º

2. Obtuse angles – greater than 90º

3. Right angle – Exactly equal to 90º

Below are some interesting facts about angles.

Adding up all angles in a 3 sided shape aka Triangle will make up to 180º
Adding up all angles in a 4 sided shape/square/rectangle/parallelogram/quadilateral qill make up to 360º
If measurements of all angles are same in any shape, lengths of all sides in the shape would be same. (all sides are equal in equilateral triangle, all side are same in a square, etc.)
A right angle/90º angle denoted by a square mark
If two angles add up to 180º then they are SUPPLEMENTARY
If two angles add up to 90º they are COMPLEMENTARY
Opposite angles formed by two intersecting lines are equal

The geometry of meandering rivers

Dave Richeson — Thu, 26 Nov 2009 04:58:05 +0000

I regularly watch the Science Friday video podcast. This week they had an interesting piece on potamology (OK, I just learned that word and wanted to use it in my post: potamology is the scientific study of rivers). The podcast showcased the work Christian Braudrick and Bill Dietrich of University of California, Berkeley, who achieved what had not been done before; they were able to create a meandering river in a laboratory setting.

Such rivers lazily snake back and forth through a gently sloping field. They change shape over time by continually eroding the outer bank of each curve and depositing silt on the inner bank. Occasionally two bends in the river come together, changing the topology of the river, and ultimately leaving behind an oxbow lake.

In passing, the narrator mentioned a very interesting mathematical fact about meandering rivers. It turns out that the size of a river cannot be determined by its shape on a map. In particular, if you looked at an aerial snapshot of a meandering river, you would not be able to tell whether it is the Amazon or a small neighborhood stream! I had to investigate this stunning fact.

It appears that nature forces certain geometric relations on the features of the river. Let denote the length of a meander (the “wavelength”) and be the width of the river (see my drawing below). The curves in a meander are roughly circular (rather than sinusoidal, for example). Let denote the radius of curvature of a bend.

It turns out that these variables are almost always related in the following way:

I was stunned to learn this. What a beautiful fact. To see a remarkable scatterplot of the first relationship, see Figure 12.16 in The environment: principles and applications, By Chris C. Park (p. 374).

From what I can tell, the observation dates back to a 1960 paper by Luna Leopold, the Chief Hydrologist of the USGS.

I have no idea why there is a linear relationship between these variables, but it seems to me that if you know that such a relationship exists and you knew one of the constants, then you could estimate the other one. For example, suppose you knew that . In each cycle, a river winds clockwise around one circle of radius , then counterclockwise around another circle of radius . In doing so it will produce a wavelength of roughly .

[Images by:prefers salt marsh (CC BY-NC-ND 2.0), petercastleton (CC BY 2.0), molas (CC BY-NC-SA 2.0), stuant63 (CC BY-NC 2.0)]

Geometry - Midterm

ecrlopez — Wed, 25 Nov 2009 19:51:57 +0000

Use this link to access the test

You must enter your full name and a valid email address in order to recieve credit. Any questions that you leave blank will be counted as wrong. If you submit your test multiple times, only your first score will be recorded.

Meditation XXI, René Descartes (1596-1650) – Meditations on the First Philosophy, and “Objections and Replies”

jamesesz — Wed, 25 Nov 2009 06:22:09 +0000

René Descartes ~ When two people meet, they unconsciously affect one another in ways the mind cannot

Triangle Congruence by AAS and ASA Part 2

Mr. Pi — Wed, 25 Nov 2009 04:33:52 +0000

If you have not read my previous post on triangle congruence, make sure you do. Anyway back to the angle-side-angle and angle-angle-side theorems. I will prove even more triangles congruent in this post! Like I wrote before, that is all I really remembering doing in high school geometry.

Example 1 Triangle Congruence Proof with AAS

The first example is example 3 in the video. If you are wondering why I am writing about something I already did in a video, let me explain. It has to do with different learning styles. It is pretty important to understand your learning style. I will have to blog about learning styles soon. Anyway, through my blog, I try to reach a variety of learning styles. Thus, I create graphics, write and create videos to get my math point across. Anyway, on with example 3 from the video.

Given: angle S is congruent to angle Q and segment RP bisects angle SRQ

Prove: triangle SRT congruent to triangle QRP

When I look at what is given and the diagram, I notice the two triangles share a side which will be a corresponding congruent side, segment PR. Since, segment RP bisects angle SRQ, that means angles SRP and QRS are congruent corresponding parts by the definition of angle bisector. Those two facts taken with the given, provides enough proof to establish AAS in both triangles SRT and QRP.

Example 2 Triangle congruence with AAS

Given: angles B and D are congruent and segment AB is parallel to segment CD

Prove: triangles ABC and CDA are congruent

As is true with any proof, you need to understand the given and how it will help you identify a pair of correspond angles or sides of a triangle congruent to use one of the methods to proving triangles congruent. From the given, we have a pair of corresponding angles congruent, D and B. From the diagram a pair of corresponding sides can be established. AC is a shared side and with the reflexive property of congruence it can stated AC is congruent to AC. Also in the given, it is stated that segments AB and DC are parallel. Whenever you hear the words parallel lines, you must remember the special angle pairs formed by two parallel lines and a transversal. In this case, angle BAC is congruent to angle DCA because of the alternate interior angles theorem. With all of that, it can be said that triangle ABC is congruent to triangle CDA by the AAS theorem.

Example 3 Proving Triangles Congruent with AAS

Given: Segments XQ and TR are parallel and segment XR bisects QT

Prove: triangle XMQ congruent to triangle RMT

This proof requires the most work of all the proofs I have done on this blog. There are no corresponding parts given as congruent, which means we have to establish three pairs of corresponding parts congruent. The two keywords in the given are parallel and bisects. Angle X and angle R are alternate interior angles and are congruent because the two angles are formed by two parallel lines and a transversal. Angles XMQ and RMT are congruent because all vertical angles are congruent. I have two angles and need to prove 1 pair of corresponding sides congruent. Those sides will be segment TM and segment QM by the definition of segment bisector. Since the sides are the non-included sides, triangle XMQ is congruent to triangle RMT by the AAS theorem.

Ethos

swanktown — Wed, 25 Nov 2009 03:13:26 +0000

I wonder if ethos would help.

I’ve lost it, you know. My window seat. We switched spots and the shades are drawn up. No one sits there anymore.

I can hear my math teacher mumbling to herself at night, scribbling on her seating chart as her husband begs her to come to turn out the light and come to bed. She tries too hard. “That Rachel girl is trouble,” she would say. “Always reading J.D. Salinger, always daydreaming. Let’s move her away from any possible distractions.” And you can’t forget the maniacal laughter.

Relax, I’m in another weird seat. This one doesn’t even have a railing. So I guess don’t relax or you’ll fall off. It’s big, which is better than the crammed-in chair next to my friend.

I miss my old seat, though. In the final days of my free-gazing fourth period, there was a bubble-letter romance beginning to unfold on my desktop. Rooted in hatred of love, I thought it ironic that they had eventually asked each other where they sat at lunch. Still, I think my teacher may have sprayed it off. She has one of those ultra-religious moral fibers.

“It’s just wrong to separate two lovers,” I would write in carefully-constructed criticism, “whether they be humans or the mind and a seat.” Which is right. How can I possibly concentrate without my alluring window seat? And how can the two lovers stay focused while they bite their nails, wondering what the teacher has to say about two Geometry students–gasp!–in a romance?

Yeah, I think ethos might work.

He Knows All the Angles

makya20 — Tue, 24 Nov 2009 09:35:43 +0000

To: Pythagoras@greece.com

From: plato@greece.com

Subject: New theorem

Dear Pythagoras,

The council has examined your new theorems. Firstly, let me reiterate that your past contributions to the world of philosophy and mathematics are greatly appreciated. Perhaps you ought to focus on your continued teaching of these past ideas, rather than obsessing with creating new theorems.

Specifically, you’re contribution – “when it comes to triangles of the right variety, b squared plus a squared is equal to c squared.” Surely you don’t think we would not notice this as a simple reorganization of your prior theorem. Ditto for “of the right triangle sort, it is always true that c squared be equal to the sums of a squared and b squared.” You had a great run, buddy, don’t push it.

Echoes

playtom — Tue, 24 Nov 2009 00:18:14 +0000

An interesting search on games like Geometry Wars gave me the result of Echoes, a free PC game inspi

adios a este maldito semestre

yakimeshi — Mon, 23 Nov 2009 22:32:57 +0000

(no , no es solo una reedicion de bye bye to el semestre) La verdad es que no es un secreto que me caga mi escuela, a la cual he asistido por más de 13 años (yuck!) pero al fin , con el termino de este semestre parece que hay un rayo de esperanza a mi aburrida estadia en el sistema de educacion mexicano, jeje y no dire más, sigan en sintonía , me encuentro trabajando y concentrado en algunas cosas como el diseño, y la música pronto sabrán algo de ello, y por mientras les dejo esta bonita imagen que podrán usar como wallpaper para su iphone, celular, zune player, etc, enjoy!

