LINES AND
ANGLES
Definitions
Free powerpoints at http://www.worldofteaching.com
Modified by Lisa Palen
PARALLEL LINES
• Definition: Parallel lines are coplanar lines that
do not intersect.
• Illustration: Use arrows to indicate lines are parallel.
B
l
m
A
• Notation: || means “is parallel to.”
l || m
D
C
AB || CD
PERPENDICULAR LINES
• Definition: Perpendicular lines are lines
that form right angles.
m
• Illustration:
n
• Notation: m  n
• Key Fact: 4 right angles are formed.
OBLIQUE LINES
• Definition: Oblique lines are lines that
intersect, but are NOT perpendicular.
• Illustration:
• Notation: m and n are oblique.
SKEW LINES
• Two lines are skew if they do not intersect and are
not in the same plane (They are noncoplanar).
A
D
B
C
E
H
F
G
PARALLEL PLANES
• All planes are either parallel or intersecting.
Parallel planes are two planes that do not
intersect.
A
D
B
C
E
H
F
G
A
D
B
C
EXAMPLES
E
H
1.
2.
3.
4.
Name all segments that are parallel to
AD
Name all segments that intersect
AD
Name all segments that are skew to AD
Name all planes that are parallel to plane ABC.
Answers:
1. Segments EH, BC & GF.
2. Segments AE, AB, DH & DC.
3. Segments CG, BF, FE & GH.
4. Plane FGH.
F
G
Review of Slope
Recall:
• Slope measures how
steep a line is.
• The slope of the
non-vertical line
through the points
(x1, y1) and (x2, y2) is
slope
m
rise

run
y 2  y1

x 2  x1
If a line goes up from left to right, then
the slope has to be positive .
Conversely, if a line goes down from left
to right, then the slope has to be negative.
Examples
Find the slope of the line through the given points
and describe the line. (rises to the right, falls to the
right, horizontal or vertical.)
1)
(1, -4) and (2, 5)
Solution
slope
y2  y1

x2  x1
5  (4)

2 1
9

1
9
This line rises to the right.
2) (5, -2) and (- 3, 1)
Solution
slope
y2  y1

x2  x1
1  (2)

 3  5
3
8
3

8

This line falls to the right.
The slope of a horizontal line is zero.
The slope of a vertical line is undefined.
Sometimes we say a vertical line has no
slope.
More Examples
Find the slope of the line through the given points
and describe the line. (rises to the right, falls to
the right, horizontal or vertical.)
3)
(7, 6) and (-4, 6)
Solution
slope
y2  y1

x2  x1
66

 4   7
0

11
0
This line is horizontal.
4) (-3, -2) and (-3, 8)
Solution
slope
y2  y1

x2  x1
8  ( 2)

 3     3 
10
No division by zero!

0
undefined
This line is vertical.
Horizontal lines have a slope of zero while
vertical lines have undefined slope.
m=0
Vertical
Horizontal
m = undefined
Slopes of Parallel lines
Postulate (Parallel lines have equal slopes.)
Two non-vertical lines are parallel if and
only if they have equal slopes.
Also:
• All horizontal lines are parallel.
• All vertical lines are parallel.
• All lines with undefined slope are
parallel. (They are all vertical.)
Slopes of Parallel lines
Example
What is the slope of this line? ____
5/12
What is the slope
y
of any line parallel
to this line?
5/12
Like
this?
x
Or
this?
because
parallel lines
have the same
slope!
Or
this?
Slopes of Perpendicular lines
Postulate
Two non-vertical lines are perpendicular if and
only if the product of their slopes is -1.
The slopes of non-vertical perpendicular lines
are negative reciprocals.
a
b
1
m and  or

and
a
b
m
Slopes of Perpendicular lines
Examples
Find the negative reciprocal of each number:
4
1.
3
1
2. 
7
3.
4.
3

