Parallel and Perpendicular Lines

Lines and Angles

Two lines are parallel line if they are coplanar and do not intersect. Lines that do not intersect and are not coplanar are called skew lines. Similarly, two planes that do not intersect are called parallel planes.

and are parallel lines. Planes U and W are parallel planes.

and are skew lines.

To write “ is parallel to .” You write || . Triangles like those on and are used on diagrams to indicate that lines are parallel.

Segments and rays are parallel if they lie on parallel lines. For example, || .

Identifying Angles Formed by Transversals

A transversal is a line that intersects two or more coplanar lines at different points. For instance, in the diagrams below, line t is a transversal. The angles formed by two lines and a transversal are given special names.

Consecutive interior angles are sometimes called same side interior angles.

When Lines and Planes Are Parallel

Definitions

Two lines that do not intersect are either parallel or skew.

Parallel lines (|| lines ) are coplanar lines that do not intersect.

Skew lines are noncoplanar lines. Therefore, they are neither parallel nor intersecting.

Theorem

If two parallel planes are cut by a third plane, then the lines of intersection are parallel.

Given : Plane X || plane Y :

Plane Z intersects X in line l :

Plane Z intersect Y in line n.

Prove: l || n

Proof:

A transversal is a line that intersects two or more coplanar lines in different points. In the next diagram, t is a transversal of h and k. The angles formed have special names.

Interior angles : angles 3, 4, 5, 6

Exterior angles : angles 1, 2, 7, 8

Alternate interior angles (alt. int. ) are two nonadjacent interior angles on opposite sides of the transversal.

Ð 3 and Ð 6 Ð 4 and Ð 5

Same - side interior angles (s - s. int. ) are two interior angles on the same side of the transversal.

Ð 3 and Ð 5 Ð 4 and Ð 6

Corresponding angles (corr. ) are two angles in corresponding positions relative to the two lines.

Ð 1 and Ð 5 Ð 2 and Ð 6 Ð 3 and Ð 7 Ð 4 and Ð 8

Proving Lines Parallel

From two angles being congruent or supplementary you will conclude that certain lines forming the angles are parallel.

Postulate: If two parallel lines are cut by a transversal, then corresponding angles ate congruent.

Postulate: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

Theorem

If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

Given: Transversal t cuts lines k and n;

Ð 1 @ Ð 2

Prove: k || n

Proof:

Theorem

It two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.

Given: Transversal t cuts lines k and n ;

Ð 1 is supplementary to Ð 2.

Prove: k || n

Theorem

In a plane two lines perpendicular to the same line are parallel.

Given: k ^ t; n ^ t

Prove: k || n

Ways to Prove Two lines Parallel

1. Show that a pair of corresponding angles are congruent.

2. Show that a pair of alternate interior angles are congruent.

3. Show that a pair of sameside interior angles are supplementary.

4. In a plane show that both lines are perpendicular to a third line.

5. Show that both lines are parallel to a third line.

Recognizing and Using Definitions

Two lines are called perpendicular lines if they intersect to form a right angle. A line perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it. The symbol ^ is read is “is perpendicular to.”

Equipment

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