Relationships Between Lines
Parallel Lines – two lines that
are coplanar and do not
intersect
Skew Lines – two lines that are
NOT coplanar and do not
intersect
Perpendicular Lines – two
lines that intersect to form a
right angle
Parallel Planes – two planes
that do not intersect
Parallel and Perpendicular Postulates
P
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.
Postulate 14: Perpendicular
Postulate
If there is a line and a point not on the
line, then there is exactly one line
through the point perpendicular to the
given point.
l
P
*
l
Transversals
• A transversal is a line that
intersects two or more
coplanar lines at different
points.
Angles Formed By Transversals
Corresponding Angles – two angles are
corresponding if they occupy corresponding
positions as a result of the intersection of the
lines. (1 & 5)
Alternate Exterior Angles – two angles are
alternate exterior if they lie outside the two
lines on opposite sides of the transversal (1 &
7)
Alternate Interior Angles – Two angles that
lie between the two lines on opposite sides of
the transversal (4 & 6)
Consecutive Interior Angles – two angles
that lie between the two lines on the same
side of the transversal (4 & 5)
Who can name another pair for each of the angle types above?
Postulates and Theorems of Parallel Lines
Postulate 15 Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the
pairs of corresponding angles are congruent.
1
2
<1 =~ <2
Theorem 3.4 Alternate Interior Angles
If two parallel lines are cut by a transversal, then the
pairs of alternate interior angles are congruent.
3
4
<3 ~
= <4
Theorem 3.5 Consecutive Interior Angles
If two parallel lines are cut by a transversal, then the
pairs of consecutive interior angles are
supplementary.
5
6
m<5 + m<6 = 180*
Theorem 3.5 Alternate Exterior Angles
If two parallel lines are cut by a transversal, then the
pairs of alternate exterior angles are congruent.
7
8
Theorem 3.7 Perpendicular Transversal
If a transversal is perpendicular to one of two parallel
lines, then it is perpendicular to the other.
<7 ~
= <8
j
h
j _|_ k
k
Proving the Alternate Interior Angles Theorem
Given: p || q
~ <2
Prove: <1 =
Statements
1. p || q
~
2. <1 =
<3
~
3. <3 = <2
4. <1 ~= <2
1
2
3
p
q
Reasons
Given
Corresponding Angles Postulate
Vertical Angles Theorem
Transitive Property of Congruence
Using Properties of Parallel Lines
Given that m<5 = 65*, find each measure.
Tell which postulate or theorem you use.
6
9
7
a.
b.
c.
d.
5
8
m<6 =
m<7 =
m<8 =
m<9 =
Use the Properties of Parallel Lines to Solve for x
125*
4
(x + 15*)
Proving Lines are Parallel
j
Postulate 16: Corresponding Angle Converse:
If two lines are cut by a transversal so that the
corresponding angles are congruent, then the lines
are parallel.
Theorem 3.8: Alternate Interior Angles Converse:
If two lines are cut by a transversal so that the
alternate interior angles are congruent, then the lines
are parallel.
Theorem 3.9 Consecutive Interior Angles
Converse:
If two lines are cut by a transversal so that the
consecutive interior angles are supplementary, then
the lines are parallel.
Theorem 3.10 Alternate Exterior Angles Converse
- If two lines are cut by a transversal so that alternate
exterior angles are congruent, then the lines are
parallel.
1
k
2
~
If <1 = <2,
Then j _|_ k
3
4
If <3 ~
= <4
Then j _|_ k
5
6
If m<5 + m<6 = 180*
Then j _|_ k
7
8
If <7 ~
= <8
Then j _|_ k
Proving the Alternate Interior Angles Converse
~
Given: <1 = <2
Prove: p _|_ q
Statements
~
1. <1 =
<2
~
2. <2 = <3
~
3. <1 =
<3
4. p _|_ q
3
2
1
p
q
Reasons
Given
Vertical Angles Theorem
Transitive Property of Congruence
Corresponding Angles Converse