Download 3-1 3-2 3-3 Parallel Lines and Angles

3-1 3-2 3-3 Parallel Lines and Angles
• Objectives
• Define transversal and the angles
associated with a transversal
• State and apply the properties of angles
associated with a transversal that cuts a
pair of parallel lines
1
3-1 Definition of transversal
• A transversal is a line that intersects two
coplanar lines at two distinct points
l
1 2
4 3
is a transversal for l and m.
c
5 6
8 7
t
t
b
1 2
4 3
m
5 6
8 7
b || c
Nonparallel Lines
lm
Parallel Lines
r
r
is a transversal for b and c.
2
3-1 Angles associated with transversals
Interior angles lie between the
two lines. Examples: ∠4, ∠6
Lines b and c
Exterior angles lie outside the
two lines. Examples: ∠1, ∠8
1 2
4 3
Alternate Interior angles are on the
opposite sides of the transversal.
Example: ∠4 and ∠6
Corresponding angles are on
the same side of the transversal and
on the same side of the lines cut by the
transversal. Example: ∠2 and ∠6
c
5 6
8 7
Alternate Exterior angles are
on the opposite sides of the
transversal. Example: ∠2 and ∠8
Same Side Interior angles are on
the same side of the transversal.
Example: ∠4 and ∠5
b
r
r
is a transversal for b and c.
3
3-1 Theorems/postulates for parallel lines
• If a transversal intersects two
parallel lines, then
Parallel Lines
– Corresponding angles are
1 2
congruent (Corr. Angles Postulate)
4 3
– Alternate interior angles are
5 6
congruent (Alt. Int. Angles Thm)
8 7
– Alternate exterior angles are
t
congruent (Alt. Ext. Angles Thm)
– Same side interior angles are
supplementary (Same Side Int.
t is a transversal for
Angles Thm)
– Same side exterior angles are
supplementary (Same Side Ext.
Angles Thm)
lm
l
m
l and m.
4
3-2 Converses of previous slide
• All converses of the previous slide are true
• Example: Converse of the Corresponding
Angle Postulate:
– If two lines and a transversal form congruent
corresponding angles, then the two lines are
parallel
• Example: Converse of the Alternate
Interior Angles Theorem:
– If two lines and a transversal form congruent
alternate interior angles, then the two lines
are parallel
5
3-3 Parallel and perpendicular lines
• If two lines are parallel to the
same line, then they are
parallel to each other.
• In a plane, if two lines are
perpendicular to the same
line, then they are parallel to
each other.
• In a plane, if a line is
perpendicular to one of two
parallel lines, then it is also
perpendicular to the other.
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