Transversal of Parallel Lines

The two lines which never meet with each other are called Parallel Lines. If we have a transversal on two parallel lines then-

a. All the pairs of corresponding angles are equal.

∠3 = ∠7

∠4 = ∠8

∠1 = ∠5

∠2 = ∠6

b. All the pairs of alternate interior angles are equal.

∠3 = ∠6

∠4 = ∠5

c. The two Interior angles which are on the same side of the transversal will always be supplementary.

∠3 + ∠5 = 180°

∠4 + ∠6 = 180°

Checking for Parallel Lines

This is the inverse of the above properties of the transversal of parallel lines.

  • If a transversal passes through two lines so that the pairs of corresponding angles are equal, then these two lines must be parallel.
  • If a transversal passes through two lines in so that the pairs of alternate interior angles are equal, then these two lines must be parallel.
  • If a transversal passes through two lines so that the pairs of interior angles on the same side of the transversal are supplementary, then these two lines must be parallel.

Example: 1

If AB ∥ PQ, Find ∠W.

Solution:

We have to draw a line CD parallel to AB and PQ passing through ∠W.

∠QPW = ∠PWC = 50° (Alternate Interior Angles)

∠BAW =∠CWA = 46°(Alternate Interior Angles)

∠PWA = ∠PWC +∠CWA

= 50°+ 46°= 96°

Example: 2

If XY ∥ QR with ∠4 = 50° and ∠5 = 45°, then find all the three angles of the ∆PQR.

Solution:

Given:  XY ∥ QR

∠4 = 50° and ∠5 = 45°

To find: ∠1, ∠2 and ∠3

Calculation: ∠1 + ∠4 + ∠5 = 180° (sum of angles making a straight angle)

∠1 = 180°- 50°- 45°

∠1 = 85°

PQ is the transversal of XY and QR, so

∠4 = ∠2 (Alternate interior angles between parallel lines)

∠2 = 50°

PR is also the transversal of XY and QR, so

∠5 = ∠3 (Alternate interior angles between parallel lines)

∠3 = 45°

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