Parallel Lines Theorem

11 Key excerpts on "Parallel Lines Theorem"

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  32. There is exactly one line perpendicular to a given line at a point on the line. *33. In a plane, if two lines are parallel to a third line, then the two lines are parallel to each other. *34. In a plane, if two lines are intersected by a transversal so that the corresponding angles are congruent, then the lines are parallel. a 2 b 2 a b x 5 x 2 25 AD AM CD MB BC — EG ! FH ! m ∠ 3 m ∠ 4 A B C D ∠ ABC BD ! ∠ ABD ∠ DBC r s ∠ 1 ∠ 5 P E Aisle F G ∠ GEF EF D C B A CD AB CD AB ∠ B ∠ A ∠ B ∠ A ∠ ABC BC AB ∠ 1 ∠ 2 m m ∠ B m ∠ A ABC AC BC 2 4 6 8 1 3 5 7 s t 3 4 G F H E A M C D B Copyright 2013 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. 82 CHAPTER 2 ■ PARALLEL LINES Unless otherwise noted, all content on this page is © Cengage Learning. For this section, here is a quick review of the relevant postulate and theorems from Section 2.1. Each theorem has the hypothesis “If two parallel lines are cut by a transversal”; each theorem has a conclusion involving an angle relationship. Proving Lines Parallel KEY CONCEPTS Proving Lines Parallel 2.3 If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. POSTULATE 11 THEOREM 2.1.2 If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. THEOREM 2.1.3 If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. THEOREM 2.1.4 If two parallel lines are cut by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Counterparts of Theorems 2.1.2–2.1.5, namely, Theorems 2.3.2–2.3.5, are proved directly but depend on Theorem 2.3.1. Except for Theorem 2.3.6, the theorems of this section require coplanar lines. 96 CHAPTER 2 ■ Parallel Lines Theorem 2.3.2 If two lines are cut by a transversal so that two alternate interior angles are congruent, then these lines are parallel. Theorem 2.3.3 If two lines are cut by a transversal so that two alternate exterior angles are congruent, then these lines are parallel. Discover When a staircase is designed, “string- ers” are cut for each side of the stairs as shown. How are angles 1 and 3 related? How are angles 1 and 2 related? 1 2 3 ANSWER Congruent, Complementary GIVEN: / and m cut by transversal t uni22201 _ uni22202 (See Figure 2.16.) PROVE: / i m PROOF: Suppose that / i m. Then a line r can be drawn through point P that is parallel to m; this follows from the Parallel Postulate. If r i m, then uni22203 _ uni22202 because these angles correspond. But uni22201 _ uni22202 by hypothesis. Now uni22203 _ uni22201 by the Transitive Property of Congruence; therefore, muni22203 5 muni22201. But muni22203 1 muni22204 5 muni22201. (See Figure 2.16.) Substituting muni22201 for muni22203 leads to muni22201 1 muni22204 5 muni22201; and by subtraction, muni22204 5 0. This contradicts the Protractor Postulate, which states that the measure of any angle must be a positive number. Then r and / must coincide, and it follows that / i m. Each claim in Theorems 2.3.2–2.3.5 is the converse of its counterpart in Section 2.1, and each claim provides a method for proving that lines are parallel. GIVEN: Lines / and m and transversal t uni22202 _ uni22203 (See Figure 2.17.) PROVE: / i m PLAN FOR THE PROOF: Show that uni22201 _ uni22202 (corresponding angles). Then apply Theorem 2.3.1, in which _ corresponding uni2220s imply parallel lines. PROOF Statements Reasons The following theorem is proved in a manner much like the proof of Theorem 2.3.2.

