OpenLLM-France/wikimedia huggingface dataset:
# Human Legacy Course/The Beginnings of Civilization (Prehistory - 1000 BC)

Early humans slowly spread from Africa to other parts of the world. These early people struggled to survive by using basic stone tools and weapons to hunt animals, catch fish, and gather plants and nuts. For hundreds of thousands of years, early people lived in this way. Then, as the last Ice Age ended, some people learned to farm. This breakthrough gave rise to villages and cities—and in time, to the first civilizations.
Scientific evidence suggests that the earliest humans lived in Africa. This week, you will learn how humans spread to other parts of the world and established the first civilizations.
Let's Begin!
# Human Legacy Course/The Early Middle Ages (800 - 1215)

At the outset of the early Middle Ages, western Europe was a land without empire. With the Roman Empire gone, new forms of community took hold. Christianity spread throughout western Europe, forging unity at a spiritual level. Strong, new political systems also arose, uniting much of Europe.
The relationships between people and their governments have changed greatly across time. This week, you will learn about sweeping political changes in Europe during the Early Middle Ages, some of which continue to shape citizenship and government today.
Let's Begin!
# Korean/Words/돌멘

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 돌멘
### Noun

* 고인돌 (큰 돌을 몇 개 둘러 세우고 그 위에 넓적한 돌을 덮어 놓은 선사 시대의 무덤).
    (loanword) dolmen

### Related terms

* 고인돌 (goindol, "dolmen"), origin unknown.
* *돌면 (돌宀, dolmyeon, "dolmen"), assumed.
* 면 (宀, myeon, Old Chinese *men, "roof; house").
* ↑ https://ko.dict.naver.com/#/search?query=돌멘
* ↑ https://en.wiktionary.org/wiki/돌멘
* ↑ A dolmen, a prehistoric megalith having a capstone supported by two or more upright stones.
* ↑ 宀 #Japanese On: べん (ben), めん (men) Kun: やね (yane, "roof"), いえ (ie, "house").
# Korean/Words/댐

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 댐
### Noun

* 발전(發電), 수리(水利) 따위의 목적으로 강이나 바닷물을 막아 두기 위하여 쌓은 둑.
    dam, cf. dyke

### Relatives

* 댐 dam < dam #English
* 발전(發電), 수리(水利) 따위의 목적으로 강이나 바닷물을 막아 두기 위하여 쌓은 둑.
* 유의어 - 둑 방죽 제방
* 둑
* 높은 길을 내려고 쌓은 언덕.
* 하천이나 호수의 물, 바닷물의 범람을 막기 위하여 설치하는, 흙이나 콘크리트 따위로 만든 구축물.
* 보를 만들거나 논밭을 보호할 목적으로 쌓은 언덕.
* 유의어: 댐 방죽 제방
* 방죽 < 방축 防築
* 물이 밀려들어 오는 것을 막기 위하여 쌓은 둑. cf. German Deich
* 파거나, 둑으로 둘러막은 못. cf. German Teich
* 유의어: 냇둑 댐 둑
* 제방 堤防
* 물가에 흙이나 돌, 콘크리트 따위로 쌓은 둑. 홍수나 해일에 물이 넘어 들어오지 못하게 하거나 물을 막아 고이게 한다.
* 유의어: 제당 강둑 댐

### Comparatives

    The following are citations.
* Afsluitdijk #
* The Afsluitdijk (... English: "Enclosure Dam") is a major dam and causeway in the Netherlands.
* The Afsluitdijk is a fundamental part of the larger Zuiderzee Works, damming off the Zuiderzee, a salt water inlet of the North Sea, and turning it into the fresh water lake of the IJsselmeer. The dam serves as a sea barrier to protect the inland against flooding.
* Afsluitdijk #Trivia
* The Dutch word 'dijk' (dike) is actually a division between land and water. The name Afsluitdijk would be correct if the entire IJsselmeer had been drained. However, due to the change of plans it was left flooded so the Afsluitdijk is a barrier between two bodies of waters. In Dutch that is a 'dam'. This structure could more properly be called 'Afsluitdam'.
* afsluiten
* (transitive) to close off, seal
* De waterkering sluit het IJsselmeer af van de Waddenzee.
* The embankment closes off the IJssel Lake from the Wadden Sea.
* (transitive) to conclude, to end, to terminate

