Mathematics
Parallel Lines
Parallel lines are two straight lines that never meet, no matter how far they are extended. In geometry, parallel lines have the same slope and will never intersect. They are always equidistant from each other and can be found in various mathematical and real-world applications, such as in architecture and engineering.
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7 Key excerpts on "Parallel Lines"
- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
OBJECTIVES 1 Identify and define parallel and perpendicular lines. 2 Identify corresponding angles, interior angles, and alternate interior angles. 3 Use properties of Parallel Lines cut by a transversal to find unknown angle measures. SECTION 9.2 Parallel and Perpendicular Lines In this section, we will consider parallel and perpendicular lines. Since Parallel Lines are always the same distance apart, the railroad tracks shown in figure (a) illustrate one application of Parallel Lines. Figure (b) shows one of the events of men’s gymnastics, the parallel bars. Since perpendicular lines meet and form right angles, the monument and the ground shown in figure (c) illustrate one application of perpendicular lines. OBJECTIVE 1 Identify and define parallel and perpendicular lines. If two lines lie in the same plane, they are called coplanar. Two coplanar lines that do not intersect are called Parallel Lines. See figure (a) on the next page. If two lines do not lie in the same plane, they are called noncoplanar. Two noncoplanar lines that do not intersect are called skew lines. The symbol indicates a right angle. (a) (b) (c) Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CHAPTER 9 • An Introduction to Geometry 722 (a) (b) l 1 l 1 l 2 l 2 Parallel Lines Perpendicular lines Parallel Lines Parallel Lines are coplanar lines that do not intersect. Some lines that intersect are perpendicular. See figure (b) above. Perpendicular Lines Perpendicular lines are lines that intersect and form right angles.
eBook - ePub- Michael Hvidsten(Author)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
In the proof of this result found in the on-line chapter on Hilbert’s axioms, we note that the proof uses the idea of extending a segment along a line, which is the focus of Euclid’s second axiom. Geometers have often intuitively understood this axiom to imply that a line has infinite extent, or length. A careful reading of the second Euclidean axiom does not support this understanding.In fact, consider the “spherical” geometry introduced in Exercise 1.6.7 of Chapter 1 . Here we interpreted the term point to mean a point on a sphere and a line to be a part of a great circle on the sphere. In this geometry segments can always be extended (you can always move around the great circle, even though you may re-trace your path), but lines are not infinite in extent.We will return to this question of the boundedness of lines in Chapter 8 , where we develop a non-Euclidean geometry akin to that of the sphere.Theorem 2.7 .(Prop. 27 of Book I) If a line n falling on two lines l and m makes the alternate interior angles congruent to one another, then the two lines l and m must be parallel (Figure 2.5 ).Figure 2.5 Alternate interior anglesTheorem 2.8 .(Prop. 28 of Book I) If a line n falling on two lines l and m makes corresponding angles congruent, or if the sum of the measures of the interior angles on the same side equal two right angles, then l and m are parallel (Figure 2.6 ).Figure 2.6 Angles and parallelsWe note here that all of the preceding theorems are independent of the fifth Euclidean postulate, the parallel postulate. That is, they can be proved from an axiom set that does not include the fifth postulate. Such results form the basis of what is called absolute or neutral geometry. All of the theorems and results found in the on-line chapter on Hilbert’s axioms, except the section on Euclidean Geometry, are independent of the parallel postulate and thus form the basis for neutral geometry.The first time that Euclid actually used the fifth postulate in a proof was for Proposition 29 of Book I. Recall that the fifth postulate states:
eBook - ePub- Bruce E. Meserve(Author)
- 2014(Publication Date)
- Dover Publications(Publisher)
Consider two distinct Parallel Lines m and m′ and a transversal t (i.e., a line that is not parallel to m). Define corresponding angles, alternate interior angles, and alternate exterior angles. 9. Prove that when Parallel Lines are cut by a transversal, any two corresponding angles are equal, alternate interior angles are equal, and alternate exterior angles are equal. 10. Prove that if the sides of one angle are parallel to the sides of another angle, then the angles are equal or supplementary. 11. Prove that two angles are equal if they are (a) supplementary to the same or equal angles, (b) complementary to the same or equal angles. 12. Given a point O on a plane, use the measures of angles with vertex O to obtain a distance relationship on the ideal line. 13. Is the distance function obtained in Exercise 12 a euclidean distance function? Explain. 