# Geometry/Chapter 3/Lesson 1

## When Lines and Planes are Parallel[edit]

Two lines that do not meet (or, *intersect*) are either parallel or skew.

**Parallel**(||, parallel lines symbol--*is parallel to*) lines do not intersect and are coplanar (please refer back to the first lesson if you have forgotten the definition of "coplanar").**Skew**lines do not intersect and are NOT coplanar.

Segments and rays in the parallel lines are also termed "parallel". Say if line *m* contained A and B, while line *l* contained C and D. AB || CD [line segments] and AB || CD [rays].

- ∦ means
*is not parallel to*. So AB ∦ CG means line AB is not parallel to line CG.

**Parallel**planes are two planes that do not intersect.**Perpendicular**lines are two lines that form 90 degrees.

## Theorems[edit]

- Theorem 2-1

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

IMG: https://www.google.com/search?q=parallel+planes&rlz=1C1GGRV_enUS760US760&source=lnms&tbm=isch&sa=X&ved=0ahUKEwj824yktNTWAhUF5yYKHYm8BAIQ_AUICigB&biw=1366&bih=637&safe=active&ssui=on#imgrc=hUqvVboZAEqRTM (Line AB and line CD are parallel. Thus AB || CD--the planes are parallel and BOTH planes contain lines AB and CD).

## Angle terms[edit]

**Transversal**lines intersect two or more coplanar lines in different points. (Line*t*intersects lines*a*and*r*)**Alternate interior angles**are two**nonadjacent**interior angles on opposite sides of the transversal [line]. (∠2 and ∠3; ∠4 and ∠1)**Same-side interior angles**are two interior angles on the same side of the transversal. (∠1 and ∠3; ∠2 and ∠4)**Corresponding angles**are two angles in corresponding positions relative to the two lines. (∠1 and ∠5; ∠2 and ∠6)

We will use the picture below to explain these terms so you may better understand this.

### Explanations[edit]

## Angles Formed by Parallel Lines[edit]

- Postulate 7-1

Corresponding Angles Postulate - If two parallel lines are cut by a transversal, then corresponding angles are congruent.

- Theorem 7-1

Alternate Interior Angles Theorem - If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

- Theorem 7-2

Same-Side (Consecutive) Interior Angles Theorem - If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary.