Transcript parallel lines

3.1 Identify Pairs of Lines and Angles
3.2 Use Parallel Lines and Transversals
1.
2.
3.
4.
Objectives:
To differentiate between parallel,
perpendicular, and skew lines
To compare Euclidean and Non-Euclidean
geometries
To identify a transversal and various angle
pairs
To find angle pair measurements with
parallel lines cut by a transversal
Vocabulary
As a group, define each
of these without your
book. Draw a picture
for each word and
leave a bit of space
for additions and
revisions.
Parallel Lines Skew Lines
Perpendicular Transversal
Lines
Example 1
Use the diagram to answer
the following.
1. Name a pair of lines that
intersect.
2. Would JM and NR ever
intersect?
3. Would JM and LQ ever
intersect?
Parallel Lines
Two lines are parallel lines if and only if
they are coplanar and never intersect.
The red arrows
indicate that the
lines are parallel.
Parallel Lines
Two lines are parallel lines if and only if
they are coplanar and never intersect.
Skew Lines
Two lines are skew lines if and only if they
are not coplanar and never intersect.
Example 2
Think of each segment in
the figure as part of a
line. Which line or plane
in the figure appear to fit
the description?
1. Line(s) parallel to CD
and containing point A.
2. Line(s) skew to CD and
containing point A.
Example 2
3. Line(s) perpendicular to
CD and containing point
A.
4. Plane(s) parallel plane
EFG and containing
point A.
Example 3
Draw line ℓ and point P. How many lines can
you draw through point P that are parallel
to line ℓ? How many lines can you draw
through point P that are perpendicular to
line ℓ?
Perpendicular Postulate
If there is a line and a point not on the line,
then there is exactly one line through the
point perpendicular to the given line.
Parallel Postulate
If there is a line and a point not on the line,
then there is exactly one line through the
point parallel to the given line.
Also referred to as Euclid’s Fifth Postulate
Euclid’s Fifth Postulate
Some mathematicians believed
that the fifth postulate was
not a postulate at all, that it
was provable. So they
assumed it was false and
tried to find something that
contradicted a basic
geometric truth.
Example 4
If the Parallel Postulate
is false, then what
must be true?
1. Through a given point
not on a given line,
you can draw more
than one line parallel
to the given line. This
makes Hyperbolic
Geometry.
Example 4
If the Parallel Postulate
is false, then what
must be true?
1. Through a given point
not on a given line,
you can draw more
than one line parallel
to the given line. This
makes Hyperbolic
Geometry.
Example 4
If the Parallel Postulate
is false, then what
must be true?
2. Through a given
point not on a given
line, you can draw
no line parallel to the
given line. This
makes Elliptic
Geometry.
Great Circles
Great Circle: The
intersection of the
sphere and a plane that
cuts through its center.
•
•
Think of the equator or
the Prime Meridian
The lines in Euclidean
geometry are
considered great circles
in elliptic geometry.
l
Great circles divide
the sphere into two
equal halves.
Example 5
1. In Elliptic geometry, how many great
circles can be drawn through any two
points?
2. Suppose points A, B, and C are collinear
in Elliptic geometry; that is, they lie on the
same great circle. If the points appear in
that order, which point is between the
other two?
Transversal
A line is a transversal
if and only if it
intersects two or
more coplanar lines.
– When a transversal
cuts two coplanar
lines, it creates 8
angles, pairs of
which have special
names
Transversal
• 1 and 5 are
corresponding angles
• 3 and 6 are
alternate interior
angles
• 1 and 8 are
alternate exterior
angles
• 3 and 5 are
consecutive interior
angles
Example 5
Classify the pair of numbered angles.
Investigation 1
Use the following
Investigation to help
you complete some
postulates and
theorems about
parallel lines and a
transversal. You will
need notebook and
patty papers.
Investigation 1
1. Using the lines on a
piece of paper as a
guide, draw a pair of
parallel lines. Now
draw a transversal
that intersects the
parallel lines. Label
the angles with
numbers.
Investigation 1
2. Place a patty paper
over the set of angles
1, 2, 3, and 4
and copy the two
intersecting lines onto
the patty paper.
3. Slide the patty paper
down and compare
angles 1 through 4 with
angles 5 through 8.
Investigation 1
Do you notice a relationship between pairs of
corresponding, alternate interior, and
alternate exterior angles?
Let’s make a four window foldable to
summarize each of the following postulates
and theorems
Four Window Foldable
Start by folding a
blank piece of paper
in half lengthwise,
and then folding it in
half in the opposite
direction. Now fold
it in half one more
time in the same
direction.
Four Window Foldable
Now unfold the paper,
and then while
holding the paper
vertically, fold down
the top one-fourth to
meet the middle.
Do the same with
the bottom onefourth.
Four Window Foldable
To finish your foldable,
cut the two vertical
fold lines to create
four windows.
Outside: Name
Inside Flap: Illustration
Inside: Postulate or
Theorem
Four Window Foldable
Corresponding Angles
Postulate
If two parallel lines are cut by
a transversal, then pairs of
corresponding angles are
congruent.
Alternate Interior Angles
Theorem
If two parallel lines are cut by
a transversal, then pairs of
alternate interior angles
are congruent.
Four Window Foldable
Alternate Exterior Angle
Theorem
If two parallel lines are cut by
a transversal, then pairs of
alternate exterior angles
are congruent.
Consecutive Interior Angles
Theorem
If two parallel lines are cut by
a transversal, then pairs of
consecutive interior angles
are supplementary.
Example 6: SAT
In the figure, if l1 || l2
and l3 || l4, what is y
in terms of x.
l2
l3
l1
x
l4
y
y
Example 7: SAT
In the figure, if l || l,
what is the value of
x?
l
3y
2y+25
x+15
m
Example 8
Prove the Alternate Interior Angle Theorem.
Example 9
Calculate each lettered angle measure.
Example 10
Find the values of x, y, and z if k || l || m.
k
7x+9
l
7y-4
11x-1
m
2y+5