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
D
A
C
B
G
E
H
FArrows on line mean they are parallel
Two planes that do not Intersect are called PARALLEL planes.
D
A
C
B
G
E
H
F
A line and a plane are parallel if they do not intersect.
Line segments and rays can be parallel too!
TY
RO
TY
R O
As long as the lines going through them are also parallel.
TY
RO
TY
RO
D
A
C
B
G
E
H
F
Let’s name some parallel planes, lines, and some skew lines.
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
1 2
5 6
A ______________ is a line that INTERSECTS two or more COPLANAR lines at different points.
Two angles are ________________ ANGLES if they occupy ________________ positions.Two angles are ________________ __________________ if they LIE _________ the two lines on __________ 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.
3 4
7 8
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
m
n
1
2
m tochange to
s' def use toneed Note,
21
,||
then
nmIf
m
n
1
4
32
5
Alternate Interior Angles Theorem (AIA Thrm)
If two lines cut by transversal are ||, then the alternate interior angles are congruent
Consecutive Interior Angles Theorem (CIA Thrm)
If two lines cut by transversal are ||, then the consecutive interior angles are supplementary
Alternate Exterior Angles Theorem (AEA Thrm)
If two lines cut by transversal are ||, then the alternate exterior angles are congruent
6
m
n
1
2tnthen
tmnmIf
,
,||
Perpendicular Transversal
If a transversal is perpendicular to one of two
|| lines, then it is perpendicular to the other.
t
1 2
5 6
3 4
7 800
Find the measure of angles 1 – 7 given the information below.
2012 x
y4414 x
Find x, y
1034
52
3243
61
xm
xm
ym
ym
Find x, y, and the measure of all angles
1234
wm
zm
ym
xm
xm
5
204
3
2452
81Find w, x, y, z, and the measure of all angles
1
2 3
4
5
3.4 – Proving Lines are Parallel
Simply stated, the postulates and theorems yesterday have TRUE converses
CAP Conv
|| are lines then the
, are ' corres theIf
Post. Corres of Converse
s
ThrmAIA Conv
|| are lines then the
, are 'int alt theIf
Thrm. Int. Alt. of Converse
s
m
n
p
12
3
p||nthen
31 If p||nthen
32 If
5
ThrmCIA Conv
|| are lines then thesupp, are
'int econsecutiv theIf
Thrm Interior
eConsecutiv of Converse
s
m
n
p
12
3
p||nthen
sup, 4,3 If
4
ThrmAEA Conv
|| are lines the
then congruent, are '
exterior Alternate theIf
Thrm Exterior
Alternate of Converse
s
p||nthen
,51 If
5
m
n
p
1 2
5
3
I show the angles, you say what theorem makes the lines parallel.
4
687
1,5 congruent
3,6 congruent
3,5 supplementary
1,8 congruent
4, 8 congruent
3, 5 congruent
5, 8 congruent
A
Which lines are parallel?
B
C
D
35
35
40
38
l m
You try it! Are l and m parallel? How?
30o
40o
110o
l m
60o
44o
66o
l mn p
40o
80o 50o
80o
discuss
Which lines are parallel? How?
l
m
You try it! What does x have to be for l and m to be parallel?
70o
xo
l
m
(x + 40)o
(3x)o
m
n
p
1 2
5
3 4
687
pn || :Prove
arysupplement
are 8 and 2 :Given
pn || :Prove
81 :Given
CAP) conv
use AEA, Conv uset (Can'
ThrmAEA Conv Proving
3.5 – Using Properties of Parallel Lines
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.
Copy an angle.
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
x
y
SLOPE = m =
SLOPE = m =y2 – y1
x2 – x1
SLOPE FORMULA!! MEMORIZE!!
Find points and label
Plug into formula
Reduce Fraction
(1, 0) (4, -1)
x1 y1 x2 y2
x
y
SLOPE = m =y2 – y1
x2 – x1
SLOPE FORMULA!! MEMORIZE!!
Find points and label
Plug into formula
Reduce Fraction
(-2, -1) (2, 5)
x1 y1 x2 y2
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, -1)
(2, 0)
(-1, 3)
(2, 3)
(-2, -1)
(-5, 2)
(-1, -2)
(-1, 3)
(-3, 0)
(1, 2)
(-8, -4)
Slope-intercept form
Point-slope form
bmxy )( 11 xxmyy
Write the equation of the line given a point and a slope in SLOPE-INTERCEPT FORM
2
1m (4,2)
11 yx
Standard form CByAx
formintercept -slopein (8,3) through going and
24xy toparallel line theofequation theWrite
formintercept -slopein 2 ofintercept -y awith
5-x5
4y parallel line theofequation theWrite
bmxy
xxmyy
FormIntercept -Slope
)(Form SlopePoint 11
formintercept -slopein
(-3,1) through going and
2x3
1y toparallel line
theofequation theWrite
)( 11 xxmyy
formintercept -slopein
5-intercept -y with 3-x2y
toparallel
line theofequation theWrite
bmxy
x
y
x
y
formintercept -slopein
(6,-1) through going and
5x3
4y toparallel line
theofequation theWrite
formintercept -slopein (10,-4)
throughgoing and 1x5
2y
toparallel
line theofequation theWrite
)( 11 xxmyy bmxy
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 =‘
)1(32 xy
Distribute
Get y by itself
)62(3
27 xy
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 Perpendicular
2
5
2
4
7
Find the slope between each set of points. See which ones match up to be perpendicular.
(4, 3)
(-2, -1)
(2, 0)
(-1, 3)
(2, 3)
(-2, -1)
(-5, 3)
(1, -2)
(-1, 3)
(-3, 0)
(3, 2)
(0, 4)
Find the slope of each line, then pair up the perpendicular and parallel lines.
33
2 xy 632 yx 023 yx 1223 yx
formintercept -slopein (8,3) through going and
24xy toLARPERPENDICU line theofequation theWrite
formintercept -slopein (-2,1) through going and
x3
2y toLARPERPENDICU line theofequation theWrite
formintercept -slopein (-3,1)
throughgoing and 2x3
1y
toLARPERPENDICU line
theofequation theWrite
formintercept -slopein (-4,1)
throughgoing and 3-x2y
toLARPERPENDICU
line theofequation theWrite
x
y
x
y
)( 11 xxmyy bmxy
formintercept -slopein (8,-1)
through going and5x3
4y
toLARPERPENDICU line
theofequation theWrite
formintercept -slopein (10,-4)
throughgoing and 1x5
2y
toLARPERPENDICU
line theofequation theWrite
)( 11 xxmyy bmxy