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Parallel Lines, Distances, and Angle Sums
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4.1 Parallel Lines
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Parallel lines are straight lines which lie in the same plane and do not intersect however far they are ex4 4 4 4 tended. The symbol for parallel is i; thus, AB iCD is read line AB is parallel to line CD. In diagrams, arrows are used to indicate that lines are parallel (see Fig. 4-1).
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Fig. 4-1
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A transversal of two or more lines is a line that cuts across these lines. Thus, EF is a transversal of AB and 4 CD, in Fig. 4-2. The interior angles formed by two lines cut by a transversal are the angles between the two lines, while 4 4 4 the exterior angles are those outside the lines. Thus, of the eight angles formed by AB and CD cut by EF in Fig. 4-2, the interior angles are j 1, j 2, j 3, and j 4; the exterior angles are j 5, j 6, j 7, and j 8.
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4.1A Pairs of Angles Formed by Two Lines Cut by a Transversal
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Corresponding angles of two lines cut by a transversal are angles on the same side of the transversal and on 4 4 the same side of the lines. Thus, j 1 and j 2 in Fig. 4-3 are corresponding angles of AB and CD cut by trans4 versal EF. Note that in this case the two angles are both to the right of the transversal and both below the lines.
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CHAPTER 4 Parallel Lines, Distances, and Angle Sums
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When two parallel lines are cut by a transversal, the sides of two corresponding angles form a capital F in varying positions, as shown in Fig. 4-4.
Fig. 4-4
Fig. 4-5
Alternate interior angles of two lines cut by a transversal are nonadjacent angles between the two lines 4 and on opposite sides of the transversal. Thus, j 1 and j 2 in Fig. 4-5 are alternate interior angles of AB and 4 4 CD cut by EF. When parallel lines are cut by a transversal, the sides of two alternate interior angles form a capital Z or N in varying positions, as shown in Fig. 4-6.
Fig. 4-6
When parallel lines are cut by a transversal, interior angles on the same side of the transversal can be readily located by noting the capital U formed by their sides (Fig. 4-7).
Fig. 4-7
4.1B Principles of Parallel Lines
PRINCIPLE
Through a given point not on a given line, one and only one line can be drawn parallel to a given line. (Parallel-Line Postulate)
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
Thus, either l1 or l2 but not both may be parallel to l3 in Fig. 4-8.
Fig. 4-8
Proving that Lines are Parallel
PRINCIPLE
Two lines are parallel if a pair of corresponding angles are congruent.
Thus, l1 i l2 if j a > j b in Fig. 4-9.
Fig. 4-9
PRINCIPLE 3:
Two lines are parallel if a pair of alternate interior angles are congruent.
Thus, l1 i l2 if j c > j d in Fig. 4-10.
Fig. 4-10
PRINCIPLE
4: Two lines are parallel if a pair of interior angles on the same side of a transversal are supplementary.
Thus, l1 i l2 if j e and j f are supplementary in Fig. 4-11.
Fig. 4-11
PRINCIPLE
Lines are parallel if they are perpendicular to the same line. (Perpendiculars to the same line are parallel.)
Thus, l1 i l2 if l1 and l2 are each perpendicular to l3 in Fig. 4-12.
Fig. 4-12
PRINCIPLE
Lines are parallel if they are parallel to the same line. (Parallels to the same line are parallel.)
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
Thus, l1 i l2 if l1 and l2 are each parallel to l3 in Fig. 4-13.
Fig. 4-13
Properties of Parallel Lines
PRINCIPLE
If two lines are parallel, each pair of corresponding angles are congruent. (Corresponding angles of parallel lines are congruent.)
Thus, if l1 i l2, then j a > j b in Fig. 4-14.
Fig. 4-14
PRINCIPLE 8:
If two lines are parallel, each pair of alternate interior angles are congruent. (Alternate interior angles of parallel lines are congruent.)
Thus, if l1 i l2, then j c > j d in Fig. 4-15.
Fig. 4-15
PRINCIPLE 9:
If two lines are parallel, each pair of interior angles on the same side of the transversal are supplementary.
Thus, if l1 i l2, j e and j f are supplementary in Fig. 4-16.
Fig. 4-16
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