Week 6 and 7

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Week 6 and 7
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3.1 Lines and Angles
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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
B
Arrows on line mean they are parallel
F
A
E
C
H
D
G
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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
E
C
H
D
G
5
Line segments and rays can be parallel too!
O
O
R
R
Y
Y
T
T
As long as the lines going through them is also
parallel.
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Lets name some parallel planes, lines, and some
skew lines.
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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.
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A TRANSVERSAL is a line that INTERSECTS two or
more COPLANAR lines at different points.
Transversal
Two angles are CORRESPONDING ANGLES if they
occupy CORRESPONDING positions.
2
1
4
3
Two angles are ALTERNATE EXTERIOR ANGLES if they
LIE OUTSIDE the two lines on OPPOSITE sides of
the TRANSVERSAL.
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6
7
8
Two angles are ALTERNATE INTERIOR ANGLES if they
LIE BETWEEN the two lines on OPPOSITE sides of
the TRANSVERSAL.
Two angles are CONSECUTIVE INTERIOR ANGLES (also
called same side interior angles) if they LIE
BETWEEN the two lines on the SAME sides of the
TRANSVERSAL.
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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.
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3.3 Parallel Lines and Transversals
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Corresponding Angles Postulate (CAP) If two
lines cut by transversal are , then the
corresponding angles are congruent
m
1
n
2
Instead of stating parallel ARROWS indicate
parallel also
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Alternate Interior Angles Theorem (AIA Thrm)
1
2
m
3
If two lines cut by transversal are , then the
alternate interior angles are congruent
4
5
n
6
Alternate Exterior Angles Theorem (AEA Thrm)
Consecutive Interior Angles Theorem (CIA Thrm)
If two lines cut by transversal are , then the
consecutive interior angles are supplementary
If two lines cut by transversal are , then the
alternate exterior angles are congruent
In groups
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Perpendicular Transversal If a transversal is
perpendicular to one of two lines, then it is
perpendicular to the other.
1
m
2
n
t
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Find the measure of angles 1 7 given the
information below.
2
1
4
3
5
6
7
800
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VERTICAL ANGLES
Find x, y
CORRESPONDING ANGLES
ALT INT ANGLES
In groups
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Find x, y, and the measure of all angles
1
3
2
4
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Find w, x, y, z, and the measure of all angles
1
4
2
3
5
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3.4 Proving Lines are Parallel
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Simply stated, the postulates and theorems
yesterday have TRUE converses
m
1
n
2
3
p
5
20
m
1
n
2
4
3
p
5
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I show the angles, you say what theorem makes the
lines parallel.
m
1,5 congruent
2
1
n
4
3
3,6 congruent
6
5
p
8
7
3,5 supplementary
1,8 congruent
4, 8 congruent
5, 8 congruent
3, 5 congruent
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Which lines are parallel?
A
C
35
40
38
35
B
D
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You try it! Are l and m parallel? How?
30o
110o
44o
40o
66o
60o
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Which lines are parallel? How?
l
m
p
n
40o
50o
80o
80o
discuss
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You try it! What does x have to be for l and m
to be parallel?
(x 40)o
70o
xo
(3x)o
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m
2
1
n
4
3
6
5
p
8
7
stuwrok
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3.5 Using Properties of Parallel Lines
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Copy an angle.
E
D
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.
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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.
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3.6 Parallel Lines in the Coordinate Plane
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SLOPE FORMULA!! MEMORIZE!!
y2 y1
Find points and label Plug into formula Reduce
Fraction
(1, 0) (4, -1)
SLOPE m
x2 x1
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SLOPE FORMULA!! MEMORIZE!!
y2 y1
Find points and label Plug into formula Reduce
Fraction
(-2, -1) (2, 5)
SLOPE m
x2 x1
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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)
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Slope-intercept form
Point-slope form
Standard form
Write the equation of the line given a point and
a slope in SLOPE-INTERCEPT FORM
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Student graph them
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student
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Grade of a road, its rise over run, then changed
into a percent.
2 grade
2
100
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3.7 Perpendicular Lines in the Coordinate Plane
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Solve for y, change it to y
Distribute Get y by itself
Notice how by solving for y, we put it in slope
intercept form, now we can find the slope.
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Parallel and Perpendicular Lines
Parallel Lines have the ___________ slope
Green
Blue
What do you notice about the lines and the slope?
Lines are perpendicular
Slopes are opposite reciprocals, or slopes
multiply to equal -1 Also, vertical and
horizontal lines are perpendicular
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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
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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)
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Find the slope of each line, then pair up the
perpendicular and parallel lines.
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Student graph them
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student