Download Alternate interior angle theorem and converse

Parallel lines: the parallel line are the lines that are coplanar meaning that they lie in
the same plane and never intersect. In the picture below the orange lines are
representing parallel lines.
Parallel planes: the parallel planes are the planes that never intersect. In the picture
below the light blue and orange represent parallel planes.
It is important to mention that their are lines named skew lines these lines are totally
the opposite to the parallel lines, the skew lines are not coplanar, not parallel and do
not intersect. In the picture
The gray and the orange lines
Represent skew lines.
Parallel postulate: through a point v not on line c there is exactly one line parallel to c.
Perpendicular postulate: if 2 lines intersect and they create a linear pair of congruent
angles, then the lines are perpendicular.
Transversal: the transversals is a line intersecting two parallel lines at two different
points.
: the corresponding angles lie on the same side of the
transversal and in the same sides of the lines in the picture in the next page the
transversal would be A and the lines would be T and S.
: the alternate exteriors lie on the opposite side of the
transversal, these means that they lie outside the two lines.
: these are non adjacent angles that lie on the opposite
side of the transversal these means that they are inside the two lines and not
outside.
: these angles lie inside the two lines and lie on the
same side of the transversal.
1
Coresponding: 2 and 6
Alternate ex: 2 and 7
Alternate int: 4 and 5
Consecutive: 4 and 6
2
4
3
5
6
7
8
Corresponding angles postulate:
If two parallel lines are cut by a transversal, then the pairs of corresponding
angles are congruent.
2. m<def
Examples:
1. m<ABC
x= 78
M<ABC=78
80
a
c
b
3. m<def
(2x– 6)=(x+3) m<def= 9+3
-x
-x
12
X– 6= 3
+6 +6
x=9
(2x-4)=(x+2)
-x
-x
X-4=2
+4 +4
m<def =X=6
6+2=8
2x-6
X+3
2x -- 4
X+2
Postulate of the converse: if two coplanar lines are cut by a transversal so that a pair
of corresponding angles are congruent, then the two lines are parallel.
1. <1 =<5
<1=<5
l m
1 2
4
3
5
7
l
6
8
m
2. m<1=(4x+16), m<8=(5x-12),x=28
1=4(28)+16=128
8=5(28)-12=128
M<1=m<8
M<4=(6x-19), m<5=(3x+14), x=11
L ll m
4=6(11)-19=47
5=3(11)+14=47
M<4=m<5
L ll m
Alternate interior angle theorem: if two parallel lines are cut by a transversal, then the
pairs of alternate interior angles are congruent.
Converse: If two parallel lines are cut by a transversal, then the pairs of corresponding
angles are congruent.
<2 = <3 (= is
congruent)
1 2
1 2
3 4
3
4
<1=<4
1
5
2
6
3
7 8
4
<3=<6
Same side interior angle theorem: if two parallel lines are cut by a transversal, then
the two pairs of same-side interior angles are supplementary.
converse: if two coplanar lines are cut by a transversal so that pair of same-side
interior angles are supplementary, then the two lines are parallel.
Alternate exterior angle theorem: If two parallel lines are cut by a transversal, then the
two pairs of alternate exterior angles are congruent.
converse: if two coplanar lines are cut by a transversal so that a pair of alternate exterior
angles are congruent, then the two lines are parallel.
In a plane, if a transversal is perpendicular to one of two parallel lines then is is
perpendicular to the other line.