notes

Several parents and students have asked me if I could post class notes on this site to help when a student misses class.
These are my notes for class--they are only an outline.
These notes will not replace the need to attend class but if a student does miss class, this should help him/her find the information missed.
The text book will have much of the necessary information.
Other students from class should be able to fill in any remaining gaps.
When all else fails, ask me.

Chapter 3--lesson plans

section 1

parallel lines--2 lines that never cross (NOT enough defintion)
skew lines--2 lines that never cross cannot have 2 different words that mean the same thing

PARALLEL--

2 lines in the same plane that do not intersect
2 lines that run the same direction and do not cross
2 lines that are the same distance apart everywhere

SKEW--

2 lines that are in different planes and do not intersect
2 non-intersecting lines that do not point in the same direction
2 lines that don't intersect but are not the same distance apart

parallel planes--planes that do not intersect

can planes be skew???

symbol for parallel lines

when writing-- ||

not the number 11 (taller and closer together than a pair of 1's)

when drawing-- V on each of the 2 lines or a small filled in triangle on each
no symbol for skew Parallel Postulate

if there is a line and a point not on the line, then there is one and only one line through the point that is parallel to the line

Perpendicular Postulate

if there is a line and a point not on the line, then there is one and only one line through the point that is perpendicular to the line

transversal--any line that crosses at least 2 other lines

corresponding angles
alternate interior angles
alternate exterior angles
consecutive interior angles (aka same side interior angles)
same side exterior angles
- these 5 types of angle pairs ALWAYS occur at different intersections
- the angles have DIFFERENT vertices
- the angles CANNOT be adjacent or vertical

section 2

if the lines are parallel and crossed by a transversal, then

corresponding angles are congruent
alternate interior angles are congruent
alternate exterior angles are congruent
consecutive interior angles are supplementary

if the lines are NOT parallel, then

NONE of the above rules can be applied

section 3

paragraph proof

allows you to shorten your proof
uses sentences--if you like that sort of thing
can use either method in this class
- there will be other methods of proof writing later in this class
basics of proofs still apply
- the order of the statements needs to be logical
- the reasons for each statement need to be obvious
you are going to have a very long list of theorems and postulates
- it might be nice to have a list
  - p. 926/927
- the really useful ones you will accidentally memorise
- do you need to memorize ALL??
  - no--some are really only useful in specific instances
  - I will tell you winch ones are likely to be addressed on tests
- do you need to know which are postulates and which are theorems??
  - no--once the theorems have been proven, there is really no difference between them

CONSTRUCT

parallel lines
perpendicular lines

section 4

slope of a line

y = mx + b from algebra
- m = slope
- rise over run
- difference in the y's over difference in the x's
kinds of slopes
- positive slope = line goes up
- negative slope = line goes down
- slope is zero = line is horizontal
- no slope exists = line is vertical
- slope is a large number (i.e. 2 or 3 or more) = line is steep
- slope is a small number (btwn 0 and 1) = line is not very steep
parallel lines
- lines have the same slope
- m₁ = m₂
- are these 2 lines parallel?
- write an equation for a line that is parallel to ____
- write the equation for a line that is parallel to ____ and passes through (__,__)
perpendicular lines
- slopes are opposite and reciprocals
- m₁ * m₂ = -1
- m₁ = -1 / m₂
- are these 2 lines perpendicular?
- write an equation for a line that is perpendicular to ____
- write the equation for a line that is perpendicular to ____ and passes through (__,__)

section 5

graph from an equation in slope intercept form

draw the line parallel to y = ___x + ___ and passes through (__,__)
draw the line perpendicular to y = ___x + ___ and passes through (__,__)

graphing from an equation in standard form

graph 2 intercepts
- finding the slope of a line from the graph
- finding the slope of a line from the intercepts
draw the line that is parallel to __x + __y = __ through (__,__)
draw the line that is perpendicular to __x + __y = __ through (__,__)

section 6

perpendicular theorems to think about

if 2 lines intersect to form a pair of congruent adjacent angles, then the lines are perpendicular
if 2 lines are perpendicular, then they form 4 right angles
if 2 sides of 2 adjacent acute angles are perpendicular, then the angles are complementary
if a transversal is perpendicular to one of 2 parallel lines, then it is perpendicular to the othere line also
if 2 lines in a plane are perpendicular to the same line, then the lines are parallel

section 6--continued

in order to accept a theorem, you must prove them

determine what the theorem states as its given--the 'if' hypothesis part
determine what the theorem states is the conclusion--the 'then' part
draw a picture
mark and label the picture

Distance from a point to a line (segment, ray, or plane)

definition--the shortest distance possible between the point and the line
must be measured perpendicular to the line

review
practice writing proofs

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