Parallel Lines and a Transversal

If a line passes through two distinct lines and intersects them at distant points then this line is called Transversal Line.

Here line “l” is transversal of line m and n.

Exterior Angles – \angle1,\angle2,\angle7 and \angle 8

Interior Angles – \angle3,\angle4,\angle5 and \angle6

Pairs of angles formed when a transversal intersects two lines.

1. Corresponding Angles :

  • \angle1 and \angle5
  • \angle2 and \angle6
  • \angle4 and  \angle8
  • \angle3 and \angle7

2. Alternate Interior Angles :

  •  \angle4 and \angle6
  • \angle3 and \angle5

3. Alternate Exterior Angles:

  • \angle1 and \angle7
  •  \angle2 and \angle8

4. Interior Angles on the same side of the transversal:

  • \angle4 and \angle5
  •  \angle3 and  \angle6

Transversal Axioms

1.  If a transversal intersects two parallel lines, then

  • Each pair of corresponding angles will be equal.
  • Each pair of alternate interior angles will be equal.
  • Each pair of interior angles on the same side of the transversal will be supplementary.

2. If a transversal intersects two lines in such a way that

  • Corresponding angles are equal then these two lines will be parallel to each other.
  • Alternate interior angles are equal then the two lines will be parallel.
  • Interior angles on the same side of the transversal are supplementary then the two lines will be parallel.

Example

Find \mathrm{\angle DGH.}

Solution

Here, AB ∥ CD and EH is transversal.

\angle EFB + \angle BFG = 180° (Linear pair)

\angle BFG = 180° – 133°  

\angle BFG = 47°

\angle BFG = \angle DGH (Corresponding Angles)

\angle DGH = 47°

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