### See also

* 담 (dam, "wall, thick barrier")
* 둑 (dug, "dyke") cf. levee instead!
* 벽 (byeog, "wall, barrier")
* ↑ https://ko.dict.naver.com/#/search?query=댐
* ↑ https://en.wiktionary.org/wiki/댐
* ↑ Synonymy is seen between dam and dyke, cf. Korean 담 (dam) and 둑 (dug).
* ↑ Simply, the dam is Korean 담 (dam, "wall") in itself.
* ↑ The English translation "Enclosure Dam" is implausible, as it literally means "close-off or shut-off dyke." [My comment]
# Korean/Words/도

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 도
### Particle

* 이미 어떤 것이 포함되고 그 위에 더함의 뜻을 나타내는 보조사.
    too, as well; either ... or, (negative) neither ... nor

### Comparatives

* da #Azerbaijani
* 都 #Cantonese (dou1)
* о́же #Russian (tóže)
* túdi #Slovene
* da #Turkish
* ↑ https://ko.dict.naver.com/#/search?query=도
* ↑ https://en.wiktionary.org/wiki/도
* ↑ Listen to the audio there!
# Korean/Words/도토리

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 도토리
### Noun

* 갈참나무, 졸참나무, 물참나무, 떡갈나무 따위의 열매를 통틀어 이르는 말. 묵을 쑤어 먹기도 한다.
    acorn

### Synonyms

* 상수리 (sansuri)
* 상실 (橡實, sangsil)
* 상자 (橡子, sanja)

### Relatives

* 참나무 (cham-namu, "oak")
* 상수리나무 (sangsuri-namu, "oak")

### See also

* 참나무 (cham-namu, "oak")
* ↑ https://ko.dict.naver.com/#/search?query=도토리
* ↑ https://en.wiktionary.org/wiki/도토리
* ↑ cf. 橡樹
# Korean/Words/담

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 담
### Noun

* 집이나 일정한 공간을 둘러막기 위하여 흙, 돌, 벽돌 따위로 쌓아 올린 것.
    Wall #Boundary wall, cf. dam as a barrier that bars passage of flood

### Synonyms

* 담장 (dam-jang)
* 울 (ul, "fence" as a thin barrier) cf. 담 (dam, "wall") as a thick barrier
* 벽 (byeog, "wall") cf. Danish væg, Norwegian vegg
* 바람벽 (baram-byeog, "enclosing wall")
* 성벽 (seong-byeog, "definsive wall")

### Compounds

* 담벼락 (-byeorag, "various walls")

### Relatives

* 댐 (daem, loanword "dam"), cf. Amsterdam, Rotterdam, etc.
* 둑 (dug, "dyke"), cf. Afsluitdijk meaning "enclosure dam"
* 못 (mot, "moat, pond")
* 해자 (haeja, "moat"), cf. The Hague
* Afsluitdijk in red. This is an "enclosure dam" and/or dyke. The Netherlands of dams and/or dykes!
* Afsluitdijk with the Wadden Sea (a part of the North Sea) on the left and the IJsselmeer on the right in the Netherlands

### Comparatives

* dam #Translations
* dæmning #Danish
* dam #Dutch
* dam #English
* Damm #German
* dam #Norway
* damm #Swedish
* ↑ https://ko.dict.naver.com/#/search?query=담
* ↑ https://en.wiktionary.org/wiki/담
* ↑ This definition, including "둘러막기" (enclosing) to stick to "boundary wall," is overdone a little, however prototypical it may be. Such is also the case with 벽 (beog, "wall" cf. Danish væg, Norwegianvegg). Otherwise, Korean 담 (dam, "wall") is more comparable with Dutch and English dam, etc.
* ↑ 깁흔 희즈 놉흔 담 쓰고 -- 교학고어사전 p.1469
* ↑ 담이나 벽 따위를 통틀어 이르는 말.
# Korean/Words/덕

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 덕
### Noun

* 도덕적ㆍ윤리적 이상을 실현해 나가는 인격적 능력. 공정하고 남을 넓게 이해하고 받아들이는 마음이나 행동. 베풀어 준 은혜나 도움. 착한 일을 하여 쌓은 업적과 어진 덕.
    virtue

### Comparatives

| Germanic | Norse |
| deugt #Dutch dought #English Tugend #German | dyd #Danish dyd #Norwegian dygð #Old Norse dygd #Swedish |
* De (Chinese) § Etymologies
* ↑ https://ko.dict.naver.com/#/search?query=덕
* ↑ https://en.wiktionary.org/wiki/덕
* ↑ (Scotland, Northern England, obsolete) might, strength
# Korean/Words/독

