Geometry/Chapter 3/Lesson 1

When Lines and Planes are Parallel

Parallel Planes

Line m and line l are parallel to each other, and are in the same plane

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.

Keith Mann's video talks about these terms listed in this image a lot more thoroughly. See his video: https://www.youtube.com/watch?v=CyydGXPpYvI

Theorems

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

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

The yellow shade is NOT a plane/intended to represent a plane

Angles Formed by Parallel Lines

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.