14. Prove that in euclidean geometry any angle ACB where AB is a diameter of a circle and C is a third point on that circle (i.e., any angle inscribed in a semicircle) is a right angle. 15. Use the property obtained in Exercise 13 and give another definition of a circle. 6–8 Common figures. We have developed euclidean plane geometry from the undefined terms (point and line), the undefined relations of incidence and separation, and the postulates of incidence, existence, projectivity, harmonic sets, separation, and continuity (based upon the isomorphism between the set of real numbers and the set of real points on a line). This synthetic development has been proved to be equivalent to an algebraic (or analytic) development in terms of triples of real numbers, the properties of numbers, and the properties of algebra. The euclidean plane has been defined by considering only real points on a projective plane with one line deleted
eBook - PDF- EISENREICH(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
In Postulate 3 distance means the length of the radius. Postulate 5 is the postulate that was subjected to so much criticism. It is illustrated in Fig. I, l.| Line k intersects lines g, /z, * Taken, with permission, from T. L. Heath, The Thirteen Books of Euclid's Elements (Cambridge University Press, New York and London, 1926). We shall refer to this work hereafter as the Elements. t In this book we are concerned only with plane geometry. The word line used alone will always mean straight line. 3. THE EXTENT OF A STRAIGHT LINE 5 forming interior angles a, b on the same side of k. If a + b is less than two right angles, then, according to Postulate 5, g and h will meet on that side of A:. As we shall see, Postulate 5 plays a decisive role in Euclid's theory of parallels. For this reason it is often called his parallel postulate. k X h a b r g Fig. I, 1 3. THE EXTENT OF A STRAIGHT LINE In addition to his ten stated assumptions Euclid used many unstated assumptions. By this we mean that in the course of his proofs he took certain geometric properties for granted, never proving them nor stating them explicitly. Just one of these unstated assumptions is relevant to our present discussion. It concerns the extent of a straight line. Euclid regarded a straight line as being infinite in extent. That he did so can be seen, for example, in Postulate 3. It can also be seen in his Proposition 12,* where the term infinite straight line is used. Further, in the proof of Proposition 16 he extends a line segment by its own length, thus obtaining a new segment twice as long as the original one. This assumes that any line segment can be doubled. By repeating this doubling one could, of course, obtain a segment whose length exceeds all bounds. In effect, then, Euclid is assuming that a line is infinite. Because of its importance to our subject we shall state Proposition 16 and give what is essentially Euclid's proof.
eBook - PDF- Patrick D Barry(Author)
- 2001(Publication Date)
- Woodhead Publishing(Publisher)
5 The parallel axiom; Euclidean geometry COMMENT . The effect of introducing any axiom is to narrow things down, and depending on the final axiom still to be taken, we can obtain two quite distinct well-known types of geometry. By introducing our final axiom, we confine ourselves to the familiar school geometry, which is known as Euclidean geometry . 5.1 THE PARALLEL AXIOM 5.1.1 Uniqueness of a parallel line We saw in 4.2 that given any line l and any point P ∈ l there is at least one line m such that P ∈ m and l m . We now assume that there is only one such line ever. AXIOM A 7 . Given any line l ∈ Λ and any point P ∈ l , there is at most one line m such that P ∈ m and l m . | COMMENT . By 4.2 and A 7 , given any line l ∈ Λ and any point P ∈ Π , there is a unique line m through P which is parallel to l . Let l ∈ Λ , P ∈ Π and n ∈ Λ be such that l = n, P ∈ n and l n . Let A and B be any distinct points of l and R a point of n such that R and B are on opposite sides of AP . Then | ∠ APR | ◦ = | ∠ PAB | ◦ , so that for Parallel Lines alternate angles must have equal degree-measures . Proof . Let m be the line PQ in 4.2.1 such that | ∠ APQ | ◦ = | ∠ BAP | ◦ . Then l m . As m and n both contain P and l is parallel to both of them, by A 7 we have m = n , so that R ∈ [ P, Q and so | ∠ APR | ◦ = | ∠ APQ | ◦ . Thus | ∠ APR | ◦ = | ∠ APQ | ◦ = | ∠ PAB | ◦ . 63 [Ch. 5 The parallel axiom; Euclidean geometry 64 Let l, n be distinct Parallel Lines, A, B ∈ l and P, T ∈ n be such that B and T are on the one side of AP , and S = P be such that P ∈ [ A, S ] . Then the angles ∠ BAP, ∠ TPS have equal degree-measures . Proof . Choose R = P so that P ∈ [ T, R ] . Then R ∈ n and B and R are on opposite sides of AP , so that ∠ BAP, ∠ AP R are alternate angles and so have equal degree-measures. But ∠ AP R and ∠ TPS are opposite angles and so have equal degree-measures. Hence | ∠ BAP | ◦ = | ∠ TPS | ◦ . A B P R Q n l ◦ ◦ Figure 5.1.