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 독
### Noun

* 간장, 술, 김치 따위를 담가 두는 데에 쓰는 큰 오지그릇이나 질그릇. 운두가 높고 중배가 조금 부르며 전이 달려 있다.
    dolium cf. Latin doga "vat, vessel"

### Synonyms

* 항아리 (hang-ari)
* 단지 (danji)

### Compounds

* 간장독 (ganjang-, "dolium for soybean source")
* 된장독 (doenjang-, "dolium for soybean paste")
* 김장독 (gimjang-, "dolium for gimjang")
* 김칫독 (gimchit-, "dolium for kimchi")
* 술독 (sul-, "dolium for liquor")

### Relatives

* 동이 (dong-i, "earthenware jar")

### Comparatives

* dolium #Descendants
* Catalan: doll
* English: dolium
* Italian: doglio
* Portuguese: talha
* ↑ https://ko.dict.naver.com/#/search?query=독
* ↑ https://en.wiktionary.org/wiki/독
* ↑ From Latin dolium "large earthenware vessel." According to John Pairman Brown, a Punic commercial loanword, comparing Hebrew דֳּלִי‎ (dŏlī) which already appears in Isaiah 40:15[2] and is from Proto-Semitic, also Arabic دَلْو‎ (dalw, “bucket”) etc.
# Korean/Words/당나귀

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 당나귀
### Noun

* 말과의 포유류. 말과 비슷한데 몸은 작고 앞머리의 긴 털이 없으며 귀가 길다. 털빛은 대부분 누런 갈색ㆍ잿빛 황색ㆍ잿빛 흑색이며, 어깨ㆍ다리에 짙은 줄무늬가 있고 허리뼈가 다섯 개이다. 병에 대한 저항력이 강하여 부리기에 적당하다. 아프리카의 야생종을 가축화한 것으로 전 세계에 분포한다.
    ass, donkey

### Synonyms

* 나귀 (nagwi)

### Relatives

* 당 (唐, dang, "Tang dynasty")

### Comparatives

* ass #Translations
* Czech: osel
* Danish: æsel
* Dutch: ezel
* English: ass, donkey
* Estonian: eesel
* Finnish: aasi
* French: âne
* German: Esel, Eselin
* Icelandic: asni
* Italian: asino
* Korean: 나귀 (nagwi), 당나귀 (dangnagwi)
* Norwegian: esel, asen
* Latin: asinus, asellus, asella
* Portuguese: asno
* Spanish: asno
* Swedish: åsna
* ↑ https://ko.dict.naver.com/#/search?query=당나귀
* ↑ https://en.wiktionary.org/wiki/당나귀
* ↑ English donkey has no cognate but possibly Korean 당나귀 (dang-nagui).
* ↑ cf. English nag and Middle English nagge, which also have few cognates.
# Korean/Words/닭

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 닭
### Noun

* 꿩과의 새. 머리에 붉은 볏이 있고 날개는 퇴화하여 잘 날지 못하며 다리는 튼튼하다. 육용과 난용으로 육종된 수많은 품종이 있으며, 가금으로 가장 많이 사육한다. 원종은 인도, 말레이시아 등지의 들꿩이다.
    chicken

### Relatives

* 달걀 (dalgyal, "hen's egg")
* 탁(啄, tag, "to peck")

### Compounds

* 첫닭 (cheot-dalg, lit. "the first chicken")
* 첫닭(이) 울다 (cheot-dalg(i) ulda, "for the first chicken to crow")

### Comparatives

| Language | to dawn #Translations | day #Translation |
| Danish: | | dag, døgn |
| Dutch: | dagen | dag |
| English: | dawn, OE: dagian | day |
| German: | tagen | Tag |
| Icelandic: | daga | dagur |
| Norwegian: | | dag, døgn |
| Swedish: | dagas | dag, dygn |
* ↑ https://ko.dict.naver.com/#/search?query=닭
* ↑ https://en.wiktionary.org/wiki/닭
* ↑ Fowl have been known in Egypt since the mid-15th century BCE, with the "bird that gives birth every day" ...
* ↑ 새벽에 맨 처음 우는 닭. the earliest chicken who's crowed at dawn
* ↑ 아침이 되다. 날이 새다. to dawn, lit. "for it to dawn or to become morning"
# Korean/Words/다래

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 다래
### Noun

* 다랫과의 낙엽 활엽 덩굴나무. 잎은 어긋나고 긴 타원형이며 톱니가 있다. 초여름에 흰 꽃이 피고 열매는 둥근 장과(漿果)로 가을에 누런 풀색으로 익는다. 열매는 식용하거나 약용한다. 깊은 산에서 자라는데 한국, 중국, 일본, 우수리강 등지에 분포한다.
    (fruit) kiwi

### Synonyms

* 참다래 (cham-, "kiwi" lit. "true kiwi")
* 키위 (kiwi, one of the worldwide loanwords)
* ↑ https://ko.dict.naver.com/#/search?query=다래
* ↑ https://en.wiktionary.org/wiki/다래
* ↑ This brand name has become the origin of worldwide loanwords; Korean 다래 (darae) may be far more worth than kiwi as such an origin.
# MDLD/inverse function

Start=inverse function Target=Mapping/Bijective/Inverse mapping/Definition
# MDLD/inverse mapping

Start=inverse mapping Target=Mapping/Bijective/Inverse mapping/Definition
# MDLD/inverse matrix

Start=inverse matrix Target=Inverse matrix/Field/Definition
# MDLD/interval bisection

Start=interval bisection Target=Real numbers/Completeness/Nested intervals/Roots/Section
# MDLD/inverse (map)

Start=inverse (map) Target=Mapping/Bijective/Inverse mapping/Definition
# MDLD/invariant linear subspace

Start=invariant linear subspace Target=Endomorphism/Invariant subspace/Definition
# MDLD/invertible (matrix)

Start=invertible (matrix) Target=Invertible matrix/Field/Definition
# MDLD/intervals (R)

Start=intervals (R) See=interval (R) MDLD/interval (R) Target=Real numbers/Intervals/Definition
# MDLD/invariant (subspace)

Start=invariant (subspace) Target=Endomorphism/Invariant subspace/Definition
# MDLD/invariant (endomorphism)

Start=invariant (endomorphism) Target=Endomorphism/Invariant subspace/Definition
# Ideas in Geometry/Instructive examples/Section 1.2

Section 1.2 Problem # 11
Find the line tangent to y = x^2 + 12x - 4 @ x=0
Well to start off, first you need to know that a tangent line to a parabola is a line that passes through the parabola in such a way that parabola(x)=line(x) has a double root. So thus, a tangent line is a line that passes through a curve such that curve(x)=line(x) has a double root.
Formula: p(x)-l(x)=(x-c)^2
Work: Start off by plugging the equation in to the formula. (x^2 + 12x - 4) would be plugged in for p(x) and 0 would be c. So the equation would be...(x^2+12x-4)-l(x)=(x-0)^2
This then is reduced down to... (x^2+12x-4)-l(x)=x^2
Now you can add l(x) to both sides to try to get it alone. (x^2+12x-4)=x^2+l(x)
From here, you want to subtract x^2 from both sides. 12x-4=l(x)
Answer: 12x-4
# Ideas in Geometry/Instructive examples/Section 1.2 problem

y=x^20-23x+4 at point x=0 f(x)=x^20-23x+4 f(x) is the slope of tangent line of this equation f(x)=20x^19-23 through calculus we that f(x)=x^a+....... (f^/)(x)=(a(x)^a-1)+...... f^/(x)is the slope of f(x) that is proven
f(x)=20x^19-23 at x=0 f(x)=(20(0)^19)-23 f(x)=-23 means the slope is -23 at x=0 y=0^20-23(0)+4 y=4
y=mx+b tangent line equation y=4 m=-23 x=0 4=-23(0)+b 4=b
so the equation is...... y=-23x+4 of the tangent line
# Ideas in Geometry/Instructive examples/Proving the Area of a Trapezoid

