3.2 Proving Lines Parallel
The secret
secret impresses
impresses no
no one.
one. The
The trick
trick you
you use
use itit for
for is
is everything.
everything.
The
-Alfred Borden
Proving Lines Parallel
l
m
1
2
Postulate 3-2: Converse of Corresponding Angles Postulate:
If two lines and a transversal form corresponding angles that are
congruent, then the two lines are parallel.
l || m
Proving Lines Parallel
We know these
two lines are
parallel!!
If Alternate Interior Angles are congruent we
can assume lines are parallel too!
Proving Lines Parallel
l
m
1
4
2
Theorem 3-5: Converse of Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are
congruent, then the two lines are parallel.
If 1  2, then l || m.
Proof of Theorem 3-5 (C of AIAT)
l
3
1
m
Given : 2  1
2
Prove : l || m
Statements
Reasons
1. 2  1
1.
2.
2. Vertical 's are 
3. 2  3
3.
l || m
4.
4.

Proving Lines Parallel
We know these
two lines are
parallel!!
If Same-Side Interior Angles are supplementary,
we can assume lines are parallel too!!
Proving Lines Parallel
l
m
1
4
2
Theorem 3-6: Converse of Same-Side Interior Angles Theorem
If two lines and a transversal form same-side interior angles that are
supplementary, then the two lines are parallel.
If 2 and 4 are supplement ary, then l || m.
Proving Lines Parallel
We know these
two lines are
parallel!!
If Alternate Exterior Angles are congruent, we
can assume lines are parallel!!!
Proving Lines Parallel
a
1
b
3 2
Theorem 3-7: Converse of Alternate Exterior Angles Theorem
If two lines and a transversal intersects form alternate exterior
angles that are congruent, then the two lines are parallel.
If 1  2, then a || b.
Proof of Theorem 3-7 (C of AEAT)
a
1
4
Given : 2  1
b
Prove : a || b
Statements
2
Reasons
1.
1.
2. 1  4
2.
3. 2  4
3.
a || b
4.
4.
Proving Lines Parallel
We know these
two lines are
parallel!!
If Same-Side Exterior Angles are supplementary,
we can assume lines are parallel!!!
Proving Lines Parallel
a
1
b
3 2
Theorem 3-8: Converse of Same-Side Exterior Angles Theorem
If two lines and a transversal intersects form same-side exterior
angles that are supplementary, then the two lines are parallel.
If 1 and 3 are supplementary, then a || b.
Let’s Apply What We Have Learned, K?
Find the value of x for which l || m
l
m
40°
(2x + 6)°
You Try One!
Find the value of x for which a || b
a
(7x - 8)°
b
62°
3.3: Parallel and Perpendicular Lines
There is no spoon.
There is no spoon.
-Spoon Boy and Neo
Relating Parallel and Perpendicular Lines
a
b
c
Theorem 3-9: If two lines are parallel to the same line, then
they are parallel to each other.
a || b

Relating Parallel and Perpendicular Lines
t
m
n
Theorem 3-10: In a plane, if two lines are perpendicular to the
same line, then they are parallel to each other.
m || n

Relating Parallel and Perpendicular Lines
t
m
1
2
n
Given : m  t, n  t
Prove : m || n
Statements
Reasons
1. m  t, n  t
1.
2. 1 and 2 are
right angles
2.
3. 1  2
3.
m || n
4.
4.
Relating Parallel and Perpendicular Lines
n
l
m
Theorem 3-11: In a plane, if a line is perpendicular to one of
two parallel lines, then it is also perpendicular to the other.
nm

Relating Parallel and Perpendicular Lines
n
1
l
2
Given : n  l, and l || m
Prove : n  m
Statements
Reasons
1. n  l, and l || m
1.
2. 1 is a right 
2.
3.
3. Corresp.  ‘s post.
4. n  m
4.
m
3.2: Proving Lines Parallel
3.3: Parallel and Perpendicular Lines
HOMEWORK: p.137 #1-8, 14-22, 24-30 even, 51-52,
p. 144 #26-27,
Checkpoint Quiz 1 (p. 153) #1-9
Get busy
busy living,
living, or
or get
get busy
busy dying.
dying.
Get
-Red