Geometry Week#12 HW

jefffromcanmore — Mon, 23 Nov 2009 05:07:50 +0000

Class #1 Section 4.7 Triangles and Coordinate Proof.GP (p.246) 1 – 5 PA (p.247 – 249) 7 – 11 odd, 12, 13, 15, 17, 19, 21, 23, 25, 27, 34 Review Section in Text.

Class#2 Review of Chapter 4 Handout Review Questions. Geometry Chapter 4 Review Makeup Tests for Unit #3.

Class#3 Test on Unit #4 All Period Surprise!

Oh, Geometry.

koreangirlxd — Mon, 23 Nov 2009 00:19:22 +0000

oh, Geometry.

why are you here

torturing us kids

giving us F’s.

Oh, Geometry,

Please go away,

I dont think Euclid meant to create you;

he never thought you would

make us rip our hair off.

Oh, geometry,

wats the use of you,

Proofs, angles, circles;

when will we use them?

Oh geometry please give me an A on my test,

Even though I dont I dont understand circles

and its supposed to be the hardest test.

Please give me an A, im trying very hard,

And help me with that final of yours.

Oh geometry, Im sorry but Algebra 2 is better

and i cant wait, cant wait.

Oh, Geometry, please dont haunt me,

in my sleep,

but most of the time,

I am without sleep studying for your tests.

SO PLEASE GIVE ME AN A I NEED 2 GO 2 SLEEP!!!!!!!!!!!!

ahhhh. Geometry is hard. hard. hard.

Actually, i think its because im just not that great at math,

because there are these 12 yr old smarties in my class that always get 100s in their tests. sighh.

I NEED 2 STUDY. Thank goodness for Thanksgiving break coming up.

Platonic Solids -- Video -- Mapping to Formation

bowman — Sun, 22 Nov 2009 22:55:35 +0000

This is a simple video that show the formation of the solids from their maps. It makes me dizzy.

Stellated Icosahedra Project Pattern and Some Examples

bowman — Sat, 21 Nov 2009 23:15:45 +0000

By their internal angles

alcyril — Sat, 21 Nov 2009 03:09:48 +0000

Let us now go to the second group of triangles which is classified by their internal angles. We will learn here four triangles namely;

Thank you wikipedia for the definition.

How can I relate the relationship of these triangles to the concept of Love? Hmmmm. . .

==============================================================

First in line is the right Love Triangle. Can a love triangle be right? When can you say that a love triangle is right? What do you think? Hey. . . let me think first of a situation. . . Hmmmmm. . . .

Eureka! Here. But I guess this one can be one those that can relate a Love triangle of becoming right. Have you ever watched Smallville which is an American television series of how Clark Kent became superman? Watch this video. . .

(Click Here)

==============================================================

===============================================================

Let’s Review the Triangle!!

from- Youtube

================================================================

4.3 Triangle Congruence by AAS and ASA

Mr. Pi — Sat, 21 Nov 2009 01:15:39 +0000

In the video, it is not labeled example 1, but the first bit of information is critical to this lesson.

Angle Side Angle Postulate

It two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Angle Angle Side Theorem

It two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent.

It would be good to remember that a postulate is something that is assumed to be true without question. Theorems are facts that have been proved true using definitions, postulates and other already proved theorems. We cannot use a theorem until we learn about it.

Example 2

In a recent post about proving The Converse of the Alternate Interior Angles Theorem, I created my first flow proof. It was a short proof, much like in this example using ASA. I color coded my markings, but on the vertical angles APX and BPY I did not get the correct shade of blue.

Example 3

As you can see, it is a fairly complicated diagram for a relatively easy proof.

Given: angle CAB is congruent to angle DAE, segment AS is congruent to segments AE and angle ABC and AED are right triangles

Prove: triangle ABC is congruent to AED

Planning this proof requires you to understand your given information and the prove statement. Since the diagram has an added dimension of difficulty because extra triangle in the middle, you need to focus on the two outer right triangles ABC and AED. These are the angles that need to be proved congruent.