Problem7: Proving the trapeziod area formula by picture:
Let's start by looking at the picture. We labeled the top segment b1 (base one) and the bottom segment b2(base two) (even though the original picture has it labeled as B1). The shorter segment must always be base 1, and the longer base must be base 2. We then broke up the trapezoid by drawing in the height at two places. The height "h", are drawn to create a square with base of b1, and height h. the remaining space of B2 we labeled with x on each side. x is the base for the triangles of either side of the square. The second picture shows that we took the triangle on the left side, and rotated it and moved it on top of the triangle on the right side in order to make another square. This square is of base x and height h. It wouldn't matter which side you would rotated the triangle to make the square.
We were out to prove that the area of a trapezoid is h(b1+b2)/2. First we labeled our picture in order to see what we could compare. We drew in the heights, labeling them "h" and also drew in Base 1, "B1", and "x" is the left over space in either side of b1 to make up b2. since the triangles on either side of the large rectangle are equal, we know that x is equal. and that 2x+b1=b2. We then solved for x. Knowing that B2=b1+2x, this means that b2-b1=2x. Thus x=(b2-B1)/2 Noting that the triangles were equal, we moved, and rotated the triangle on top of the other triangle to crate a rectangle with height "h" and width "x". We now have two rectangles next to each other. One of height "h" and length "b1", and the other of height "h" and length "x"
Using this logic, we then solved for the entire rectangle created. We know that the area of a rectangle is base times height. The height of the rectangle we looked for was "h" and the base was b1+x. So Area = h(b1+x). We then plugged in our previous x value (x= (b2-b1)/2) and go that area=h(b1-(b2-b1)/2). We then distributed the h value to get that area=hb1+(b2h/2)+(b1h/2). We then pulled out a 1/2 giving us that area=1/2 h (2b1+b2-b1) we then combined like terms and distributed giving us that area=(h(b1+b2))/2
Picture is Below
By: Becky Paul paul18@illinois.edu, Heather Brozowski hbrozowski@gmail.com, Tim McCormick tmccorm2@illinois.edu, Brianne Kilgallon, Sarah, and Shannon
# Ideas in Geometry/Instructive examples/Proving the Pythagorean Theorem By Picture

The problem stated that we were allowed to assume triangles 1 & 2 are similar triangles. And on the larger triangle, the two legs (the sides that are not the hypotenuse) are equal. If you reflect the image, you can create a square within a larger square. After reflecting the image, two more similar triangles are formed (making 4 in total) and one more larger triangle is formed, thus creating a tilted square in the middle.
If the outer, smaller triangles are folded in, toward the center of the square, they do not overlap. The only part of the square showing will now be a small square formed by the longer leg (b) of each triangle. Label this length x.
x = b – a
We know this because a, the shorter side of the triangle, and x together are the same length as b, the longer leg of the triangle. If side a is taken away from side b, x is the measurement of the side that remains.
To prove the Pythagorean theorem: a$^2$ + b$^2$ = c$^2$
c =hypotenuse of the smaller triangles and is also the length of the side of the square
so, c$^2$ = the area of the square
Now we are looking for c$^2$ , or the area of the square: (Area of the four triangles) + (Area of inner square) = Area $_{Square}$ = c$^2$
Area $_{fourtriangles}$ = 4( ½ ab) Area $_{innersquare}$ = (b-a)$^2$
Area$_{square}$ = 4( ½ ab) + (b-a)$^2$ = c$^2$ = 2ab + b$^2$ – 2ab + a$^2$ = c$^2$ = b$^2$ + a$^2$ = c$^2$
a$^2$ + b$^2$ = c$^2$
By looking for the area of the square (c$^2$), we find that a$^2$ + b$^2$ = c$^2$ , which is the Pythagorean Theorem.
Group Members: Jamie Balayti, Christine Yu, Loren Martell, Kara, Katlin, Amy Nadell
# Ideas in Geometry/Instructive examples/Proving Parallel Bisectors in Quadrilaterals

