Geometry/Chapter 3/Lesson 1

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When Lines and Planes are Parallel[edit]

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[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]

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

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.