Parallels

§ 4.1 Parallel Lines and Planes

§ 4.2 Parallel Lines and Transversals

§ 4.3 Transversals and Corresponding Angles

§ 4.4 Proving Lines Parallel

§ 4.5 Slope

§ 4.6 Equations of Lines
Parallel Lines and Planes
You will learn to describe relationships among lines,
parts of lines, and planes.
In geometry, two lines in a plane that are always the same
parallel lines
distance apart are ____________.
No two parallel lines intersect, no matter how far you extend them.
Parallel Lines and Planes
Definition of
Parallel
Lines
Two lines are parallel iff they are in the same plane and
intersect
do not ________.
Parallel Lines and Planes
Planes can also be parallel.
The shelves of a bookcase are examples of parts of planes.
The shelves are the same distance apart at all points, and do not appear to
intersect.
parallel
They are _______.
parallel planes
In geometry, planes that do not intersect are called _____________.
Q
R
Plane PSR || plane JML
Plane JPQ || plane MLR
Plane PJM || plane QRL
P
J
K
S
L
M
Parallel Lines and Planes
Sometimes lines that do not intersect are not in the same plane.
skew lines
These lines are called __________.
Definition of
Skew
Lines
Two lines that are not in the same plane are skew iff
they do not intersect.
Parallel Lines and Planes
Name the parts of the figure:
1) All planes parallel to plane ABF
Plane DCG
B
2) All segments that intersect DH
AD, CD, GH, AH, EH
C
A
D
3) All segments parallel to CD
AB, GH, EF
4) All segments skew to AB
DH, CG, FG, EH
F
E
G
H
Parallel Lines and Transversals
You will learn to identify the relationships among pairs of
interior and exterior angles formed by two parallel lines
and a transversal.
Parallel Lines and Transversals
In geometry, a line, line segment, or ray that intersects two or more lines at
transversal
different points is called a __________
A
2
1
4
5
8
6
7
3
l
m
B
AB
is an example of a transversal. It intercepts lines l and m.
Note all of the different angles formed at the points of intersection.
Parallel Lines and Transversals
Definition of
Transversal
In a plane, a line is a transversal iff it intersects two or more
Lines, each at a different point.
The lines cut by a transversal may or may not be parallel.
Parallel Lines
Nonparallel Lines
l
1 2
4 3
lm
t
1 2
4 3
m
5 6
8 7
c
5 6
8 7
b || c
t
is a transversal for l and m.
b
r
r
is a transversal for b and c.
Parallel Lines and Transversals
Two lines divide the plane into three regions.
The region between the lines is referred to as the interior.
The two regions not between the lines is referred to as the exterior.
Exterior
Interior
Exterior
Parallel Lines and Transversals
eight angles are formed.
When a transversal intersects two lines, _____
These angles are given special names.
l
1 2
4 3
m
5 6
8 7
t
Interior angles lie between the
two lines.
Exterior angles lie outside the
two lines.
Alternate Interior angles are on the
opposite sides of the transversal.
Alternate Exterior angles are
on the opposite sides of the
transversal.
Consectutive Interior angles are on
the same side of the transversal.
Parallel Lines and Transversals
Theorem 4-1 If two parallel lines are cut by a transversal, then each pair of
congruent
Alternate
Alternate interior angles is _________.
Interior
Angles
1 2
4 3
5 6
8 7
 4  6
3  5
Parallel Lines and Transversals
Theorem 4-2 If two parallel lines are cut by a transversal, then each pair of
supplementary
Consecutive consecutive interior angles is _____________.
Interior
Angles
1 2
4 3
5 6
8 7
4  5  180
3  6  180
Parallel Lines and Transversals
Theorem 4-3 If two parallel lines are cut by a transversal, then each pair of
congruent
Alternate
alternate exterior angles is _________.
Exterior
Angles
1 2
4 3
5 6
8 7
1  7
 2  8
Practice Problems:
1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36,
38, 40, 42, 44, and 46 (total = 23)
Transversals and Corresponding Angles
You will learn to identify the relationships among pairs of
corresponding angles formed by two parallel lines and a
transversal.
Transversals and Corresponding Angles
When a transversal crosses two lines, the intersection creates a number of
angles that are related to each other.
Note 1 and 5 below. Although one is an exterior angle and the other is an
interior angle, both lie on the same side of the transversal.
corresponding angles
Angle 1 and 5 are called __________________.
l
1 2
4 3
m
5 6
8 7
t
Give three other pairs of corresponding angles that are formed:
4 and 8
3 and 7
2 and 6
Transversals and Corresponding Angles
