Download Document

Parallel Lines and
Transversals
Angle pairs formed by parallel lines and a
transversal are congruent, supplementary,
or both.
t
12
43
56
87
l
m
If two parallel lines are cut by a transversal,
then alternate interior angles are
t
congruent.
If l || m, then
4  6 and 3  5.
12
43
56
87
l
m
If two parallel lines are cut by a transversal,
then alternate exterior angles are
t
congruent.
If l || m, then
1  7 and 2  8.
12
43
56
87
l
m
If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
t
If l || m, then
1  5, 2  6,
4  8, and 3  7.
12
43
56
87
l
m
If two parallel lines are cut by a transversal,
then same side interior angles
t
(consecutive interior angles) are
supplementary.
If l || m, then 4 and 5
are supplementary,
3 and 6 are
supplementary.
12
43
56
87
l
m
Are any other angle pairs congruent?
t
12
43
56
87
l
m
Are any other angle pairs congruent?
Vertical angle
pairs are
congruent.
t
12
43
56
87
l
m
Are any other angle pairs supplementary?
t
12
43
56
87
l
m
Are any other angle pairs supplementary?
Angles that form
a linear pair are
supplementary.
t
12
43
56
87
l
m
There are only two angle measures formed
when two parallel lines are cut by a
transversal.
t
12
43
56
87
l
m
If l || m, and m1 = 110°, find the remainder
of the angle measures.
t
12
43
56
87
l
m
If l || m, and m1 = 110°, find the remainder
of the angle measures.
t
110°
70°
12
70° 4 3 110°
110° 5 6 70°
8 7110°
70°
l
m
Proving Lines Parallel
Angle pairs that are congruent or
supplementary can be used to determine
whether two lines are parallel.
t
12
43
56
87
l
m
If two lines are cut by a transversal and
alternate interior angles are congruent,
t
then the lines are parallel.
If 4  6 or 3  5, then
l || m.
12
43
56
87
l
m
If two lines are cut by a transversal and
alternate exterior angles are congruent,
t
then the lines are parallel.
If 1  7 or 2  8, then
l || m.
12
43
56
87
l
m
If two lines are cut by a transversal and
corresponding angles are congruent, then
t
the lines are parallel.
If 1  5, 2  6,
4  8, or 3  7,
then l || m.
12
43
56
87
l
m
If two lines are cut by a transversal and
same side interior angles (consecutive
t
interior angles) are supplementary, then
the lines are parallel.
If 4 and 5 are
supplementary or 3 and
6 are supplementary,
then l || m.
12
43
56
87
l
m
Parallel and Perpendicular
Lines
Use the relationship of two lines to a third
line to determine whether the two lines are
parallel or perpendicular.
If two lines are parallel to the same line, then
they are parallel to each other.
If a || c and b || c,
then a || b.
a
b
c
In a plane, if two lines are perpendicular to
the same line, then they are parallel.
t
If m  t and n  t,
then m || n.
m
n
In a plane, if a line is perpendicular to one of
two parallel lines, then it is perpendicular
to the other.
If n  l and l \\ m,
then n  m.
Parallel Lines and Triangles
Through a point not on a line, there is one
and only one line parallel to the given line.
P
Through a point not on a line, there is one
and only one line parallel to the given line.
P
Sum of the Interior Angles of a
Triangle
Lets use this concept to explore the angles
of a triangle.
Given ABC , construct a line through point
B, parallel to AC.
B
A
C
Given ABC , construct a line through point
B, parallel to AC.
B
A
C
What do we know about m1, m2, and
m3?
B
1
A
2
3
C
What do we know about m1, m2, and
m3? m1 + m2 + m3 = 180°, these
angles together form a straight angle, straight
angles measure 180.
B
1
A
2
3
C
What do we know about m1 and mA, m3
and mC?
B
1
A
2
3
C
What do we know about m1 and mA, m3
and mC? m1 = mA, m3 = mC, If two
parallel lines are cut by a transversal,
alternate interior angles are congruent.
B
1
A
2
3
C
If m1 + m2 + m3 = 180°, and
m1 = mA, m3 = mC, by substitution,
mA + m2 + mC = 180°.
B
1
A
2
3
C
This means the sum of the interior angles of
a triangle is 180°.
B
1
A
2
3
C
Exterior Angle of a Triangle
An exterior angle of a triangle is formed by
extending a side of the triangle.
exterior
angle
For each exterior angle, the two nonadjacent
interior angles are called its remote interior
angles.
remote interior angle
exterior
angle
remote interior angle
Exterior Angle Theorem
The measure of each exterior angle of a
triangle equals the sum of its two remote
interior angles.
m4 = m1 + m2
remote interior angle
1
2
remote interior angle
3
exterior
angle
4