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  Geometry Transformed

  Euclidean Plane Geometry Based on Rigid Motions

  • James R. King(Author)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  Transversals and Parallel Lines Definition 7.3. A line that intersects each of a pair of lines at two distinct points is called a transversal of the lines. The Parallel Postulate says something important about transversals: Theorem 7.4 (Transversal of Parallels) . Suppose 𝑚 and 𝑛 are parallel lines. If ? is a line distinct from 𝑚 that intersects 𝑚 , then ? also intersects 𝑛 , so ? is a transversal. Proof. If ? intersects 𝑚 at ? and does not intersect 𝑛 , then ? and 𝑚 are two lines through ? that are parallel to 𝑛 . By the Parallel Postulate, the lines cannot be distinct. Therefore, ? must intersect 𝑛 . □ 86 7. Parallel Lines and Translations In the next theorem, a half-turn is used to establish important properties of transversals. C' M A B C D E Figure 4. Transversal with Half-turn at Midpoint Theorem 7.5 (Transversal Theorem) . Suppose ?? is a transversal of ? and ? , with ? on ? and ? on ? . Let ℋ ? be the half-turn with center ? , the midpoint of ?? . (1) The line ? = ℋ ? (?) if and only if ? and ? are parallel. (2) If ? is not on ?? , let ? ′ = ℋ ? (?) . Then ℋ ? ( ⃗ ??) = ⃗ ?? ′ is a parallel ray on the opposite side of ?? . Also ?? is parallel to ? ′ ? . (3) If ? and ? are on opposite sides of ?? , then 𝑚∠??? = 𝑚∠??? if and only if ?? is parallel to ?? . Proof. The line ℋ ? (?) is parallel to ? by Theorem 6.1. The Parallel Postulate says that if ? is also a line through ? parallel to ? , then it must be the same line. This proves the first statement. If ? ′ = ℋ ? (?) , then ?? ′ intersects ?? at ? . So if ? is not on ?? , then ? and ? ′ are in opposite half-planes. Since ?? ′ = ℋ ? (??) , the lines ?? and ? ′ ? are parallel. Also, since ℋ ? is a rigid motion, 𝑚∠??? = 𝑚∠??? ′ . If 𝑚∠??? = 𝑚∠??? ′ = 𝑚∠??? , with ? and ? on opposite sides of ?? , then ⃗ ?? = ⃗ ?? ′ . Therefore ?? is parallel to ?? = ?? ′ . Conversely, if the lines are parallel and ? is on the opposite side of ?? , then ⃗ ?? = ⃗ ?? ′ since both are on the unique parallel to ?? through ? .

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  Basic Mathematics for College Students with Early Integers

  • 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.

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  Prealgebra

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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 820 OBJECTIVE 3 Use properties of parallel lines cut by a transversal to find unknown angle measures. Lines that are cut by a transversal may or may not be parallel. When a pair of parallel lines are cut by a transversal, we can make several important observations about the angles that are formed. 1. Corresponding angles property: If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. In the figure below, if l 1 i l 2 , then 1 > 5, 3 > 7, 2 > 6, and 4 > 8. 2. Alternate interior angles property: If two parallel lines are cut by a transversal, alternate interior angles are congruent. In the figure below, if l 1 i l 2 , then 3 > 6 and 4 > 5. 3. Interior angles property: If two parallel lines are cut by a transversal, interior angles on the same side of the transversal are supplementary. In the figure below, if l 1 i l 2 , then 3 is supplementary to 5 and 4 is supplementary to 6. 3 4 1 2 5 6 7 8 l 3 l 2 l 1 Transversal l 1 l 2 4. If a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other line. In figure (a) below, if l 1 i l 2 and l 3 ⊥ l 1 , then l 3 ⊥ l 2 . 5. If two lines are parallel to a third line, they are parallel to each other. In figure (b) below, if l 1 i l 2 and l 1 i l 3 , then l 2 i l 3 . l 3 l 2 l 1 (a) l 2 l 3 l 1 (b) Refer to the figure. If l 1 i l 2 and m(3) 5 120° , find the measures of the other seven angles that are labeled. Strategy We will look for vertical angles, supplementary angles, and alternate interior angles in the figure.

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  Introduction to Non-Euclidean Geometry

