3.1 Lines and Angles
3.3 Parallel Lines and Transversals
3.4 Proving Lines Parallel
3.5 Using Properties of Parallel Lines
Objectives:
 Be able to identify relationships between lines.
 Be able to identify angles formed by transversals.
 Be able to prove and use results about parallel lines
and transversals.
 Be able to use properties of parallel lines.
Definitions
• Parallel lines – Two lines are parallel
lines if they are coplanar and do not
intersect.
• Skew lines—Lines that do not intersect
and are not coplanar.
• Parallel planes—two planes that do not
intersect.
Identifying relationships in space
Example
1) Think of each segment in
the diagram. Which appear to
fit the description?
a.
Parallel to AB and contains
D CD
b. Perpendicular to AB and
contains D AD
c.
Skew to AB and contains D
DG, DH , DE
a.
Name the plane(s) that
contains D and appear to
be parallel to plane ABE
DCH
B
C
D
A
F
E
G
H
Postulate 13: 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.
P
l
Parallel ConstructionCopying an Angle
Use the following steps to construct an angle
that is congruent to a given angle A.
1. Using a straight edge, draw an angle A.
2. Using a straight edge, draw a line below
angle A. Label a point on the line D.
3. Draw an arc with center A. Label B and C.
With the same radius, draw an arc with
center D. Label E.
4. Draw DF.
EDF  BAC
Parallel ConstructionParallel Lines
Use the following steps to construct a line that
passes through a given point P and is parallel to
a given line m.
1. Using a straight edge, draw line m.
2. Draw points Q and R on line m.
3. Draw PQ.
4. Draw an arc with a compass point at Q so
that it crosses QP and QR.
5. Copy angle PQR on QP. Be sure the two
angles are corresponding. Label the new
angle TPS.
PS QR
6. Draw PS.
Exterior Angles
Interior Angles
1, 2, 7, 8
3, 4, 5, 6
Consecutive
Interior Angles or
Same Side Interior
3 and 5
Alternate
Exterior Angles
1 and 8
Alternate
Interior Angles
Corresponding
Angles
1
4 and 6
2 and 7
3 and 6
4 and 5
1 and 5, 2 and 6
3 and 7, 4 and 8
3
6
5
7
8
2
4
Corresponding Angles
Postulate 15
If 2  lines are cut by
a transversal, then the
pairs of corresponding
s are .
l
1
2
m
1  2
If
, then corr 's  .
Postulate 16
Corresponding Angle
Converse
If corr 's  , then
.
Alternate Interior Angles
Theorem 3.4
If 2  lines are cut by
a transversal, then
the pairs of alternate
interior s are .
l
2
1
m
1  2
If
, then alt int 's  .
Theorem 3.8
Alternate Interior Angle
Converse
If alt ext 's  , then
.
Consecutive Interior Angles
Theorem 3.5
If 2  lines are cut by a
transversal, then the pairs
of consecutive interior s
are supplementary.
l
12
m
m1  m2  180
If
, then con int 's supp.
Theorem 3.9
Consecutive Interior
Angle Converse
If , con int 's supp, then
.
Alternate Exterior Angles
Theorem 3.6
If 2  lines are cut by
a transversal, then
the pairs of alternate
interior s are .
1
l
m
2
1  2
If
, then alt ext 's  .
Theorem 3.10
Alternate Exterior Angle
Converse
If alt ext 's  , then
.
Parallel Lines Theorem
• Theorem 3.11: If two lines are parallel
to the same line, then they are parallel
to each other.
p
q
r
If p║q and q║r,
then p║r.
ASSIGNMENT
• Read 129-131, 143-145, and 150-152
• 3X5 Cards: Parallel Lines, Skew Lines,
Parallel Planes, Postulate 15 and 16,
Theorems 3.4, 3.5, 3.6, 3.8, 3.9, 3.10.
3.11
3.1
3.3
3.4
3.5
Work Day
• Pages 132-134 #10-13, 21-31, 41-47
• Pages 146-149 #8-26
• Pages 153-156 #10-26