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Transcript
Helvetica is the font
used in the Subways
and in gap adds. You
can tell it is Helvetica
because the “a” looks
like a drop.
Geometry
Journal 3
Helvetica was founded in Switzerland,
there is a move about it on Netflix, you
should watch it, its called “Helvetica”. It
is a really good film.
This font is
Helvetica,
the most
used font in
the world at
this time.
Maria Maldonado Hempstead
9-4
Parallel Lines, Parallel Planes, and Skew
Lines
What are Parallel Lines?
Parallel lines are two lines on the same plane that never intersect.
What are Parallel planes?
Parallel Planes are planes that Never Intersect.
What are Skew Lines?
Slew Lines are lines that are
not coplanar and
will never intersect.
What is a Transversal?
A Transversal is any line that intersects two
other lines at any 2 points.
Angles That Get Formed When You
Have a Transversal
Corresponding Angles: <1 and <5
1
Alternate Exterior Angles:
<1 and <8 and <2 and <7
2
3
4
5
6
7 8
Alternate Interior Angles: <5 and <4
and <3 and <6
Consecutive Interior Angles: <3 and <5 and <4 and <6
Corresponding angle Postulate and
Converse
If two Parallel lines are cut by a transversal, then the
pairs of corresponding angles are congruent.
Converse
If corresponding angles are congruent, then two
parallels were cut by a transversal.
Angle one and 5 are corresponding, there fore
Since angle one has a measure of 80 degrees, so does angle 5.
1
5
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate
interior angles are congruent.
Converse
If two coplanar lines are cut by a transversal, so that a pair of alternate
interior angles are congruent, then the two lines are parallel.
Angle 3 and angle 6 are congruent, because they
are alternate interior angles.
Angle 4 and angle 5 are congruent, because
they are alternate interior angles.
3
4
5
6
Same Side Interior Angles Theorem
If Two parallel lines are cut by a transversal, then
the two pairs of alternate exterior angles are
congruent.
If two coplanar lines are cut by a transversal so that
a pair of same side interior angles are
supplementary, then the two lines are parallel.
Angle 3 and angle 5 are congruent
Angle 4 and angle 6 are
Congruent
3
4
5
6
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the two pairs of alternate
exterior angles are congruent.
Converse
If two coplanar lines are cut by a transversal so that a pair of alternate
exterior angles are congruent, then the two lines are parallel.
Angle 1 and angle 8 are Congruent
Because they are alternate exterior.
Angle 2 and angle 7 are congruent too,
Because they are alternate exterior.
1
2
7
8
Perpendicular Transversal Theorems
In a plane, if a transversal is congruent to one of
two parallel lines, then it is perpendicular to
the other line. Since Perpendicular lines create
a 90 degree angle with the line they intersect,
then
the other lines will be
have the
same slope but it will be
Opposite
and reciprocal.
Transitive Property
Transitive property: a=b and b=c so a=c
How does it apply to parallel lines?
If one line is the same as the other, then it will
be parallel.
How does it apply to perpendicular lines?
Perpendicular lines are the same line, but they
are opposite reciprocals.
Black and Yellow
• How Transitive property relates to parallel
lines:
Since these two lines are equal.
• How Transitive property relates to
perpendicular lines: Since these are
perpendicular, they have opposite, reciprocal
slopes.
How Do you Find Slope?
Slope is the measure of the steepness of a line.
Y1-Y2
X1-X2 This is the formula to find the slope of a
line.
Parallel Lines will always have the exact Same slope,
and two lines that are perpendicular will have
opposite, reciprocal slopes.
3 Examples of Slopes of Lines
Coordinates:
(7, 9) (5, 3)
m= 9-5
7-3
m= 4/4, or 1
Coordinates:
(4, -2) (-6, -3)
m= -2-3
4-6
m= 5/2
Coordinates:
(8, 4) (6, 2)
m=4-2
8-6
m= 2
Slope Intercept Form
Slope Intercept form of a line is Y=mx+b.
m= slope
b= y intercept
Ex.
Y=4/3x+7
Ex.
Y=7x+5
Ex.
Y=56x-3
Point Slope Form
Y-Y1=m(X-X1)
How do you graph that????!!!
M= slope
Y1= your Y1 coordinate, X1= your X1 coordinate.
So, put your first point at your X1, Y1 point and then
graph the slope (m).
Ex.
Ex.
Ex.
Y-2=-3/5(x+3)
Y-5=9/1(x+4) Y+4=7/8(x-3)
Point: (-3, 2)
Point: (-4, 5)
Point: (3, -4)
Slope:-3/5
Slope: 9/1
Slope:7/8
Note: To graph they x and y values in the equation, you
must reciprocalize them.