4
7
6
1
6
0
1
0
Undefined
!
Slopes of Perpendicular Lines
Also
•All horizontal lines are perpendicular
to all vertical lines.
•The slope of a line perpendicular to a
Undefined
line with slope 0 is undefined.
!
0 and
1

0
Examples
Any line parallel to a line with slope
has slope _____.
2
7
4
Any line perpendicular to a line with slope 
3
has slope _____.
Any line parallel to a line with slope 0
has slope _____.
Any line perpendicular to a line with undefined slope
has slope _____.
Any line parallel to a line with slope 2
has slope _____.
Transversal
• Def: a line that intersects two lines (that
are coplanar) at different points
t
• Illustration:
30
Vertical Angles
• Two non-adjacent angles formed by
intersecting lines. They are opposite
t
angles.
1 2
3 4
5
7
6
8
1   4
2   3
5   8
6   7
Vertical Angles
• Find the measures of the missing angles.
t
125 
x
125
y
55
55 
x = 125
y = 55
Linear Pair
• Supplementary adjacent angles. They form a line and
their sum = 180)
t
m1 + m2 = 180º
m2 + m4 = 180º
m4 + m3 = 180º
m3 + m1 = 180º
1 2
3 4
5
7
6
8
m5 + m6 = 180º
m6 + m8 = 180º
m8 + m7 = 180º
m7 + m5 = 180º
Supplementary Angles/
Linear Pair
• Find the measures of the missing angles.
t
x 72 
108
108
y 
x = 180 – 72
y = x = 108
Corresponding Angles
• Two angles that occupy corresponding
positions.
t
Top Left
Top Right
1
3
Bottom Left
Top Left
Bottom Left
2
4
Bottom Right
5
6 Top Right
7
8
Bottom Right
1 and  5
2 and  6
3 and  7
4 and  8
Corresponding Angles Postulate
• If two parallel lines are crossed by a transversal,
then corresponding angles are congruent.
t
Top Left
Top Right
1
3
Bottom Left
Top Left
Bottom Left
2
4
Bottom Right
5
6 Top Right
7
8
Bottom Right
1   5
2   6
3   7
4   8
Corresponding Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
t
145 
z = 145
145
z 
Alternate Interior Angles
• Two angles that lie between the two lines
on opposite sides of the transversal
t
1
2
3
4
5 6
7
8
3 and  6
4 and  5
Alternate Interior Angles Theorem
If two parallel lines are crossed by a transversal,
then alternate interior angles are congruent.
t
1
2
3
4
5 6
7
8
3   6
4   5
Alternate Interior Angles Theorem
If two parallel lines are crossed by a transversal,
then alternate interior angles are congruent.
Given: l  m
Prove: 4  5
t
1
l
4
m
5
Statements
Reasons
1. Given
1. l  m
2. Vertical
2.  4   1
AnglesThm
3. Corres3.  1   5
ponding
Angles Post.
4. Transitive
Property of
4.  4   5
Congruence
Alternate Interior Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
t
z = 82
82 
82
z 
Alternate Exterior Angles
• Two angles that lie outside the two lines
on opposite sides of the transversal
t
1
2
3
4
5 6
7
8
2 and  7
1 and  8
Alternate Exterior Angles
Theorem
If two parallel lines are crossed by a transversal,
then alternate exterior angles are congruent.
t
1
2
3
4
5 6
7
8
2   7
1   8
Alternate Exterior Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
t
120 
w = 120
w
120 
Consecutive Interior Angles
• Two angles that lie between the two lines
on the same sides of the transversal
t
1
2
3
4
5 6
7
8
m3 and m5
m4 and m6
Consecutive Interior Angles
Theorem
• If two parallel lines are crossed by a transversal,
then consecutive interior angles are
supplementary.t
1
2
3
4
5 6
7
8
m3 +m5 = 180º
m4 +m6 = 180º
Consecutive Interior Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
t
135 
?45 
180º - 135º
Angles and Parallel Lines
If two parallel lines are crossed by a transversal,