  • EISENREICH(Author)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  (c) A line that meets one of two parallels also meets the other. (d) If two parallels are cut by a transversal, the alternate interior angles are equal. (e) There exists a triangle whose angle-sum is two right angles. (f) Parallel lines are equidistant from one another.* (g) There exist two parallel lines whose distance apart never exceeds some finite value. (h) Similar triangles exist which are not congruent. (i) Through any three noncollinear points there passes a circle. (j) Through any point within any angle a line can be drawn which meets both sides of the angle. (k) There exists a quadrilateral whose angle-sum is four right angles. (1) Any two parallel lines have a common perpendicular. We shall prove two of these substitutes now, leave others as exercises, and verify the remaining ones in our study of non-Euclidean geometry. * All perpendicular distances from either line to the other are equal. 14 I. EUCLID'S FIFTH POSTULATE Proof of Substitute (a). Since we already know from Section 4 that statement (a) can be proved when Postulate 5 and the basis E are used, it only remains to prove Postulate 5 when statement (a) and the basis E are used. Hence, let g, h (Fig. I, 10) be two lines such that, when they are cut by a transversal PQ, a + b < two right angles. (1) Fig. I, 10 We must show that g, h meet to the right of line PQ. Through Q take line y so that a + c equals two right angles. Then h, j are distinct since b Φ c. Also, j is parallel to g (Prop. 28). Because of statement (a), which we are assuming, h cannot also be parallel to g. Hence g and h must meet in some point R. If R were to the left of line PQ, the sum of the angles at P and Q in triangle PQR would exceed two right angles in view of (1). This contradicts Proposition 17. Hence R is to the right of line PQ. Proof of Substitute (f). We have already seen that statement (f) can be proved when Postulate 5 and the basis E are assumed (§4, Ex. 5).

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  Two-Dimensional Geometries

  A Problem-Solving Approach

  • C. Herbert Clemens(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  Part II Euclidean (plane) geometry Chapter 3 Rectangles and cartesian coordinates 3.1. Euclid’s Fifth Postulate, the Parallel Postulate We are finally ready to introduce Euclid’s fifth and final axiom, the so-called Parallel Postulate . Axiom (E5) . Through a point not on a line there passes one and only one parallel line. NG together with E5 is called Euclidean geometry ( EG ). As mentioned above, we will see later that there is another geometry HG that satisfies all the postulates of NG but not E5. In it, an infinite number of parallel lines will pass through a given point not on a line. Furthermore the sum of the interior angles of a triangle will always be less than 180 ° [ MJG , 134]! Now suppose we have two parallel lines, L and M , in the Euclidean plane ( EG ). Suppose we have a third line that is transversal to L and M , that is, a line that meets L at a point A and meets M a point B . Using Exercise 11, construct a line M through B that is parallel to L . Exercise 21 ( EG ) . a) Show that if two parallel lines L and M are cut by a transversal line, opposite interior angles are equal. Hint: Concentrate on the words “only one” in E5. b) Show that if two parallel lines L and M are cut by a transversal line, corre-sponding angles are equal. Hint: Use a) and the fact that vertical angles are equal. Exercise 22. a) Show that if a line T crosses one of two parallel lines L and M , it must cross the other. b) Show that if two distinct lines L and M are both parallel to another line, then they are parallel to each other. 19 20 3. Rectangles and cartesian coordinates Exercise 23. a) Show that angles ∠ BAC and ∠ B A C in the Euclidean plane are equal (i.e., have the same measure) if their corresponding rays are parallel. Hint: Construct a transversal through A and A . b) Show that ∠ BAC and ∠ B A C are equal (i.e., have the same measure) if ∠ B A C can be rotated around A to obtain an angle ∠ B A C with corresponding rays parallel to those of ∠ BAC .

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  Geometry with Trigonometry

  • 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.

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  In this way, we conclude that CD CB | | | | = . The triangles ACD and ACB are now seen to be congruent by the sss condition and, therefore, BAC ˆ must be a right angle. A B C D FIGURE 9.21. The converse of the Pythagorean theorem. 310 • Foundations of Mathematics 9.5.3 Parallelograms and Parallel Lines Theorem 9.5.9. (a) The opposite sides of a parallelogram have the same length. (b) (Converse) If the opposite sides of a quadrilateral have the same length, it is a parallelogram. (c) The opposite angles of a parallelogram are equal. (d) (Converse) If the opposite angles of a quadrilateral are equal, then it is a parallelogram. (e) The diagonals of a parallelogram bisect each other. (f) (Converse) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. (g) If both members of one pair of opposite sides of a quadrilateral are parallel and have the same length, it is a parallelogram. (h) The diagonals of a rectangle have the same length. (i) The diagonals of a rhombus bisect each other at right angles and bisect the angles of the rhombus. Theorem 9.5.9(a) and (c) can be proved by joining a pair of opposite vertices of a parallelogram with a diagonal line to create a pair of triangles. The two pairs of alternate angles formed in this way are equal, and so the triangles are congruent by the s ∠∠ criterion. Thus, both members of a pair of opposite angles of the parallelogram are equal, and both members of a pair of opposite sides are equal. Similarly, the members of the other pairs of opposite angles and opposite sides are equal. The proof of theorem 9.5.9(e) is also a consequence of the formation of congruent triangles, when both diagonals are drawn. The proofs of the converse statements, that is, theorem 9.5.9(b), (d), (f), and (g), are left as exercises. Theorem 9.5.9(h) and (i) are two special cases that can be verified easily.

