Definition of parallel lines
Two lines are parallel if and only if they are coplanar
lines that do not intersect.
Definition of perpendicular lines
Two lines are perpendicular if and only if they are
intersect to form a right angle.
Parallel Postulate
If you have a line and a point NOT on the line, then there
is exactly ONE line through the point that is parallel to
the line.
Perpendicular Postulate
If you have a line and a point NOT on the line, then there
is exactly ONE line through the point that is parallel to
the line.
Corresponding Angles Postulate
1
l
2
m
If two parallel lines are cut by a transversal, then the
corresponding angles are congruent.
l∥m  1  2.
Corresponding Angles Converse
(Postulate)
1
l
2
m
If two lines are cut by a transversal AND corresponding
angles are congruent, then the lines are parallel.
1  2  l∥m.
Alternate Interior Angles Theorem
l
1
2
m
If two parallel lines are cut by a transversal, then the
alternate interior angles are congruent.
l∥m  1  2.
Alternate Interior Angles Converse
(Theorem)
l
1
2
m
If two lines are cut by a transversal AND alternate
interior angles are congruent, then the lines are parallel.
1  2  l∥m.
Alternate Exterior Angles Theorem
1
l
m
2
If two parallel lines are cut by a transversal, then the
alternate exterior angles are congruent.
l∥m  1  2.
Alternate Exterior Angles Converse
(Theorem)
1
l
m
2
If two lines are cut by a transversal AND alternate
exterior angles are congruent, then the lines are parallel.
1  2  l∥m.
Consecutive Interior Angles Theorem
l
1
2
3
m
m 1 m 2 1800
If two parallel lines are cut by a transversal, then the
consecutive interior angles are supplementary.
l∥m  m1 + m2 =1800.
Consecutive Interior Angles Converse
(Theorem)
m 1 m 2 1800
l
1
2
3
m
If two lines are cut by a transversal AND consecutive
interior angles are congruent, then the lines are parallel.
m1 + m2 =1800  l∥m.
Transitive Property of Parallel Lines
Theorem
l
m
n
If two parallel lines are parallel to a third line, then they
are parallel to each other.
l∥m and m∥n  l∥n.
Lines Perpendicular to a Transversal
Theorem
l
m
n
(In a Plane,) if two lines are perpendicular to the same
line, then the lines are parallel to each other.
l⊥m and l⊥n  m∥n.
Perpendicular Transversal Theorem
l
m
n
If a transversal is perpendicular to one of two parallel
lines, then it is perpendicular to the other line.
If l⊥n and m∥n, then l⊥m.
Congruent Linear Pairs Theorem
l
1
2
m
If two lines intersect to form a linear pair of congruent
angles, then the lines are perpendicular.
If 1 and 2 are a linear pair and 1  2, then l⊥m.
Four Right Angles Theorem
l
1
3
2
m
4
Two lines are perpendicular if and only if they intersect
to form four right angles.
l⊥m  m1 = m2 = m3 = m4= 900.
Adjacent Angles and Perpendicular Lines
Theorem
l
1
2
m
1 and 2 are complementary
If two sides of acute adjacent angles are perpendicular,
then, angles are complementary.
If l⊥m AND 1 & 2 are adjacent, then 1 & 2 are
complementary
Slope of Parallel Lines Postulate
l
l∥m
m
Slope of line l = slope of line m
(In the coordinate plane,) two lines are parallel if and
only if they have the same slope.
l∥m  slope of line l = slope of line m.
*vertical lines are parallel
Slope of Perpendicular Lines Postulate
l
m
l⊥m
(Slope of line l)(slope of line m) = –1
(In the coordinate plane,) two lines are perpendicular if
and only if the product of their slopes is negative 1.
l⊥m  the product of their slopes is negative 1.
*vertical lines are perpendicular to horizontal lines