Pairs of Lines

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

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.

A * transversal* is any line that intersects two or more lines in the same plane but at

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

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

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

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*.

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

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.

- 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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