It is given that angles CAB and DAE are congruent and are corresponding angles in each triangle. Also, the corresponding sides AB and AE are congruent. Finally, there is information relating to the corresponding angles ABC and AED. They are both right angles. Angles ABC and AED are the angle that gives Angle-Side-Angle in each triangle. Since all right angles are congruent, angles ABC and AED are congruent. Therefore, both triangles ABC and AED are congruent by ASA. QED

Example 4

Given: segments NM and NP are congruent and angles M and P are congruent

Prove: triangles NML and NPO congruent

There are many ways to prove the triangles congruent in this diagram, but I like to produce the most concise proof possible. Hopefully you can see the ASA relationship that will be able to be proved from this set up. Angles LNM and ONP can be proved congruent to establish the ASA congruence in these two triangles.

Since it is given that segments NM and NP are congruent and angles M and P are congruent, a ASA congruence can be established with triangles NML and NPO. Angle LNM and angle ONP are congruent because they are vertical are congruent. Finally, it can be stated that triangles NML and NPO congruent because of the Angle-Side-Angle Theorem. QED

As usual, I hope this has been helpful. Be sure to check back for more geometry.

Regards,

Mr. Pi

Relative length of their sides

alcyril — Sat, 21 Nov 2009 00:55:09 +0000

There are three main classification of Triangles with regards to the relative length of their side namely;

thank you wikipedia for the definition

As you can see, each triangle has corresponding persons that makes each triangle a Love Triangle. Here are the stories behind those Love Triangles.

=============================================================

First we have the famous love triangle of Kevin Federline, Britney Spears and Shar Jackson. Let us consider this an Equilateral Love Triangle wherein both Britney and Shar indeed had an equal relationship with Kevin Federline since they have both tried marrying Kevin and had some sad times during at some point in both relationships. See how Equilateral?

See full article. . .Thanks Castina for this article.

==========================================================

And then next is this Isoceles Love Triangle. This is a romantic love triangle wherein a girl, for example, really can’t say whether which of the two guys really has the edge over her heart.

Have you ever watched Boys Over Flowers? I really love Boys Over flowers a lot because of its unique love triangle story between Gu Jum Pyo, Geum Jan Di and Yoon Ji Hoo. At first, Geum Jan Di really is seeking for the love of his “soulmate” Yoon Ji Hoo but later on, this rude guy, Gu Jun Pyo came in and then changed everything. Here in this TV drama, you will see how Geum Jan Di really had a hard time to choose the real one. . . since both sides are equal!

===========================================================

Lastly is the Scalene Love Triangle. Who knows about the love triangle of the Kapuso Stars Kris Bernal, Marky Cielo and Aljur Abrenica? Eventhough both of Marky and Aljur are very consistent pleasing this lovely lady, none of them really have tried courting her. Maybe there’s something that is happening, but as what Kris said, there’s no instance that both tried courting her and then that they are only good friends. As what Kris said, there’s no edge between Aljur and Marky in her heart. Both of them really has a special place in her heart and that theirs no one parallel to her. See how this love triangle goes?

Read more. . .

=============================================================

Construcciones Imposibles con regla y compás (parte 2)

slavador86 — Sat, 21 Nov 2009 00:47:53 +0000

Duplicación del Cubo Duplicar el cubo consiste en construir el lado de un cubo que tenga el doble de

Properties of Triangles and Quadrilaterals

Samantha Starke — Fri, 20 Nov 2009 22:20:47 +0000

Here’s a handy cheat sheet for remembering the properties of some common polygons.

Three Sided Shapes (Triangles):

Area=1/2 b*h

Sum of Interior Angles =180°

Equilateral

-All sides of equal length

-All angles equal 60°

Isosceles

-Two sides of equal length

-Two angles are congruent

Scalene

-No sides of equal length

-No angles are congruent

Right Triangle

-One 90° angle

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Four Sided Shapes (Quadrilaterals):

Sum of Interior Angles =360°

Area = b*h

Rectangles

-All interior angles = 90°

-Opposite sides are congruent and parallel.

Parallologram

-Opposite sides and congruent and parallel

Square

-All interior angles = 90°

-Opposite sides

Rhombus

-All sides are congruent

-Opposite sides are parallel

Trapezoid

-At least one pair of parallel sides

-At least one pair of congruent sides

Geometry Assignment -- Book 6.4

bowman — Fri, 20 Nov 2009 17:00:55 +0000

By their internal angles

alcyril — Fri, 20 Nov 2009 06:36:34 +0000

The other group of triangle is classified by their internal angles. They are namely;

Thanks to wikipedia for this definition

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Let’s review the different types of triangle with this very simple video.

This would really help you in recalling the concepts about the triangle.

from- Youtube