then the following pairs of angles are congruent.
•
•
•
Corresponding angles
Alternate interior angles
Alternate exterior angles
If two parallel lines are crossed by a transversal,
then the following pairs of angles are
supplementary.
•
Consecutive interior angles
Review Angles and Parallel Lines
A
1 2
4 3
C
5 6
8 7
t
Alternate interior angles
Alternate exterior angles
Corresponding angles
Consecutive interior angles
Consecutive exterior angles
B
D
Examples
A
1
4
C
5
8
s
2
3
6
9 10
12 11
13 14
16 15
7
B
D
t
If line AB is parallel to line CD and s is parallel to t,
find the measure of all the angles when m1 = 100º.
Justify your answers.
Click for Answers
m 5=100º
m 9=100º
m13=100º
m 2=80º
m 6=80º
m10=80º
m14=80º
m 3=100º
m 7=100º
m11=100º
m15=100º
m 4=80º
m 8=80º
m12=80º
m16=80º
More Examples
A
1
4
C
5
8
s
2
3
6
9 10
12 11
13 14
16 15
7
B
D
t
If line AB is parallel to line CD and s is parallel to t, find:
1. The value of x, if m3 = (4x + 6)º and the m11 = 126º.
2. The value of x, if m1 = 100º and m8 = (2x + 10)º.
3. The value of y, if m11 = (3y – 5)º and m16 = (2y + 20)º.
Click
for Answers
ANSWERS:
1. 30
2. 35
3. 33
Proving Lines Parallel
Recall: Corresponding Angles Postulate
If two lines cut by a transversal are parallel,
then corresponding angles are congruent.
So what can you say about the lines here?
145
144
NOT
PARALLEL!
Contrapositive:
If corresponding angles are NOT congruent,
then two lines cut by a transversal are NOT
parallel.
Proving Lines Parallel
So what can you say about the lines here?
144
144
PARALLEL!
Corresponding Angles Postulate:
If two lines cut by a transversal are parallel,
then corresponding angles are congruent.
Converse of the Corresponding Angles Postulate:
If corresponding angles are congruent,
then two lines cut by a transversal are parallel.
Proving Lines Parallel
Converse of the Corresponding Angles Postulate
If corresponding angles are congruent, then
two lines cut by a transversal are parallel.
A
B
C
D
two lines cut by a transversal are parallel
AB CD
Proving Lines Parallel
Converse of the Alternate Interior Angles Theorem
If alternate interior angles are congruent, then
two lines cut by a transversal are parallel.
A
B
C
D
AB CD
Converse of the Alternate Interior
Angles Theorem
If alternate interior angles are congruent, then two
lines cut by a transversal are parallel.
Given: 5  4
Prove: l  m
t
1
l
4
m
5
Statements
Reasons
1.  5   4 1. Given
2. Vertical
2.  4   1
AnglesThm
3. Transitive
3.  1   5
Property of
Congruence
4. l  m
4. Converse
of the Corresponding
Angles Post/
Proving Lines Parallel
Converse of the Alternate Exterior Angles Theorem
If alternate exterior angles are congruent,
then two lines cut by a transversal are parallel.
A
B
C
D
AB CD
Proving Lines Parallel
Converse of the Consecutive Interior Angles Theorem
If consecutive interior angles are
SUPPLEMENTARY,
then two lines cut by a transversal are parallel.
A
B
C
D
AB CD
Proving Lines Parallel
Examples
• Find the value of x which will make lines a
and lines b parallel.
1.
b
a
3.
3x
a
b
2.
70
ANSWERS: 20; 50; 45; 20
2x
b
60
(x - 20)
a
4.
80
a
3x
b
60
Ways to Prove Two Lines Parallel
• Show that corresponding angles are
congruent.
• Show that alternative interior angles are
congruent.
• Show that alternate exterior angles are
congruent.
• Show that consecutive interior angles are
supplementary.
• In a plane, show that the lines are
perpendicular to the same line.