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  Kant's Philosophy of Mathematics: Volume 1, The Critical Philosophy and its Roots

  • Carl Posy, Ofra Rechter(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

    Space and Geometry  Kant on Parallel Lines Definitions, Postulates, and Axioms Jeremy Heis* The fifth postulate in Euclid’s Elements is the notorious Axiom of Parallels: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.  This axiom was criticized already in the ancient world. By the late nineteenth century, Eugenio Beltrami and Felix Klein showed that it is consistent to deny the axiom, and of course Einstein showed that it is not even true of physical space. Kant believed that the axioms of geometry are synthetic a priori truths. This view sits uncomfortably with the now consensus view among geometers that both Euclidean and non-Euclidean geometries are equally legitimate subjects of mathematical study, and it sits very uncomfortably with the consensus among physicists that the axiom of parallels is false of physical space. Not surprisingly, then, some philoso- phers have argued that Kant’s philosophy of mathematics (indeed, his entire theoretical philosophy) has been refuted by the real possibility of non-Euclidean physical spaces.  Moreover, some of Kant’s critics, surprised that Kant says nothing about the well-known problems with Euclid’s axiom of parallels, have viewed * This essay was greatly improved by comments and conversations with Katherine Dunlop, Stephen Engstrom, Michael Friedman, Penelope Maddy, Bennett McNulty, Cailin O’Connor, Konstantin Pollock, Lisa Shabel, Marius Stan, Clinton Tolley, Eric Watkins, and audiences at HOPOS , Cal State Long Beach, and the  Pacific meeting of the NAKS. I am especially grateful to Ian Proops and Clinton Tolley for providing detailed comments on an earlier draft, and to Ofra Rechter for giving me feedback that helped greatly improve the clarity and presentation of the essay.

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  Analytical Geometry

  The Commonwealth and International Library of Science, Technology, Engineering and Liberal Studies: Mathematics Division

  • Barry Spain, W. J. Langford, E. A. Maxwell, I. N. Sneddon(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  59. Show that the medians of a triangle are concurrent. 60. Prove that the three perpendicular bisectors of the sides of a triangle are concurrent. (The point of concurrency is called the circumcentre.) 61. Prove that the straight line x{2+t)+y{l+t) = 5+lt always passes through a fixed point, whatever the value of /, and find the coordinates of this point. (U.L.) 20. Parametric equations of a straight line The straight line through (x l9 y x ) of gradient tan φ has the equa-tion y—yi=(x—Xi) tan φ, which can be written *-*i = y-y cos φ sin φ Equating each fraction to t, we obtain x = x x +t cos φ, y = y x +t sin φ. Elimination of φ yields t* = (x-xj* + (y-ytf and so / represents the distance between a variable point (x, y) of the line and the fixed point (x l9 y^). Hence, as t varies from — oo to + oo, the point (x, y) traces out all points of the straight line. The equations x = x ± +t cos φ, y =yi+t sin φ are called the parametric equations of the straight line and t is called the para-meter. The equations x =x 1 +lt 9 y =y 1 +mt also represent a straight line since the elimination of t yields the linear equation m(x—x^ = l{y—y-j). In this case /, in general, does not represent the distance between (.v, y) and (x 1? y). The parametric equations are often useful in solving problems and we show their application in the following problems : 34 ANALYTICAL GEOMETRY Illustration I: Find the coordinates of the mirror image of (α, β) in the straight line ax+by+c = 0. The straight line through (α, β) perpendicular to ax+by+c = 0 has gradient b/a and so its parametric equations can be taken as x = a--t cos 0, y = ß+t sin 0 where tan 0 = b/a. Let the mirror image correspond to the value t of the parameter. The mid-point of the line joining (α, β) to its mirror image is (α+Jf cos 0, ß+t sin 0). This point lies on the given line and so a{a+U cos 0) + biß+it sin 0) + c = 0, from which / = -2{aa+bß+c)l(a cos θ+b sin 0).

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