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PROPERTIES OF PARALLEL LINES

 
      In Geometry and other branches of Mathematics, properties of parallel lines are most useful. Especially when a transverse line meets two parallel lines are intersect, this line gives useful results. Earlier we have told that parallel ines run parallel to each other and they maintain same distance at every point. We must remember that distance is to be measured in perpedicular direction. Let us assume that two parallel lines are running in East-West direction. Then any line perpendicular to both lines will run in North -South direction. Any line perpendicular to one of the lines will also be perpendicuilar to the other line.
       
In the figure below, AB & CD are two parallel lines. At ponts F & H on line CD, FE & HG are two perpendiculars which meet line AB on E & G points. By definition of parallel lines, perpendicular distances at each point are equal in length. Therefore, we reach the conclusion that both perpendiculars FE & HG are of equal length.
      Transversal line is the line which meets two or more parallel lines. It is not necessary that transversal line should cross each line. A line which only meets two parallel lines is also a transversal line. In both figures below, line EF is transversal line.
       
RECALL OF STRAIGHT ANGLE
Here it will usefull to recall the meaning of right angle. Straight angle is the angle whose both arms make a straight line. One can say that straight angle is the angle which covers one side of a straight line.
In the above figure angle CAB is made at A and its both arms CA & BA lie in straight line CB. Therefore, < CAB is straight angle and its value is 1800. Now notice straight angles made at point O in the following figure.
Here     <COB + <BOA =straight angle COA =1800
And     <BOA + < AOD =straight angle BOD =1800
Right hand side of above both equations is equal and is 1800. Hence left hand sides will also be equal. It means:
    <COB + <BOA =<BOA + < AOD
Now <BOA is on both sides. So if we delete it from both sides (we subtract this angle BOA from both sides, then equation gives us:
    <COB =< AOD
Similarly taking other angles, we can prove that
    < COD = < AOB
This proves that when two straight lines intersect each other, opposite vertex angles made at the common meeting point of lines are equal in value. This result in combination with properties of parallel lines gives us useful results.
Now look at the following figure:
Whenever a transversal line intersects two parallel lines, exterior angles are formed in the area outside both lines and interior angles are formed in the area in between the both lines. In the figure above < EGA, <EGB are outside and above line AB and < FHC & < FHD are outside and below line CD. All these four angles are exterior angles. < AGH, < BGH on line AB are in the area inside the lines and < CHG, <DHG on line CD are also in the area inside the lines. These four angles are interior angles. Out of four exterior angles two are based on one line and remaining two on the second line. Also on each line one angle is to the right of tranversal line EF and one is to the right of the transversal line. We can make two stes of exterior angles each set containing one angle on first line and other angle on second line in a manner that angle on first line is on one side of transversal line EFand second angle is on second line and on the other side of transversal line EF'. Each of two exterior angles, one on each line, situated on opposite sides of transversal line are called alternate exterior angles. Similarly, there are 4 angles situated in area inside the parallel lines. All these four angles are interior angles. Two angles AGH & GHD are on opposite sides of tranversal line one on each line. Similarly two angles BGH & GHC are also on opposite sides of transversal line one on each line. Each set of these two interior angles, one on each line, situated on opposite sides of transversal line is called alternate interior angles.
       
       
 At this stage we know that:
(i) slope or titlt of transversal line EF from parallel lines AB & CD is equal. This gives us < EGB = < GHD
(ii) at a vertex opposite angles are equal. At G this gives us < AGE = < HGB & <EGB = < AGH. Similarly at H we find that < GHC = < FHD & < GHD =< CHF
From above two logics, we can prove that alternate exterior < EGB = alternate exterior < CHF & alternate exterior < AGE = alternate exterior < FHD. Similarly, for interior angles we can prove that alternate interior < AGH = alternate interior < GHD & alternate interior < BGH = alternate interior < GHC.
This will give important results. Alternate exterior angles are equal in measurement, and alternate interior angles are equal in measurement.
Students are expected to prove themselves that (i) alternate exterior angles are equal in measurement and (ii) alternate interior angles are equal in measurement. If any student wants our help, he may ask but he will have to send details of efforts made by him.
 
 

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