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Chapter 3 Notes
3.1 Lines and Angles
Two lines are PARALLEL if they are COPLANAR and
do not INTERSECT
Two lines are SKEW if they are NOT COPLANAR and
do not INTERSECT
Arrows on
line mean
they are
parallel
B
F
A
C
E
H
D
G
Two planes that do not Intersect are called
PARALLEL planes.
A line and a plane are parallel if they do not
intersect.
B
F
A
C
E
H
D
G
Line segments and rays can be parallel too!
O
R
Y
T
O
R
Y
T
As long as the lines going through them are
also parallel.
R
T
O
Y
O
R
T
Y
Let’s name some parallel planes, lines, and some skew
lines.
B
F
A
C
E
H
D
G
Parallel line postulate If there is a line and a point not on
the line, there is EXACTLY one parallel line through the
given point.
Perpendicular line postulate  If there is a line and a point
not on the line, there is EXACTLY one perpendicular line
through the given point.
Not Parallel
A ______________ is a line that
INTERSECTS two or more COPLANAR lines
at different points.
Two angles are ________________
ANGLES if they occupy
________________ positions.
1
2
3
4
Two angles are ________________
__________________ if they LIE
5
6
_________ the two lines on
7
8
__________ sides of the
________________.
Two angles are _____________________________ if they LIE
__________ the two lines on ___________ sides of the TRANSVERSAL.
Two angles are ______________________________ (also called same
side interior angles) if they LIE _________ the two lines on the
_______ sides of the TRANSVERSAL.
Given a point off a line, draw a line perpendicular to
line from given point.
1) From the given point, pick any arc and mark the circle left
and right.
2) Those two marks are your
endpoints, and construct a
perpendicular bisector just like
the previous slide.
3.3 – Parallel Lines and
Transversals
Corresponding Angles Postulate (CAP)
If two lines cut by transversal are ||, then the
corresponding angles are congruent
If m || n,
then 1  2
m
Note, need to use def  ' s
1
to change to m
n
2
Alternate Interior Angles
Theorem (AIA Thrm)
m
If two lines cut by transversal
are ||, then the alternate
interior angles are congruent
n
Consecutive Interior Angles
Theorem (CIA Thrm)
If two lines cut by transversal
are ||, then the consecutive
interior angles are
supplementary
1 2
3
4 5
6
Alternate Exterior Angles
Theorem (AEA Thrm)
If two lines cut by transversal
are ||, then the alternate
exterior angles are congruent
Perpendicular Transversal
If a transversal is
perpendicular to one of two
|| lines, then it is
perpendicular to the other.
If m || n, m  t
then, n  t
m
1
n
2
t
Find the measure of angles 1 – 7 given the
information below.
1
2
3
4
5
7
6
800
12x  20
Find x, y
14x  4
4y
Find x, y, and the
measure of all angles
4
3
2
1
m1  6 y
m3  4 y  32
m2  5 x
m4  3x  10
Find w, x, y, z, and the
measure of all angles
1
4
2
3
5
m1  8 x
m2  5 x  24
m3  y
m4  z  20
m5  w
3.4 – Proving Lines are
Parallel
Simply stated, the postulates and theorems
yesterday have TRUE converses
Converse of Alt. Int.  Thrm.
If the alt int ' s are ,
then the lines are ||
Conv AIA Thrm
Converse of Corres  Post.
If the corres ' s are ,
then the lines are ||
Conv CAP
If 1  3
then n || p
m
1
If 2  3
then n || p
n
2
3
5
p
Converse of Consecutiv e
Interior  Thrm
If the consecutiv e int ' s
are supp, then the lines are ||
Conv CIA Thrm
Converse of Alternate
Exterior  Thrm
If the Alternate exterior
' s are congruent, then
the lines are ||
Conv AEA Thrm
If 3, 4 sup,
then n || p
If 1  5,
then n || p
m
1
4
n
2
3
5
p
I show the angles, you say what theorem
makes the lines parallel.
m
1,5 congruent
3,6 congruent
3,5
supplementary
1 2
3 4
5 6
7 8
n
p
1,8 congruent
4, 8 congruent
3, 5 congruent
5, 8 congruent
Which lines are parallel?
A
35
40
B
38
35
D
C
You try it! Are l and m parallel? How?
l
l
m
30o
40o
m
110o
44o
60o
66o
Which lines are parallel? How?
l
m
p
n
40o
80o
discuss
50o
80o
You try it! What does x have to be for l and m to be parallel?
l
m
(x + 40)o
l
70o
xo
m
(3x)o
m
Proving Conv AEA Thrm
(Can' t use Conv AEA, use
conv CAP)
Given : 1  8
Prove : n || p
1 2
3 4
5 6
7 8
n
p
Given : 2 and 8 are
supplement ary
Prove : n || p
3.5 – Using Properties of
Parallel Lines
Copy an angle.
