Download Parallel lines

Parallel and Perpendicular
Lines
Today’s Learning Goals
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We will investigate the properties of angle
measurements when you have perpendicular or
parallel lines.
Definitions
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Consider the following two lines:
These two lines are called parallel lines. Parallel
lines are lines that never meet.
The arrows at either end signify that lines go on
forever in both directions.
How many lines are parallel to the two lines above?
Nice…an infinite amount of parallel lines.
Notation for Lines
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When given more than one line (like below), we need
some way of distinguishing them apart.
A
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B
D
C
Similar to what we did with angles, sometimes we put
points on the lines and use the letters to describe lines.
How could we name the top line? Yes…AB
How could we name the bottom line? Good…CD
Notice how there are arrows in both directions to
denote a line.
Line Segments vs. Lines
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Line segments are only parts of a line.
A
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B
D
C
What are the two line segments shown above?
Yes…AB and CD.
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What is the difference between how we denoted a
line from a line segment?
Great…Lines have a double arrow over the letters
while a line segment does not have the arrows.
Rays vs. Line Segments and
Lines
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Rays are lines that only go in one direction from a
starting point.
B
A
D
C
 Using the starting point to name the rays, what are
some rays shown above?
Excellent…A, B, C, D, A, B, C, and D.
 What is the difference between how we denoted a
ray from a line segment?
Great…Line segments needed two letters while rays only
need the starting point and rays have single arrows.
Notation for Lines
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Another way to denote a line is to give it one letter to
the side of the line.
p
q
t
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Above, we could say we have parallel lines p and q.
The symbol “||” means parallel. So, p || q.
A transversal is a line that intersects two or more
parallel lines. Line t is a transversal above.
Parallel Lines
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When a transversal intersects two or more parallel lines,
we create angles.
Corresponding angles are angles that are on the same
side of the transversal and same side of the parallel lines.
Which angles are corresponding in the figure below?
1&3
8 1
p
7 2
6 3
5 4
t
q
2&4
5&7
6&8
Parallel Lines
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Imagine dropping one of the parallel lines down and
plopping them one on top of the other.
When we plop line p onto line q, 1 plops on top of
which angle?
Yes…1 plops onto 3.
If 1 plops perfectly on top of 3, what does that
mean about the angle measurements?
Nice…m1 = m3.
8 1
7 2
6 3
5 4
t
p
q
Parallel Lines
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We just saw that when we plop line p on top of line
q, m1 = m3. What other angles will plop on top
of each other?
Great… 1 plops onto 3
2 plops onto 4
8 plops onto 6
8 1
7 2
6 3
5 4
t
7 plops onto 5
p
q
Parallel Lines
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If corresponding angles plop on top of each other
perfectly, what does that mean about corresponding
angles when we have parallel lines?
Excellent…corresponding angles have the same
measurement (are congruent).
8 1
7 2
6 3
5 4
t
p
q
Definitions
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Now, consider these:
These lines are called perpendicular lines.
Perpendicular lines are lines that form a 90 angle
between them.
A square between the lines is used to denote the 90.
Perpendicular Lines
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The symbol  means “perpendicular”.
t
L
M
Q
s
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P
If the lines above are labeled as shown, how would
we say which lines are perpendicular?
Nice…s  t
and
LM  PQ
Construction
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Parallel and perpendicular line segments are often
used in construction, art, and other real-world
applications.
Why do you think I said parallel and perpendicular
line segments are often used instead of parallel and
perpendicular lines?
Nice…lines go on forever in both directions and
that is impossible in most real-world applications.
Equal Angles with Parallel
Lines
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When we have parallel lines with a transversal cutting
through them, there are some interesting properties
of equality for the angles that are made.
Consider the following. Which angles did we say are
equal if p || q?
Nice…all corresponding
angles are equal. So,
8 1
p 1 = 3,
7 2
6 3
5 4
t
q
2 = 4,
8 = 6,
7 = 5.
Equal Angles with Parallel
Lines
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What angles are also equal because they are vertical
angles?
Nice…
1 = 7,
2 = 8,
8 1
7 2
6 3
5 4
t
p
q
3 = 5, and
4 = 6.
Equal Angles with Parallel
Lines
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We said that 1 = 3 and 1 = 7. What does that mean
about angles 3 and 7?
Great… 3 = 7.
3 and 7 are called alternate interior angles
because they are on different sides of the transversal
and both between (inside) the parallel lines.
8 1
7 2
6 3
5 4
t
p
q
Equal Angles with Parallel
Lines
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There is another pair of alternate interior angles
different than angles 3 and 7. What are they?
Yes… 2 and 6 are alternate interior angles.
Is 2 = 6? Explain how you know.
Excellent… 2 = 6 because 2 = 4 and
4 = 6.
8 1
7 2
6 3
5 4
t
p
q
Partner Work
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You have 15 minutes to work on the following
problems with your partner.
For those that finish early
1. Look around the room. Explain where you see each
of these in the classroom.
a) parallel line segments
b) perpendicular line segments
c) transversals
d) corresponding angles
2. Is there such a thing as parallel lines or
perpendicular lines in the real world? Explain.
For those that finish early
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Consider the following diagram where p || q || r.
Determine the measures of all 19 missing angles.
1 2
110 5
3 4
60
50
7 6
9
8
10
11
15 14
16 17
13 12
18 19
p
q
r
Big Idea from Today’s Lesson
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Corresponding Angles are congruent.
Alternate interior angles are congruent.
Alternate exterior angles are congruent.
Homework
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Pgs. 220 – 221 (1 – 20)