14. Properties of Parallel Lines and Transversal

 

14. PROPERTIES OF PARALLEL LINES AND TRANSVERSAL

Learn and Remember :

 1. Transversal : If a line intersects two or more coplanar lines in distinct points, it is called a transversal of the given lines.

e.g. in the figure, line p intersects the lines land min two distinct points A and B. Hence the line p is the transversal of the lines l and m.

 2. Interior angles : If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are supplementary.

e.g. parallel lines l and m are intersected by a transversal p. ∠a and ∠b are the interior angles on the same side of the transversal. :: ∠a and ∠b are supplementary.

i.e. m ∠a + m∠b = 180°.

Also ∠c and ∠d are the interior angles on the same side of the transversal.

∴ ∠c and ∠d are supplementary.

i.e. m∠c + m∠d = 180°. 

3. (a) Interior alternate angles : If two parallel lines are intersected by a transversal, then the pairs of alternate angles so formed are congruent.

e.g. in the adjoining figure, parallel lines land mare intersected by the transversal n. The two pairs of alternate angles are congruent. i.e. ∠g = ∠b and ∠e = ∠d.

   (b) Exterior alternate angles : In the figure, ∠a = ∠h and ∠c = ∠f.

4. Corresponding angles : If two parallel lines are intersected by a transversal, then the pairs of corresponding angles thus formed to are congruent.

e.g. in the adjoining figure, parallel lines land mare intersected by the transversal n.

The corresponding angles thus formed are congruent.

i.e. ∠a = ∠b, ∠c =∠d, ∠e =∠f and ∠g = ∠h.

5. The ratio of the intercepts made on a transversal by three parallel lines is equal to the ratio of the corresponding intercepts made on any other transversal by the same parallel lines.

Line 1 || line m || line n and line x and line y are the transversals.

  ∴  AB ∣  PQ =  BC ∣  QR

  [This is known as the property of intercepts made by three parallel lines. ]