Maths – Parallel lines and properties relating to parallel lines

Properties of Parallel Lines –

  • The basic characteristic of Parallel Lines is they never meet at any point
  • The distance between two Parallel Lines would always be same at every point
  • Any third line will cut Parallel Lines at same angles
  • Any straight line cutting Parallel Lines is called Transversal
  • Two Transversals cutting Parallel lines at same angle would also be Parallel in themselves
  • Any line Parallel to one Parallel Line would be Parallel to all Parallel Lines

Please assume following as shown in above figure –

  • P1 & P2 are two Parallel Lines
  • P3 as third Parallel Line to P2
  • T1 as Transversal
  • T2 as a Parallel Line to T1

Let us first consider just two Parallel Lines P1 & P2 and one Transversal T1

Exterior Area –

Any area below bottom line and above top line is called Exterior Area

Interior Area –

Any area between both the lines is called Interior Area

Same Distant –

For any two Parallel Lines their distances would always remain same at all points i.e. D1 = D2

Corresponding Angles –

For any two Parallel Lines Corresponding angles are the ones which are marked with same colors and they would always be equal –

  • Angle 1 = Angle 3
  • Angle 2 = Angle 4 
  • Angle a = Angle c 
  • Angle b = Angle d

Alternate Interior Angles –

These are the angles which are non-adjacent and on opposite sides of the Transversal and between the two Parallel Lines and they would always be equal –

 

  • Angle a = Angle 4              =  
  • Angle b = Angle 3             =  

 

Alternate Exterior Angles –

These are the angles which are non-adjacent and on opposite sides of the Transversal and outside the two Parallel Lines and they would always be equal –

  • Angle 1 = Angle d   
  • Angle 2 = Angle c       =

Same-Side Interior Angles –

These are the angles which are non-adjacent and on same side of the Transversal and between the two Parallel Lines and they would always be Supplementary –

  • Angle a + Angle 3 = 180          = 180
  • Angle b + Angle 4 = 180          = 180

Vertically Opposite Angles –

These are the angles which are non-adjacent and on Opposite side of the point where Transversal cuts Parallel Lines and they would always be equal –

  • Angle 1 = Angle b    =
  • Angle 3 = Angle d    =
  • Angle 2 = Angle a     =
  • Angle 4 = Angle c     =

Adding one more Parallel Line P3 –

Assume there is one more line P3 which is parallel to P2, it would always be parallel to P1 as well and its distance would remain same from P1 as well at all points.

Adding one more Transversal T2 –

Assume there is more line T2 which cuts all three Parallel lines at same angles as it is done by T1, if the angle is same then both T1 & T2 are parallel with each other and all laws which are applied to T1 are applied to T2

Congruent Angles –­­­

All angles formed by T1 & T2 with Parallel Lines P1, P2 & P3 corresponding to each other at respective positions would always be equal, Assuming there are two Parallel Lines P1 & P3 and two Transversals T1 & T2 –

  • Angle 1 = Angle 7    
    Angle 5 = Angle 11

 

  • Angle 2 = Angle 8     
    Angle 6 = Angle 12

 

  • Angle a = Angle g    
    Angle e = Angle k

 

  • Angle b = Angle h   
    Angle f = Angle i

Alternate Interior Angles –

These are the angles which are non-adjacent and on opposite sides of the Transversal and between the two Parallel Lines and they would always be equal –

  • Angle a = Angle 6              = 
    Angle g = Angle 12
  • Angle b = Angle 5             = 
    Angle h = Angle 11

Alternate Exterior Angles –

These are the angles which are non-adjacent and on opposite sides of the Transversal and outside the two Parallel Lines and they would always be equal –

  • Angle 1 = Angle f            =
    Angle 7 = Angle l
  • Angle 2 = Angle e            =
    Angle 8 = Angle k

Same-Side Interior Angles –

These are the angles which are non-adjacent and on same side of the Transversal and between the two Parallel Lines and they would always be Supplementary –

  • Angle a + Angle 5 = 180        +       = 180
    Angle g + Angle 11 = 180
  • Angle b + Angle 6 = 180      +        = 180
    Angle h + Angle 12 = 180

Vertically Opposite Angles –

These are the angles which are non-adjacent and on Opposite side of the point where Transversal cuts Parallel Lines and they would always be equal –

  • Angle 7 = Angle h             =
  • Angle 11 = Angle l            =
  • Angle 8 = Angle g             =
  • Angle 12 = Angle k           =