- Carl Posy, Ofra Rechter(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
. ., but no definition of the distance of a straight line in general in the same plane.” So, Wolff’s definition of Parallel Lines is only useable if we have a definition of distance between straight lines, and this definition – if we choose it right – might fix the hole in Wolff’s proof. Kant proposes the following definition: The distance of two straight lines from one another is the perpendicular line that is drawn from a point of one to the other, insofar as it is congruent with the line that is erected perpendicularly from the same point to the first. In other words, the distance between two lines is a mutual perpendicular between them. It follows immediately that two lines have a distance at a point only if there is a mutual perpendicular on both lines at that point, and that the two lines are everywhere equidistant only if they have mutual perpendiculars everywhere. Pairs of straight lines with no mutual perpen- diculars thus have no distance anywhere. This definition removes the ambiguity in Wolff’s proof of I.– and it plugs one hole in the reasoning (how do we infer from HK being the distance of CD to AB at K to its being the distance of AB to CD as well?). As Kant clearly recognizes, however, removing the ambiguity just makes explicit the remaining hole in the proof that cannot be filled. Even more interesting than Kant’s criticism of the mathematical flaws in Wolff’s proof of I.–, though, is that Kant furthermore uses his notion of distance to criticize Wolff’s proof of I. (the converse of I.–) on philosophical grounds. Wolff proves I. from a special case of it, which – using again the Figure on p. above, – is this: Elementa §: If CD is parallel and HK perpendicular to AB, then HK will also be perpendicular to CD. also intersects CD at a right angle at H. Wolff proves that KH is a distance in the weaker (not necessarily symmetric) sense, but then helps himself to the stronger claim that KH is a distance in the symmetric sense.
eBook - PDFCollege Geometry
A Unified Development
- David C. Kay(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
1 1 Lines,.Distance,. Segments,.and.Rays 1 .1. Intended.Goals It is appropriate to take a moment to address the goals and style of this book. It is written primarily with you, the student, in mind. The attempt is made as much as possible to explain what is being done (defining terms or proving theorems) and why (the historical background). The appendices are to fill in the gaps where necessary. Being a book about geometry, natu-rally the objects of study are sets of points in the plane or in space, either realized in actuality in coordinate geometry (models and examples) or as imaginary entities in the axiomatic development. There are many ways to approach the subject of geometry, but virtually all secondary school books on geometry remain traditional in nature. There is a very good reason for this; Euclid’s axiomatic approach has been found to be the best way to learn the subject of geometry, in spite of various attempts at innovation. As Euclid himself is supposed to have once said, in so many words, that there is no “quick and easy” way to learn geometry. But it is perhaps the best way. It is certainly the most elegant. Euclid began with a set of 10 principles he called common notions and postulates . The five common notions were algebraic in nature (“if equals are added to equals, the wholes are equal”), and the five postulates were those dealing with geometric objects (e.g., points, lines, circles, angles). For example, one of the postulates reads “a finite straight line [a segment in our development] can be produced continuously in a line.” The famous Postulate Five basically states: If two lines do not make either equal or supplementary angles with a transversal, they intersect . This fifth postu-late of parallels started a controversy in mathematics about which much will be said later. Despite the many shortcomings found in the Elements , this work serves as the beginners’ handbook.
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