Parallel Lines and Transversal

Parallel Lines and Transversal

Parallel lines are lines which extend in the same direction and remain the same distance apart.

In the following figure, AB and CD are parallel lines. We denote them by “AB // CD” and say “AB is parallel to CD”.

The rails on railway track are parallel.

In geometry, parallel lines are indicated with equal numbers of arrow heads on each line.

Transversal

When a line intersects two other lines, the line is called a transversal of those two lines. In the figure, there are 8 angles formed at the two intersections where TS cuts lines AB and CD.

These angles are named in pairs according to their relative positions as follows:

1.      ∠a and ∠p, ∠b and ∠q, ∠c and ∠r, ∠d and ∠s are pairs of corresponding angles.

 2.      ∠a and ∠r, ∠b and ∠s are pairs of alternate angles. More specifically, these angles are referred to as alternate interior angles.

 3.      ∠a and ∠s, ∠b and ∠r, are pairs of interior angles on the same side of the transversal, or simply consecutive interior angles.

Conditions When Parallel Lines are Cut by a Transversal

When two parallel lines are cut by a transversal, then

1. their corresponding angles are equal,

2. their alternate angles are equal,

3. their consecutive interior angles are supplementary.

The converse statements for the above are also true. That is, when two straight lines AB and CD are cut by a transversal TS, then

1. if the corresponding angles are equal, then AB // CD,

2. if the alternate angles are equal, then AB // CD,

3. if the consecutive interior angles are supplementary, then AB // CD.

Example 1: In the figure, AB // CD. Find the measures of angles x, y and z.

Solution: ∠x = 70° (corr. ∠s, AB // CD)

∠y = 70° (alt. ∠s, AB // CD)

∠z + 70° = 180° (int. ∠s, AB // CD)

∴ ∠z = 110°

Example 2: In the figure, BA // DE. Find the measure of angle x.

Solution: Construct FC // BA // DE.

∠x₁ = 35° (alt. ∠s, FC // DE)

∠x₂ = 30° (alt. ∠s, FC // BA)

∠x = ∠x₁ + ∠x₂ 

      = 35° + 30°

      = 65°