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10: If lines are parallel, a line perpendicular to one of them is perpendicular to the others also.
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Thus, if l1 i l2 and l3 ' l1, then l3 ' l2 in Fig. 4-17.
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Fig. 4-17
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CHAPTER 4 Parallel Lines, Distances, and Angle Sums
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11: If lines are parallel, a line parallel to one of them is parallel to the others also.
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Thus, if l1 i l2 and l3 i l1, then l3 i l2 in Fig. 4-18.
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12: If the sides of two angles are respectively parallel to each other, the angles are either congruent or supplementary.
Thus, if l1 i l3 and l2 i l4 in Fig. 4-19, then j a > j b and j a and j c are supplementary.
Fig. 4-19
SOLVED PROBLEMS
Numerical applications of parallel lines In each part of Fig. 4-20, find the measure x and the measure y of the indicated angles.
Fig. 4-20
Solutions
(a) x (b) x (c) x (d) x (e) x (f) x 130 (Principle 8). y 80 (Principle 8). y 75 (Principle 7). y 180 130 50 (Principle 9).
70 (Principle 7). 180 75 105 (Principle 9). mj A, mj B 70 ) 65 . Hence, y 65 (Principle 8).
65 (Principle 7). Since mj B 30 (Principle 8). y 180 110
180 (30
80 (Principle 9).
70 (Principle 9). y
110 (Principle 12).
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
Applying parallel line principles and their converses The following short proofs refer to Fig. 4-21. In each, the first statement is given. State the parallelline principle needed as the reason for each of the remaining statements.
Fig. 4-21
(a) 1. j 1 > j 2 2. AB i CD 3. (b) 1. 3. j3 > j4 j2 > j3 j 4 sup. j 5
1. Given 2. _____ 3. _____ 1. Given 2. _____ 3. _____
(c) 1. 3.
j 5 sup. j 4 j3 > j6 EF ' CD
2. EF i GH (d) 1. EF ' AB, GH ' AB, 2. EF i GH
1. Given 2. _____ 3. _____ 1. Given 2. _____ 3. _____
2. EF i GH
3. CD ' GH
Solutions
(a) 2: Principle 3; 3: Principle 7. (b) 2: Principle 2; 3: Principle 9. (c) 2: Principle 4; 3: Principle 8. (d ) 2: Principle 5; 3: Principle 10.
Algebraic applications of parallel lines In each part of Fig. 4-22, find x and y. Provide the reason for each equation obtained from the diagram.
Fig. 4-22
Solutions
(a) 3x 20 x y 10 y 10 y 2x (Principle 8) 20 2x (Principle 7) 40 30 (b) x x 4y y 2y 44 x 180 92 22 92 92 48 88 (Principle 9) (Principle 7)
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
(c) (1) x y (2) x 2y 3y y x 40 x
150 (Principle 8) 30 (Principle 9) 120 (Subt. Postulate) 40 150 110
Proving a parallel-line problem
Given: AB > AC AE i BC To Prove: AE bisects jDAC Plan: Show that j1 and j2 are congruent to the congruent angles B and C.
PROOF:
Statements 1. 2. 3. 4. 5. 6. AE i BC j1 > jB j2 > jC AB > AC jB > jC j1 > j2
Reasons 1. 2. 3. 4. 5. 6. Given Corresponding of i lines are >. Alternate interior of i lines are >. Given In a ^, opposite > sides are >. Things > to > things are > to each other.
7. AE bisects jDAC.
7. To divide into two congruent parts is to bisect.
Proving a parallel-line problem stated in words
Prove that if the diagonals of a quadrilateral bisect each other, the opposite sides are parallel. Given: Quadrilateral ABCD AC and BD bisect each other. To Prove: AB i CD AD i BC Plan: Prove j1 > j4 by showing ^ I > ^ II. Prove j2 > j3 by showing ^ III > ^IV.
PROOF:
Statements 1. 2. 3. 4. 5. 6.
AC and BD bisect each other. BE > ED AE > EC
Reasons 1. 2. 3. 4. 5. 6. Given To bisect is to divide into two congruent parts. Vertical are >. SAS s Corresponding parts of congruent ^ are >. Lines cut by a transversal are i if alternate interior are >.
j5 > j6, j7 > j8 ^I > ^ II, ^ III > ^ IV j1 > j4, j2 > j3 AB i CD, BC i AD
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
4.2 Distances
4.2A Distances Between Two Geometric Figures
The distance between two geometric figures is the straight line segment which is the shortest segment between the figures. 1. The distance between two points, such as P and Q in Fig. 4-23(a), is the line segment PQ between them.
Fig. 4-23
2. The distance between a point and a line, such as P and AB in (b), is the line segment PQ, the perpendicular from the point to the line. 4 4 3. The distance between two parallels, such as AB and CD in (c), is the segment PQ, a perpendicular between the two parallels. 4. The distance between a point and a circle, such as P and circle O in (d ), is PQ, the segment of OP between the point and the center of the circle. 5. The distance between two concentric circles, such as two circles whose center is O, is PQ, the segment of the larger radius that lies between the two circles, as shown in (e).
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