Proving Parallel Bisectors in Quadrilaterals:
Problem 9 :Explain how the following picture "proves" that if a quadrilateral has two opposite angles that are equal, then the bisectors of the other two angles are parallel or top of each other. In order for this to be true within our picture we first found that each shape has two right angles in them. This means that these angles have equal measures of 90 degrees. This satisfies the first part of the statement ( two angles that are equal) Our next step was to prove that the lines drawn in the picture are parallel. To do this we wanted to work with interior/exterior angles. We need to prove that two angles are equal in order to show that the lines within the picture are indeed parallel.
We looked at different shapes to see how this statement would work. In a square, and within a parallelagram, the lines are on top of one another. In a rectangle the lines are parallel. - we haven't exactly figured out how to do this, we have a number of different ideas of how to go about finding our solution. We are assuming that the line is bisecting the small angle within the dark grey shape and the white grey shape. This angles total measure is 45 degrees. We figured this out because when we bisected the top right angle into 45 degrees. We knew that the two angles added up to 180, and we had the measures of 90 degrees and 45 degrees leaving us with an angle measure of 45 degrees for the missing angle. We assumed that the line is a bisector, leaving us bisecting this 45 degree angle bisected into two 22.5 degree angles. This is the measure of the exterior angle that we are using to prove the lines are parallel. We repeated this process on the lower quadrilaterals, proving that the other exterior angle also equals 22.5 degrees. These two angles have the same measure, and they are exterior angles. If two exterior angles are equal, then the lines creating them are parallel. We proved that these two lines are parallel. Because these lines are parallel, and our quadrilaterals have two opposite angles of equal measure, this original statement holds true. The bisectors of the remaining two angles within the quadrilateral are parallel ( in this case) or on top of one another.
By: molly.mueller12@gmail.com, michelleh7575@hotmail.com, balk1@illinois.edu, eck1@illinois.edu, east1@illinois.edu
# Ideas in Geometry/Instructive examples/Proving Pythagorean Theorem Using Similar Angels

On the picture of the triangle with the segment of ab cut in the middle of the triangle, we see that the small triangle with sides bc, b^2 and ab were multiplied by b. If we divide this triangle by b again, the sides would become side c, side b and side a. The other small triangle that is attached was multiplied by a, forming sides ab, ac, and a^2. Again if we divide these triangles up by a, the sides would be side b, side c and side a. The angle across from Side length of b should both have the same angle. Same goes for the angle across from side length a, and c. For example, the angles across from sides b in both the small triangles should have the same angle. The angle across from sides a in both the triangles should also have the same angles. Same goes for angles across from side c.
Since the angles across from sides a, and the angles across from sides b are the same, the angles across from side C should be 90 degrees. This means that there are angles of 30, 60 and 90 degrees in each of the small triangles, which proves that this that the segment ab splitting the whole triangle is actually cutting a 90 degrees angle. This proves that the triangle is 90 degrees. Then we can assume that the sides of the whole triangle are equal to the triangle of the big triangle. There for the hypotenuse of the triangle with the segment ab, should be a^2 + b^2, which should be equal to the hypotenuse of the other large triangle which is c^2. So a^2+b^2=c^2
Completed by: Joe Delinski joe.delinski, Cynthia Gonzalez cgnozal4, Suzie Moon smoo1244, Charmaine Ramos cramos2, Susan Zurawski sue652
# Ideas in Geometry/Instructive examples/Section 1.3

http://commons.wikimedia.org/wiki/File:Miset_1.3.jpg
In city geometry and regular geometry, a parabola is a set of points such that each of those points is the same distance from a given point, A, as it is from a given line, L.
To find the midset (a midset is a set of points that are equal distance between two points A and B) of two points in city geometry first find the distance between the two points. To find the distance between 2 points you subtract the 2 y's (y-y2) and add the difference of the x's (x-x2) so using this formula we can see d(a-b)= (-2 - 3) + (2-2) = 5 (find absolute value). Next find the halfway point between the distance of the points: 5/2 = 2.5 (Draw a city geometry circle with radius 2.5 to find where the two points will intersect. Once you find that intersection line you must check to see if more points will be an equal distance away from each point by moving up and down or right and left. Once you have checked all directions and have found the overlapping points you have found the midset you are looking for.
# Ideas in Geometry/Instructive examples/Proving Sets Equal, Section 2.1

Click to make image bigger.
In order to prove that X∩(Y∪Z) = (X∩Y)∪(X∩Z), you must first learn what the symbols mean. When there is an intersection symbol, ∩, it means that you want to shade in the common area between the two areas intersecting. When there is a union symbol, ∪, it means that you want to shade in everything included in both areas. To prove that X∩(Y∪Z) = (X∩Y)∪(X∩Z), it is best to break it into parts. As shown in the picture above, for X∩(Y∪Z) I first shaded in X. Then I shaded in Y∪Z. After doing those parts separately, I shaded where they had common shaded areas because of the intersection symbol. I did the same thing for (X∩Y)∪(X∩Z) by first shading in X∩Y. Then I shaded in X∩Z. Since there is a union symbol for this one, I then combined the shaded areas for both parts. After breaking up both sides in order to find the shaded region, you can see that the two ultimately equal each other.
By: Katy Anderson