Pairs of Lines

Objective

In geometry, lines often occur in pairs.

This topic gives an overview of;

  • Intersecting Lines
  • Transversal
  • Angles made by a Transversal
  • Transversal of Parallel Lines
  • Checking for Parallel Lines 

Intersecting Lines

Two lines are said to be intersecting lines if they have a common point. It is to be noted that if two straight lines intersect each other, they can intersect at exactly point and that point is said to be the Intersecting point of those two lines.

In the figure, there are two lines and are two straight lines and P is a common point. Here, we say that the straight lines are intersecting at the point P and the point P is called the Intersecting point of those two lines.

The blackboard on its stand, the letter Y made up of line segments and the grill-door of a window are examples of Intersecting Lines.

Transversal

A transversal is any line that intersects two or more lines in the same plane but at distinct points. Road crossing two or more roads or a railway line crossing several other lines are examples of transversal.

The transversal is said to cut the two lines that it crosses.

 

Angles made by a Transversal

If we draw two parallel lines and then draw a line transversal through them we will get eight different angles.The eight angles will together form four pairs of corresponding angles.

Angles F and B in the figure above constitutes one of the pairs. Corresponding angles are congruent  if the two lines are parallel. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs. Angles that are in the area between the parallel lines like angle H and C above are called Interior angles whereas the angles that are on the outside of the two parallel lines like D and G are called Exterior angles.Angles that are on the opposite sides of the transversal are called Alternate angles e.g. H and B.

Angles that share the same  vertex and have a common ray, like angles G and F or C and B in the figure above are called Adjacent angles. As in this case where the adjacent angles are formed by two lines intersecting we will get two pairs of adjacent angles (G + F and H + E) that are both supplementary. Two angles that are opposite each other as D and B in the figure above are called vertical angles. Vertical angles are always congruent

Transversal of Parallel Lines

If two parallel lines are cut by a transversal, each pair of corresponding angles is equal in measure. 

When t cuts the parallel lines, l, m, we get, ∠3 = ∠7 (vertically opposite angles). But ∠7 = ∠8 (corresponding angles). Therefore, ∠3 = ∠8, similarly ∠1 = ∠6. Thus If two parallel lines are cut by a transversal, each pair of alternate interior angles is equal.

The second result is  ∠3 + ∠1 = 180° (∠3 and ∠1 form a linear pair), But ∠1 = ∠6 (A pair of alternate interior angles) Therefore, we can say that ∠3 + ∠6 = 180°. Similarly, ∠1 + ∠8 = 180°.

Thus, if two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are supplementary.

Checking for Parallel Lines

If two lines are parallel, then we know that a transversal gives rise to pairs of equal corresponding angles, equal alternate interior angles and interior angles on the same side of the transversal being supplementary. When a transversal cuts two lines, such that pairs of corresponding angles are equal, then the lines have to be parallel.

Look at the letter Z, the horizontal segments here are parallel, because the alternate angles are equal. When a transversal cuts two lines, such that pairs of alternate interior angles are equal, the lines have to be parallel.

Draw a line l. Draw a line m, perpendicular to l. Again draw a line p, such that p is perpendicular to m. Thus, p is perpendicular to a perpendicular to l. We find p is parallel to l. This is because p such that ∠1 + ∠2 = 180 degree.

 

Thus, when a transversal cuts two lines, such that pairs of  interior angles on the same side of the transversal are supplementary, the lines have to be parallel.

Summary

  • When two lines meet, we say they intersect; the meeting point is called the point of intersection.
  • When lines drawn on a sheet of paper do not meet, however far produced, we call them to be parallel lines. 
  • When two lines intersect (looking like the letter X) we have two pairs of opposite angles. They are called vertically opposite angles. They are equal in measure.
  • A transversal is a line that intersects two or more lines at distinct points.
  • A transversal gives rise to several types of angles.
  • We have learnt about six types of angles Interior, Exterior, Corresponding, Alternate interior, Alternate exterior, Interior on the same side of transversal.
  • When a transversal cuts two parallel lines, we have the following interesting relationships:
    • Each pair of corresponding angles is equal.
    • Each pair of alternate interior angles is equal.
    • Each pair of interior angles on the same side of transversal is supplementary.

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