1) Draw a ray
2) Use original vertex, make radius.
3) Transfer radius to the ray you drew, and draw an arc.
4) Set radius from D and E, and transfer it to the new lines,
setting the point on F and draw an intersection on the arc,
then connect the dots.
Given a line and a point, construct a line parallel to
the given line through the given point.
1) Pick any point on the line,
draw a line from there through
the given point.
2) Using the angle formed by the
given line and the drawn line,
make a congruent angle using
the given point as the vertex.
3.6 – Parallel Lines in the
Coordinate Plane
SLOPE FORMULA!! MEMORIZE!!
SLOPE = m =
y2 – y1
x2 – x1
Find points (1, 0) (4, -1)
and label
x1 y1 x2 y2
Plug into
formula
y
Reduce
Fraction
x SLOPE = m =
SLOPE FORMULA!! MEMORIZE!!
SLOPE = m =
y2 – y1
Find points (-2, -1) (2, 5)
and label
x1 y1 x2 y2
x2 – x1
Plug into
formula
Reduce
Fraction
y
x
SLOPE = m =
Postulate: Slopes of Parallel Lines
In a coordinate plane, two nonvertical lines are parallel IFF they
have the same slope. Any two vertical lines are parallel.
Basically  Same slope means parallel.
Find the slope between each set of points. See which ones
match up to be parallel.
(4, 3)
(2, 0)
(2, 3)
(-5, 2)
(-1, 3)
(1, 2)
(-2, -1)
(-1, 3)
(-2, -1)
(-1, -2)
(-3, 0)
(-8, -4)
y  mx  b
y  y1  m( x  x1 )
Ax  By  C
Slope-intercept form
Point-slope form
Standard form
Write the equation of the line given a point and a slope in
SLOPE-INTERCEPT FORM
1
(4,2) m 
2
x1 y1
Point  Slope Form  y  y1  m( x  x1 )
Slope - Intercept Form  y  mx  b
Write the equation of the line parallel to y  4x  2
and going through (8,3) in slope - intercept form
4
Write the equation of the line parallel y  x - 5
5
with a y - intercept of 2 in slope - intercept form
y  y1  m( x  x1 )
Write the equation of the
1
line parallel to y  x  2
3
and going through (-3,1)
in slope - intercept form
y  mx  b
Write the equation of the line
parallel to
y  2 x - 3 with y - intercept - 5
in slope - intercept form
y
y
x
x
y  y1  m( x  x1 )
Write the equation of the
4
line parallel to y  x  5
3
and going through (6,-1)
in slope - intercept form
y  mx  b
Write the equation of the line
parallel to
2
y   x  1 and going through
5
(10,-4) in slope - intercept form
Grade of a road, it’s
rise over run, then
changed into a
percent.
2% grade
2
100
3.7 – Perpendicular Lines in
the Coordinate Plane
Solve for y, change it to ‘y =‘
2
y  7  (2 x  6)
3
y  2  3( x  1)
Distribute
Get y by itself
Notice how by solving for y, we put it in slope
intercept form, now we can find the slope.
Parallel and Perpendicular Lines
Parallel Lines have the ___________ slope
Blue
Green
What do you notice about
the lines and the slope?
Slopes are opposite
reciprocals, or slopes
multiply to equal -1
Also, vertical and
horizontal lines are
perpendicular
Parallel Lines, SAME SLOPE
Perpendicular Lines, opposite reciprocal.
State the slopes of the line parallel and
perpendicular to the slopes on the left.
Slope
Parallel
2
2

5
7
4
Perpendicular
Find the slope between each set of points. See which ones
match up to be perpendicular.
(4, 3)
(2, 0)
(2, 3)
(-5, 3)
(-1, 3)
(3, 2)
(-2, -1)
(-1, 3)
(-2, -1)
(1, -2)
(-3, 0)
(0, 4)
Find the slope of each line, then pair up the perpendicular
and parallel lines.
2
y  x3
3
2x  3y  6
3x  2 y  0
3 x  2 y  12
Write the equation of the line PERPENDICU LAR to y  4x  2
and going through (8,3) in slope - intercept form
2
Write the equation of the line PERPENDICU LAR to y 
x
3
and going through (-2,1) in slope - intercept form
y  mx  b
Write the equation of the line
y  y1  m( x  x1 )
Write the equation of the
line PERPENDICU LAR to
1
y  x  2 and going through
3
(-3,1) in slope - intercept form
PERPENDICU LAR to
y  2 x - 3 and going through
(-4,1) in slope - intercept form
y
y
x
x
y  y1  m( x  x1 )
Write the equation of the
line PERPENDICU LAR to
4
y  x  5and going through
3
(8,-1) in slope - intercept form
y  mx  b
Write the equation of the line
PERPENDICU LAR to
2
y   x  1 and going through
5
(10,-4) in slope - intercept form