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. What You'll Learn.
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Parallels
Parallel Lines and Planes
What You'll Learn
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 distance apart are ____________.
parallel lines
No two parallel lines intersect, no matter how far you extend them.
Parallel Lines and Planes
intersect
Q
R
Parallel Lines and Planes
S
P
K
L
M
J
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 _____________.
Plane PSR || plane JML
Plane JPQ || plane MLR
Plane PJM || plane QRL
Parallel Lines and Planes
Sometimes lines that do not intersect are not in the same plane.
These lines are called __________.
skew lines
2) All segments that intersect
B
C
A
D
3) All segments parallel to
4) All segments skew to
Parallel Lines and Planes
F
G
DH, CG, FG, EH
AB, GH, EF
AD, CD, GH, AH, EH
E
H
Name the parts of the figure:
1) All planes parallel to plane ABF
Plane DCG
End of Section 4.1
Parallel Lines and Transversals
What You'll Learn
You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel linesand a transversal.
A
l
m
B
is an example of a transversal. It intercepts lines l and m.
Parallel Lines and Transversals
In geometry, a line, line segment, or ray that intersects two or more lines at
different points is called a __________
transversal
2
1
4
3
5
6
8
7
Note all of the different angles formed at the points of intersection.
Parallel Lines
Nonparallel Lines
b
l
2
1
2
1
3
4
4
3
m
c
6
5
6
5
8
7
7
8
t
r
t is a transversal for l and m.
r is a transversal for b and c.
Parallel Lines and Transversals
The lines cut by a transversal may or may not be parallel.
Exterior
Interior
Exterior
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.
l
2
1
4
3
m
6
5
8
7
Parallel Lines and Transversals
When a transversal intersects two lines, _____ angles are formed.
eight
These angles are given special names.
t
Exterior angles lie outside the
two lines.
Interior angles lie between 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
congruent
2
1
4
3
6
5
7
8
2
1
4
3
6
5
8
7
Parallel Lines and Transversals
supplementary
2
1
4
3
6
5
8
7
Parallel Lines and Transversals
congruent
?
End of Lesson
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
What You'll Learn
You will learn to identify the relationships among pairs of
corresponding angles formed by two parallel lines and a
transversal.
l
2
1
4
3
m
6
5
8
7
t
Transversals and Corresponding Angles
When a transversal crosses two lines, the intersection creates a number ofangles 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 __________________.
Give three other pairs of corresponding angles that are formed:
4 and 8
3 and 7
2 and 6
Transversals and Corresponding Angles
congruent
Transversals and Corresponding Angles
Types of angle pairs formed when
a transversal cuts two parallel lines.
consecutive interior
alternate interior
alternate exterior
corresponding
s
t
c
1
3
4
2
5
6
7
8
9
12
11
d
10
14
13
15
16
Transversals and Corresponding Angles
s || t and c || d.
Name all the angles that arecongruent to 1.
Give a reason for each answer.
corresponding angles
3 1
vertical angles
6 1
alternate exterior angles
8 1
corresponding angles
9 1
alternate exterior angles
14 1
corresponding angles
11 9 1
corresponding angles
16 14 1
End of Lesson
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
What You'll Learn
You will learn to identify conditions that produce parallel lines.
Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24).
hypothesis
Within those statements, we identified the “__________” and 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):
IF ___________________________________,
THEN ________________________________________.
two parallel lines are cut by a transversal
two parallel lines are cut by a transversal
each pair of corresponding angles is congruent
each pair of corresponding angles is congruent
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 ____________________________________.
1
a
2
b
Proving Lines Parallel
parallel
If 1 2,
then _____
a || b
a
1
2
b
Proving Lines Parallel
parallel
If 1 2,
then _____
a || b
1
a
b
2
Proving Lines Parallel
parallel
If 1 2,
then _____
a || b
a
1
2
b
Proving Lines Parallel
parallel
If 1 + 2 = 180,
then _____
a || b
t
a
b
Proving Lines Parallel
parallel
If a t andb t,
then _____
a || b
Proving Lines Parallel
We now have five ways to prove that two lines are parallel.
Y
G
90°
R
D
90°
A
Proving Lines Parallel
Identify any parallel segments. Explain your reasoning.
Find the value for x so BE || TS.
E
B
(2x + 10)°
(6x - 26)°
(5x + 2)°
T
S
ES is a transversal for BE and TS.
If mBES + mEST = 180, then BE || TS by Theorem 4 – 7.
Thus, if x = 24, then BE || TS.
Proving Lines Parallel
mBES + mEST = 180
(2x + 10) + (5x + 2) = 180
consecutive interior
BES and EST are _________________ angles.
7x + 12 = 180
7x = 168
x = 24
End of Lesson
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
What You'll Learn
You will learn to find the slopes of lines and use slope to
identify parallel and perpendicular lines.
There has got to be some “measurable” way to get this aircraftto clear such obstacles.
Discuss how you might radio a pilot and tell him or her how toadjust the slope of their flight path in order to clear the mountain.
If the pilot doesn’t change something, he / she will not make it home for Christmas. Would you agree?
Consider the options:
1) Keep the same slope of his / her path.
Not a good choice!
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” requiredand divide that by the “horizontal change” required.
y
10000
x
0
0
10000
y
10
10
5
x
-10
-10
-5
5
10
10
-5
-10
-10
Slope
The steepness of a line is called the _____.
slope
Slope is defined as the ratio of the ____, or vertical change, to the ___, orhorizontal change, as you move from one point on the line to another.
run
rise
y
x
Slope
The slope m of the non-vertical line passing through the pointsand is
Slope
y
(3, 6)
(1, 1)
?
6 & 7
x
Slope
The slopem of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.
rise = 6 - 1 = 5 units
run = 3 - 1 = 2 units
Slope
the same slope
?
8 & 9
Slope
the product of their slope is -1
End of Lesson
Practice Problems:
1, 3, 4, 5, 6, 7, 8, 9, 10, 12,
14, 16, 17, 20, 22, 24, 26, 30, and 32 (total = 19)
y
8
8
7
6
5
4
(3, 5)
(2, 3)
3
(1, 1)
2
1
x
0
-1
-1
-1
-1
1
2
3
4
5
6
7
8
8
0
Equations of Lines
What You'll Learn
You will learn to write and graph equations of lines.
linear equation
The equation y = 2x – 1 is called a _____________ because its graph is
a straight line.
We can substitute different values for x in the graph to find corresponding
values for y.
There are many more points whose ordered
pairs are solutions of y = 2x – 1. These points also lie on the line.
1
1
y = 2(1) -1
2
3
y = 2(2) -1
3
5
y = 2(3) -1
y = 2x – 1
y
5
5
y - intercept
slope
4
(0, -1)
3
2
1
x
0
-1
-2
-3
-3
-3
-3
-2
-1
1
2
3
4
5
5
0
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
This value is called the ____________ of the line.
y = mx + b
Most linear equations can be written in the form __________.
slope – intercept form
This form is called the ___________________.
y = mx + b
Equations of Lines
y
5
5
4
(0, 3)
(1, 1)
3
2
1
x
0
-1
-2
-3
-3
-3
-3
-2
-1
1
2
3
4
5
5
0
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
1) Identify and graph the y-intercept.
2) Follow the slope a second point on the line.
3) Draw the line between the two points.
3) Write an equation of the line perpendicualr to the graph of
that passes through the point ( - 3, 8).
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
y = -4x -4
End of Lesson
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)