Download Topic 3 Notes - Lines and Planes

Topic 3 Lecture Notes
Exploring Lines and Planes
I. Definitions A. Parallel Planes - planes that do not intersect.
B. Parallel Lines - lines that are coplanar and do not intersect.
<>
C. Skew Lines - lines that are not coplanar and do not intersect.
D. Oblique Lines – Lines that are coplanar and intersecting
II. Theorem: If two lines are parallel to the same line, then they are parallel to each other.
III. Theorem: If two coplanar lines are perpendicular to the same line, then they are
parallel to each other.
IV. Theorem: If a line is perpendicular and coplanar to one of two parallel lines, then it is
perpendicular
to the second line
V. Slopes of Parallel and Perpendicular Lines A. Concept - Two lines are parallel iff their slopes are equal (m1 = m2)
B. Concept - Two lines are perpendicular iff the product of their slopes are
negative reciprocals (m1 * m2 = -1).
Connections to Algebra
I. Linear Functions - Slope-Intercept Form, Point-Slope Form, Standard Form
A. Slope-Intercept Form: y = mx + b where, m is the slope and b is the yintercept.
Example #1: Write the equation of the function that contains the point (0,
4) and has a slope of 1/2, and graph the function.
B. Point-Slope Form: The function that passes through point (x1,y1) and has slope
m is
y - y1 = m(x - x1)
Example #1: Write an equation of the function that contains the point (2, -
5) and has a slope of 4/3.
C. Standard Form: AX + BY - C = 0 where, A is the coefficient of X, B is the
coefficient of Y and, C is the constant.
D. Additional Examples:
Example #1:
Example #2:
II. Systems of Equations
A. Concept - In Geometry, used to determine the intersection of linear functions.
1. Substitution Method:
2. Elimination Method:
Styles of Proofs
I. Theorems A. Theo. 3.3 - If two lines are perpendicular, then they intersect to form four right
angles.
B. Theo. 3.4 - All right angles are congruent.
C. Theo. 3.5 - If two lines intersect to form a pair of adjacent, congruent angles,
then the lines are perpendicular.
II. Types of Proofs A. Paragraph Proof - a proof written as a narrative; often referred to as an
"informal proof").
1. Example #1: Given that "M" is the midpoint of segment AB, show that
segment AM is congruent to segment MB.
Answer: From the definition of midpoint, we know that AM = MB. By the
definition of congruent segments, if segments AM and MB have the same
measure, then they are congruent segments. Therefore, segment AM is
congruent to segment MB.
B. Flow Proof - uses a flow diagram style to show the "flow" of the logical
statements.
1. Example #1:
C. Two-Column - uses a deductive reasoning process that lists formal statements
that are accepted as true (definitions, previously proven theorems, postulates,
etc.).
1. Four essential elements:
a. state the theorem to be proved
b. list the given information
c. if possible, draw a diagram to show the given information
d. develop a valid proof using a system of deductive reasoning
2. Format:
III. Proof of Theorem: Compliments of congruent angles are congruent
<>
IV. Additional Example -
<>
Sec.3.5: Properties of Parallel Lines
I. Transversal - a line that intersects two coplanar lines at two different points.
A. A transversal (and two coplanar lines) forms eight angles.
1. alternate interior angles - two non adjacent interior angles on opposite
sides of the "t".
2. alternate exterior angles - two nonadjacent exterior angles on opposite
sides of "t".
3. corresponding angles - two nonadjacent angles on the same side of "t"
such that one is an exterior angle and one is an interior angle.
4. consecutive (same-side) interior angles - two interior angles on the same
side of "t".
B. Example:Identify each pair of angles below.
II. Theorem - If two parallel lines are cut by a transversal, then each pair of
corresponding angles is congruent.
III. Theorem - If two parallel lines are cut by a transversal, then each pair of alternate
interior angles is congruent.
IV. Theorem - If two parallel lines are cut by a transversal, then each pair of consecutive
interior angles is supplementary.
V. Theorem - If two parallel lines are cut by a transversal, then each pair of alternate
exterior angles is congruent.
A. Proof:
VI. Theorem - If a line is perpendicular to one of two parallel lines, then it is
perpendicular to the other line.
VII. Algebraic Application:
Example:
Sec.3.6: Proving Lines Parallel
I. Postulate - If two coplanar lines are cut by a transversal such that corresponding angles
are congruent, then the lines are parallel.
II. Postulate - If there is a line and a point not on the line, then there exists exactly one
line through the point that is parallel to the given line.
III. Theorem - If two coplanar lines are cut by a transversal such that a pair of alternate
exterior angles are congruent, then the two lines are parallel.
IV. Theorem - If two coplanar lines are cut by a transversal such that a pair of
consecutive interior (same-side) angles are supplementary, then the lines are parallel.
V. Theorem - If two coplanar lines are cut by a transversal such that a pair of alternate
interior angles is congruent, then the lines are parallel.
VI. Theorem - Two lines parallel to a third line are parallel to each other.
VII. Ways to Prove Lines are Parallel Using a Transversal
A. Show that a pair of corresponding angles are congruent.
B. Show that a pair of alternate interior angles are congruent.
C. Show that a pair of alternate exterior angles are congruent.
D. Show that a pair of consecutive interior angles are supplementary
E. Show that two coplanar lines are perpendicular to the same line.
F. E. Show that two coplanar lines are parallel to the same line.
VIII. Algebraic Applications
A. Example #1:
B. Example #2:
C. Example #3:
IX. Additional Proofs
A. Example #1:
B. Example #2: