Download 3.3 Prove Lines are Parallel

3.3 Prove Lines are Parallel
Objectives:
1. To use angle pair
relationships to
prove that two lines
are parallel
2. To construct parallel
lines with a compass
and straightedge
Assignment:
• P. 165-169: 2-8 even,
9, 19-22, 27, 28, 31,
38, 39, 40, 42, 44, 46,
47
• Challenge Problems
Warm-Up
What is the converse of the Corresponding
Angles Postulate?
If two parallel lines are cut by a transversal,
then pairs of corresponding angles are
congruent.
Is this converse necessarily true?
Objective 1
You will be able to use angle pair
relationships to prove that two lines
are parallel
Investigation 1
Use the following
Investigation to help
you test the converse
of the Corresponding
Angles Postulate.
You will need patty
papers.
Investigation 1
1. Draw 2 intersecting
lines on a patty
paper and copy
these lines onto a
second patty paper.
Slide the top copy
so that the two
transversals stay
lined up.
Investigation 1
2. Trace the lines and the
angles from the bottom
original onto the top
copy. When you do
this, you are
constructing sets of
congruent
corresponding, alternate
interior, and alternate
exterior angles.
Investigation 1
Do the lines appear to be parallel? You can
test this with the right angle of a patty
paper to see if the distance between the
two lines remains the same.
Let’s test this another way, but first let’s learn
how to do a basic compass and
straightedge construction.
Objective 2
You will be able to construct parallel lines with a
compass and straightedge
Copying an Angle
Draw angle A on your paper. How could you
copy that angle to another part of your
paper using only a
compass and a
straightedge?
Copying an Angle
1. Draw angle A.
Copying an Angle
2. Draw a ray with endpoint A’.
Copying an Angle
3. Put point of compass on A and draw an
arc that intersects both sides of the angle.
Label these points
B and C.
Copying an Angle
4. Put point of compass on A’ and use the
compass setting from Step 3 to draw a
similar arc on the ray.
Label point B’ where
the arc intersects
the ray.
Copying an Angle
5. Put point of compass on B and pencil on
C. Make a small arc.
Copying an Angle
6. Put point of compass on B’ and use the
compass setting from Step 5 to draw an
arc that intersects the
arc from Step 4.
Label the
new point
C’.
Copying an Angle
7. Draw ray A’C’.
Constructing Parallel Lines
Now let’s apply
the construction
for copying an
angle to create
parallel lines by
making
congruent
corresponding
angles.
Constructing Parallel Lines
1. Draw line l and
point P not on l.
Constructing Parallel Lines
2. Draw a
transversal
through point P
intersecting line l.
Constructing Parallel Lines
3. Copy the angle
formed by the
transversal and
line l at point P.
Objective 1
You will be able to use angle pair
relationships to prove that two lines
are parallel
Proving Lines Parallel
Converse of Corresponding Angles
Postulate
If two lines are cut by a transversal
so that corresponding angles are
congruent, then the lines are
parallel.
Converse of Alternate Interior
Angles Theorem
If two lines are cut by a transversal
so that alternate interior angles
are congruent, then the lines are
parallel.
Proving Lines Parallel
Converse of Alternate Exterior
Angles Theorem
If two lines are cut by a transversal
so that alternate exterior angles
are congruent, then the lines are
parallel.
Converse of Consecutive Interior
Angles Theorem
If two lines are cut by a transversal
so that consecutive interior
angles are supplementary, then
the lines are parallel.
Example 1
Can you prove that lines a and b are
parallel? Explain why or why not.
Example 2
Find the value of x that makes m||n.
Example 3
Prove the Converse of the Alternate Interior
Angles Theorem.
Given: 3  6
Prove: l m
Example 4
Given: 1 and 3 are supplementary
2  3
Prove: RA TP
Example 5
Find the values of x and y so that l||m.
l
5x+3
m
10y+2
15y+6
2x-6
o
n
Oh, My, That’s Obvious!
Transitive Property of
Parallel Lines
If two lines are parallel
to the same line, then
they are parallel to
each other.
3.3 Prove Lines are Parallel
Objectives:
1. To use angle pair
relationships to
prove that two lines
are parallel
2. To construct parallel
lines with a compass
and straightedge
Assignment:
• P. 165-169: 2-8 even,
9, 19-22, 27, 28, 31,
38, 39, 40, 42, 44, 46,
47
• Challenge Problems