Parallel lines are straight lines which, being in the same plane, and being produced to infinity in each direction, meet with one another in neither (of these directions).
According to Corollary to Proposition 1.16, there exist straight lines in the plane, which do not have any point in common. According to the Post. 1, they are also unique. Since they exist and are unique, we can use those facts to define the concept of parallel straight lines:
Two straight lines in a plane are called parallel, if and only if they do not have any point in common. Any ray, or segments contained in parallel straight lines are also said to be parallel.
Axioms: 1
Corollaries: 2
Definitions: 3
Examples: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85
Propositions: 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111
Solutions: 112 113