Postulate 4-1 If two parallel lines are cut by a transversal, then each pair of
congruent
Corresponding corresponding angles is _________.
Angles
Transversals and Corresponding Angles
Types of angle pairs formed when
a transversal cuts two parallel lines.
Concept
Summary
Congruent
Supplementary
alternate interior
consecutive interior
alternate exterior
corresponding
Transversals and Corresponding Angles
s
s || t and c || d.
Name all the angles that are
congruent to 1.
Give a reason for each answer.
1 2
5 6
9
10
13 14
3  1
corresponding angles
6  1
vertical angles
8  1
alternate exterior angles
9  1
corresponding angles
14  1
alternate exterior angles
11  9  1
corresponding angles
16  14  1
corresponding angles
t
3
7
11 12
15 16
c
4
8
d
Practice Problems:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20,
22, 24, 26, 28, 30, 32, 34, 36, and 38 (total = 19)
Proving Lines Parallel
You will learn to identify conditions that produce parallel lines.
Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24).
hypothesis and the
Within those statements, we identified the “__________”
conclusion
“_________”.
I said then that in mathematics, we only use the term
“if and only if”
if the converse of the statement is true.
Proving Lines Parallel
Postulate 4 – 1 (pg. 156):
two parallel lines are cut by a transversal
IF ___________________________________,
each pair of corresponding angles is congruent
THEN ________________________________________.
The postulates used in §4 - 4 are the converse of postulates that you already
know. COOL, HUH?
§4 – 4, Postulate 4 – 2 (pg. 162):
IF ________________________________________,
THEN ____________________________________.
Proving Lines Parallel
In a plane, if two lines are cut by a transversal so that a pair
of corresponding angles is congruent, then the lines are
parallel
_______.
Postulate 4-2
1
2
If 1 2,
a
b
a || b
then _____
Proving Lines Parallel
In a plane, if two lines are cut by a transversal so that a pair
of alternate interior angles is congruent, then the two lines
parallel
are _______.
Theorem 4-5
If 1 2,
a
2
1
b
a || b
then _____
Proving Lines Parallel
In a plane, if two lines are cut by a transversal so that a pair
of alternate exterior angles is congruent, then the two lines
parallel
are _______.
Theorem 4-6
1
2
If 1 2,
a
b
a || b
then _____
Proving Lines Parallel
In a plane, if two lines are cut by a transversal so that a pair
of consecutive interior angles is supplementary, then the two
parallel
lines are _______.
Theorem 4-7
If 1 + 2 = 180,
1
2
a
b
a || b
then _____
Proving Lines Parallel
In a plane, if two lines are cut by a transversal so that a pair
of consecutive interior angles is supplementary, then the two
parallel
lines are _______.
Theorem 4-8
If a  t and b  t,
t
a
b
a || b
then _____
Proving Lines Parallel
We now have five ways to prove that two lines are parallel.
Show that a pair of corresponding angles is congruent.
Show that a pair of alternate interior angles is congruent.
Concept
Summary Show that a pair of alternate exterior angles is congruent.
Show that a pair of consecutive interior angles is
supplementary.
Show that two lines in a plane are perpendicular to a
third line.
Proving Lines Parallel
Identify any parallel segments. Explain your reasoning.
GY and RD are both perpendicular toGA
therefore, GY
RD by T heorem4 - 8.
G
R
A
Y
90°
90°
D
Proving Lines Parallel
B
Find the value for x so BE || TS.
T
(6x - 26)°
(2x + 10)°
(5x + 2)°
ES is
a transversal
for BE= and
mBES
+ mEST
180TS.
(2x + and
10) EST
+ (5x are
+ 2)_________________
= 180
consecutive
interior angles.
BES
7x + 12 = 180
If mBES + mEST = 180, then BE || TS by Theorem 4 – 7.
7x = 168
x = 24
Thus, if x = 24, then BE || TS.
S
E
Practice Problems:
1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 21, 25, and 26 (total = 19)
Slope
You will learn to find the slopes of lines and use slope to
identify parallel and perpendicular lines.
There
has the
got
to bechange
some “measurable”
to will
get not
this make
aircraft
If
the pilot
doesn’t
something, heway
/ she
it
Consider
options:
to
clearfor
such
obstacles.
home
Christmas.
Would you agree?
1) Keep the same slope of his / her path.
Discuss
you
might radio a pilot and tell him or her how to
Not ahow
good
choice!
adjust the slope of their flight path in order to clear the mountain.
2) Go straight up.
Not possible! This is an airplane, not a helicopter.
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” required
and divide that by the “horizontal change” required.
vertical change
y
Slope 

horizontal change x
x
y
vertical change
y 4, 000 ft
4 2
Slope 


 
horizontal change x 10, 000 ft 10 5
y
10000
0
0
10000
x
Slope
slope
The steepness of a line is called the _____.
Slope is defined as the ratio of the ____,
run or
rise or vertical change, to the ___,
horizontal change, as you move from one point on the line to another.
y
1
0
5
-10
-5
5
-5
-10
1
0
x
Slope
The slope m of the non-vertical line passing through the points ( x1 , y1 )
and ( x2 , y2 ) is
y
( x2 , y2 )
y2  y1
( x1 , y1 )
x2  x1
x
m 
rise change in y y2  y1


run change in x x2  x1
Slope
The slope “m” of a line containing two points with coordinates
(x1, y1), and (x2, y2), is given by the formula
Definition
of
Slope
differenceof they - coordinates
slope 
differenceof thecorresponding x - coordinates
y2  y1
m 
,
x 2  x1
where x2  x1
Slope
The slope m of a non-vertical line is the number of units the line rises or falls
for each unit of horizontal change from left to right.
y
rise (y )
m 
run (x)
(3, 6)
6 1
m 
3 1
5
m 
2
rise = 6 - 1
= 5 units
(1, 1)
x
run = 3 - 1
= 2 units
6 & 7
Slope
Two distinct nonvertical lines are parallel iff they have
the same slope
_____________.
Postulate
4–3
y1  m1 x  b1
y2  m2 x  b2
L1 L2 iff m1  m2
Slope
Two nonvertical lines are perpendicular iff
the product of their slope is -1
___________________________.
Postulate
4–4
y1  m1 x  b1
L1  L2 iff
y2  m2 x  b2
m1 m2   1
8 & 9
Practice Problems:
1, 3, 4, 5, 6, 7, 8, 9, 10, 12,
14, 16, 17, 20, 22, 24, 26, 30, and 32 (total = 19)
Equations of Lines
You will learn to write and graph equations of lines.
linear equation because its graph is
The equation y = 2x – 1 is called a _____________
a straight line.
We can substitute different values for x in the graph to find corresponding
values for y.
y
8
7
6
x
y = 2x -1
y
There are many more points whose ordered
1
pairs1are solutions
y = 2(1) of
-1 y = 2x – 1.
These points also lie on the line.
2
3
y = 2(2) -1
3
y = 2(3) -1
5
5
(3, 5)
4
3
(2, 3)
2
1
(1, 1)
x
0
-1
-1
0
1
2
3
4
5
6
7
8
Equations of Lines
Look at the graph of y = 2x – 1 .
-1
The y – value of the point where the line crosses the y-axis is ___.
y - intercept of the line.
This value is called the ____________
y = mx + b
Most linear equations can be written in the form __________.
– intercept form
This form is called the slope
___________________.
y = mx + b
slope
y = 2x – 1
y
5
4
3
y - intercept
2
1
x
0
-1
(0, -1)
-2
-3
-3
-2
-1
0
1
2
3
4
5
Equations of Lines
Slope –
Intercept
Form
An equation of the line having slope m and y-intercept b is
y = mx + b
Equations of Lines
1) Rewrite the equation in slope – intercept form by solving for y.
2x – 3 y = 18
2) Graph 2x + y = 3 using the slope and y – intercept.
y = –2x + 3
y
5
4
1) Identify and graph the y-intercept.
3
2) Follow the slope a second point on
the line.
3) Draw the line between the two
points.
(0, 3)
2
1
(1, 1)
x
0
-1
-2
-3
-3
-2
-1
0
1
2
3
4
5
Equations of Lines
1) Write an equation of the line parallel to the graph of y = 2x – 5 that
passes through the point (3, 7).
y = 2x + 1
2) Write an equation of the line parallel to the graph of 3x + y = 6 that
passes through the point (1, 4).
y = -3x + 7
3) Write an equation of the line perpendicualr to the graph of
that passes through the point ( - 3, 8).
y = -4x -4
1
y  x5
4
Practice Problems:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 20, 22,
24, 26, 28, 30, 32, 34, 40, and 42 (total = 24)