Geometry/Chapter 3/Lesson 1
When Lines and Planes are Parallel[edit | edit source]
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 | edit source]
- 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 | edit source]
- 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 | edit source]
Angles Formed by Parallel Lines[edit | edit source]
- 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.