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34(4)
34(4)

Chapter 5 Congruence Based on Triangles
Chapter 5 Congruence Based on Triangles

... 10. In a scalene triangle, LNM, show that an altitude, NO, cannot be an angle bisector. (Hint: Use an indirect proof.) 11. A telephone pole is braced by two wires that are fastened to the pole at point C and to the ground at points A and B. The base of the pole is at point D, the midpoint of AB. If ...
Congruence Through Transformations
Congruence Through Transformations

... corresponding sides of congruent triangles. • Explore the relationship between corresponding angles of ...
Elementary Number Theory
Elementary Number Theory

Exploring Advanced Euclidean Geometry
Exploring Advanced Euclidean Geometry

... The book consists mostly of exercises, tied together by short explanations. The user of the book should work through the exercises while reading the book. That way he or she will be guided through the discovery process. Any exercise that is marked with a star (*) is meant to be worked at the compute ...
arXiv:math/0510054v2 [math.HO] 17 Aug 2006
arXiv:math/0510054v2 [math.HO] 17 Aug 2006

Theory of L-functions - Institut für Mathematik
Theory of L-functions - Institut für Mathematik

Quadrilaterals
Quadrilaterals

Quadrilaterals
Quadrilaterals

Diskrete Mathematik für Informatik (SS 2017)
Diskrete Mathematik für Informatik (SS 2017)

Galois Theory - University of Oregon
Galois Theory - University of Oregon

Circles - Central CUSD 4
Circles - Central CUSD 4

Rank statistics for a family of elliptic curves over a function field
Rank statistics for a family of elliptic curves over a function field

Part 1
Part 1

Chapter 5: Relationships in Triangles
Chapter 5: Relationships in Triangles

Additive decompositions of sets with restricted prime factors
Additive decompositions of sets with restricted prime factors

Chapter 8: Quadrilaterals
Chapter 8: Quadrilaterals

Chapter 3 Proving Statements in Geometry
Chapter 3 Proving Statements in Geometry

... Using the definition of a scalene triangle, we know that: 1. The definition contains a hidden conditional statement and can be rewritten using the words If . . . then as follows: t: A triangle is scalene. p: A triangle has no congruent sides. t → p: If a triangle is scalene, then the triangle has no ...
TO CONSTRUCT AN ANGLE CONGRUENT TO A GIVEN ANGLE
TO CONSTRUCT AN ANGLE CONGRUENT TO A GIVEN ANGLE

... Determine which formulas (Distance, midpoint, or slope) you need to answer the question. Write out the formulas. You MUST use distance, midpoint, or slope formulas to receive credit for the problem. Substitute the numbers into the formulas to show your work. Be organized and neat when showing your w ...
4-5 Isosceles and Equilateral Triangles
4-5 Isosceles and Equilateral Triangles

Practice B Triangle Congruence: CPCTC
Practice B Triangle Congruence: CPCTC

... perpendicular bisectors that bisect the vertex angles of the square. ...
Geometry M1: Unit 4 Practice Exam
Geometry M1: Unit 4 Practice Exam

2nd Edition - Discrete Mathematics: An Open Introduction
2nd Edition - Discrete Mathematics: An Open Introduction

2nd Edition (printable) - Discrete Mathematics: An Open Introduction
2nd Edition (printable) - Discrete Mathematics: An Open Introduction

... The topics covered in this text were chosen to match the needs of the students I teach at UNC. The main areas of study are combinatorics, sequences, logic and proofs, and graph theory, in that order. Induction is covered at the end of the chapter on sequences. Most discrete books put logic first as ...
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Four color theorem



In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not.Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May (Wilson 2014, 2), “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property.”Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century (Heawood 1890); however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. (If they did appear, you could make a smaller counter-example.) Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exist because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand (Swart 1980). Since then the proof has gained wider acceptance, although doubts remain (Wilson 2014, 216–222).To dispel remaining doubt about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally in 2005, the theorem was proven by Georges Gonthier with general purpose theorem proving software.
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