Mathematics
Perpendicular Lines
Perpendicular lines are two straight lines that intersect at a 90-degree angle. In a coordinate plane, the slopes of perpendicular lines are negative reciprocals of each other. This relationship is a fundamental concept in geometry and is used in various mathematical applications, such as finding the equation of a line perpendicular to a given line.
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5 Key excerpts on "Perpendicular 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 - PDF- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
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 818 (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. OBJECTIVE 2 Identify corresponding angles, interior angles, and alternate interior angles. A line that intersects two coplanar lines in two distinct (different) points is called a transversal. For example, line l 1 in the figure to the right is a transversal intersecting lines l 2 and l 3 . When two lines are cut by a transversal, all eight angles that are formed are important in the study of parallel lines. Descriptive names are given to several pairs of these angles. In the figure below, four pairs of corresponding angles are formed. Corresponding angles n 1 and 5 n 3 and 7 n 2 and 6 n 4 and 8 Corresponding Angles If two lines are cut by a transversal, then the angles on the same side of the transversal and in corresponding positions with respect to the lines are called corresponding angles. In the figure below, four interior angles are formed.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
Using this definition, we proved the theorem stating that “Perpendicular Lines meet to form right angles.” We can also say that two rays or line segments are perpendicular if they are parts of Perpendicular Lines. We now consider a method for constructing a line perpendicular to a given line. CONSTRUCTION 6 To construct the line that is perpendicular to a given line from a point not on the given line. GIVEN: In Figure 2.1(a), line / and point P not on / CONSTRUCT: < PQ > # / CONSTRUCTION: Figure 2.1(b): With P as the center, open the compass to a length great enough to intersect / in two points A and B. Figure 2.1(c): With A and B as centers, mark off arcs of equal radii (using the same compass opening) to intersect at a point Q, as shown. Draw < PQ > to complete the desired line. In this construction, uni2220PRA and uni2220PRB are right angles. Greater accuracy is achieved if the arcs drawn from A and B intersect on the opposite side of line / from point P. Construction 6 suggests a uniqueness relationship that can be proved. P (a) P (b) A B P Q (c) A B R Figure 2.1 The term perpendicular includes line-ray, line-plane, and plane-plane relationships. In Figure 2.1(c), RP > # /. The drawings in Figure 2.2 indicate two Perpendicular Lines, a line perpendicular to a plane, and two perpendicular planes. PARALLEL LINES Just as the word perpendicular can relate lines and planes, the word parallel can also be used to describe relationships among lines and planes. However, parallel lines must lie in the same plane, as the following definition emphasizes. 2.1 ■ The Parallel Postulate and Special Angles 81 Parallel lines are lines in the same plane that do not intersect. DEFINITION Discover In the sketch below, lines / and m lie in the same plane with line t and are perpendicular to line t. How are the lines / and m related to each other? t m ANSWER These lines are said to be parallel.
No longer available |Learn more- 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.
- Allen Ma, Amber Kuang(Authors)
- 2022(Publication Date)
- For Dummies(Publisher)
624. Perpendicular Lines that are parallel have equal slopes. Lines that are perpendicular have slopes that are negative reciprocals of each other. The line x 3 is a vertical line. Pick any two points on this line. Because every coordinate along this line has an x value of 3, you can match the x value of 3 with any y value you would like. Use ( , ) 3 1 and ( , ) 3 4 . Now you can find the slope of this line using the slope formula: m y y x x 2 1 2 1 4 1 3 3 3 0 The slope is undefined. CHAPTER 18 Answers and Explanations 313 ANSWERS 601–700 The line y 4 is a horizontal line. Pick any two points that land on this line. Because every coordinate along this line has a y value of –4, you can match the y value of –4 with any x value you would like. Use ( , ) 1 4 and ( , ) 3 4 . Now you can find the slope of this line using the slope formula: m y y x x 2 1 2 1 4 4 3 1 0 2 ( ) The slope is 0. The slopes of the lines are reciprocals of each other, so the lines are perpendicular to each other. (Because you’re dealing with a value of 0, you’d never be able to see a negative reciprocal.) 625. Perpendicular If two lines are parallel, their slopes are equal. If two lines are perpendicular, their slopes are negative reciprocals of each other. Use the slope formula to find the slope of the line containivng the points ( , ) 0 1 and ( , ): 5 6 m y y x x m 2 1 2 1 6 1 5 0 5 5 1 Use the slope formula to find the slope of the line containing the points ( , ) 1 5 and ( , ) 3 1 : m y y x x m 2 1 2 1 1 5 3 1 4 4 1 ( ) The slopes are negative reciprocals of each other, which means that the two lines are perpendicular. 626. Neither If two lines are parallel, their slopes are equal. If two lines are perpendicular, their slopes are negative reciprocals of each other.
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