Antiparallel (mathematics)

In geometry, anti-parallel lines can be defined with respect to either lines or angles. In geometry, anti-parallel lines can be defined with respect to either lines or angles. Given two lines m 1 {displaystyle m_{1},} and m 2 {displaystyle m_{2},} , lines l 1 {displaystyle l_{1},} and l 2 {displaystyle l_{2},} are anti-parallel with respect to m 1 {displaystyle m_{1},} and m 2 {displaystyle m_{2},} if ∠ 1 = ∠ 2 {displaystyle angle 1=angle 2,} in Fig.1. If l 1 {displaystyle l_{1},} and l 2 {displaystyle l_{2},} are anti-parallel with respect to m 1 {displaystyle m_{1},} and m 2 {displaystyle m_{2},} , then m 1 {displaystyle m_{1},} and m 2 {displaystyle m_{2},} are also anti-parallel with respect to l 1 {displaystyle l_{1},} and l 2 {displaystyle l_{2},} . In any quadrilateral inscribed in a circle, any two opposite sides are anti-parallel with respect to the other two sides (Fig.2). Two lines l 1 {displaystyle l_{1}} and l 2 {displaystyle l_{2}} are antiparallel with respect to the sides of an angle if and only if they make the same angle ∠ A P C {displaystyle angle APC} in the opposite senses with the bisector of that angle (Fig.3). In a Euclidean space, two directed line segments, often called vectors in applied mathematics, are antiparallel, if they are supported by parallel lines and have opposite directions. In that case, one of the associated Euclidean vectors is the product of the other by a negative number.