None

1. LINES AND ANGLES
Definitions
Free powerpoints at http://www.worldofteaching.com
Modified by Lisa Palen

2. PARALLEL LINES
• Definition: Parallel lines are coplanar lines that
do not intersect.
• Illustration: Use arrows to indicate lines are parallel.
• Notation: || means “is parallel to.”
l || m AB || CD
l
m A
B
C
D

3. PERPENDICULAR LINES
• Definition: Perpendicular lines are lines th
at form right angles.
• Illustration:
• Notation: m  n
• Key Fact: 4 right angles are formed.
m
n

4. OBLIQUE LINES
• Definition: Oblique lines are lines that inter
sect, but are NOT perpendicular.
• Illustration:
• Notation: m and n are oblique.

5. SKEW LINES
• Two lines are skew if they do not intersect and are not in the same p
lane (They are noncoplanar).
H
E
G
D
C
B
A
F

6. PARALLEL PLANES
• All planes are either parallel or intersecting. Parallel planes a
re two planes that do not intersect.
H
E
G
D
C
B
A
F

7. EXAMPLES
1. Name all segments that are parallel to
2. Name all segments that intersect
3. Name all segments that are skew to
4. 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.
AD
AD
AD
H
E
G
D
C
B
A
F

8. Recall:
• Slope measures how steep a l
ine is.
• The slope of the non-vertical
line through the points (x1, y
1) and (x2, y2) is


m
ris e
ru n
slope
2 1
2 1
y y
x x



Review of Slope

9. 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 s
lope has to be negative.

10. Examples
Find the slope of the line through the given points a
nd describe the line. (rises to the right, falls to the ri
ght, horizontal or vertical.)
1) (1, -4) and (2, 5) 2) (5, -2) and (- 3, 1)
2 1
2 1
5 ( 4 )
2 1
9
1
9



 




y y
x x
 
2 1
2 1
1 ( 2 )
3 5
3
8
3
8



 

 


 
y y
x x
Solution
slope
Solution
slope
This line rises to the right. This line falls to the right.

11. 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.

12. 3) (7, 6) and (-4, 6) 4) (-3, -2) and (-3, 8)
 
2 1
2 1
6 6
4 7
0
1 1
0





 



y y
x x
   
2 1
2 1
8 ( 2 )
3 3
1 0
0



 

  

y y
x x
u n d e f i n e d
Solution
slope
More Examples
Find the slope of the line through the given points
and describe the line. (rises to the right, falls to th
e right, horizontal or vertical.)
This line is horizontal. This line is vertical.
Solution
slope
No division by zero!

13. Horizontal lines have a slope of zero while vertical lines have
undefinedslope.
Horizontal
Vertical
m = 0
m = undefined

14. 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.)

15. x
y
Slopes of Parallel lines
Like t
his?
Or thi
s?
Or thi
s?
____
Example
What is the slope of this line?
What is the slope
of any line parallel
to this line?
5/12
5/12
because parall
el lines have t
he same slope
!

16. Slopes of Perpendicular lines
Postulate
Two non-vertical lines are perpendicular if and onl
y if the product of their slopes is -1.
The slopes of non-vertical perpendicular lines are
negative reciprocals.
a
b
1
m

m and or and 
b
a

17. Undefined
!
Slopes of Perpendicular lines
Examples
Find the negative reciprocal of each number:
1.
2.
3.
4.
4
3
1
7

3
4

7
6

1
6
0
1
0

18. Slopes of Perpendicular Lines
1
0

0 and
Also
•All horizontal lines are perpendicular t
o all vertical lines.
•The slope of a line perpendicular to a l
ine with slope 0 is undefined. Undefined
!

19. Examples
Anylineparallel to a line withslope
has slope _____.
Anyline perpendicular to a line withslope
has slope _____.
Anylineparallel to a line withslope 0
has slope _____.
Anylineperpendicular to a linewithundefinedslope
has slope _____.
Anylineparallel to a line withslope 2
has slope _____.
2
7
4
3


20. • Def: a line that intersects two lines (that ar
e coplanar) at different points
• Illustration:
Transversal
t
30

21. Vertical Angles
• Two non-adjacent angles formed by inters
ecting lines. They are opposite angles.
1 2
3 4
5 6
7 8
t
1   4
2   3
5   8
6   7

22. Vertical Angles
• Find the measures of the missing angles.
125 
x
y
55 
t
55
125 x = 125
y = 55

23. Linear Pair
• Supplementary adjacent angles. They form a line and the
ir sum = 180)
1 2
3 4
5 6
7 8
t
m1 + m2 = 180º
m2 + m4 = 180º
m4 + m3 = 180º
m3 + m1 = 180º
m5 + m6 = 180º
m6 + m8 = 180º
m8 + m7 = 180º
m7 + m5 = 180º

24. Supplementary Angles/
Linear Pair
• Find the measures of the missing angles.
x 72 
y
t
108
108 
x = 180 – 72
y = x = 108

25. Corresponding Angles
• Two angles that occupy corresponding po
sitions.
Top Left
t
Top Left
Top Right
Top Right
Bottom Right
Bottom Right
Bottom Left
Bottom Left
1 and  5
2 and  6
3 and  7
4 and  8
1 2
3 4
5 6
7 8

26. Corresponding Angles Postulate
• If two parallel lines are crossed by a transversal,
then corresponding angles are congruent.
Top Left
t
Top Left
Top Right
Top Right
Bottom Right
Bottom Right
Bottom Left
Bottom Left
1   5
2   6
3   7
4   8
1 2
3 4
5 6
7 8

27. Corresponding Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
145 
z
t
145 
z = 145

28. Alternate Interior Angles
• Two angles that lie between the two lines
on opposite sides of the transversal
t
3 and  6
4 and  5
1 2
3 4
5 6
7 8

29. If two parallel lines are crossed by a transversal, th
en alternate interior angles are congruent.
Alternate Interior Angles Theorem
t
3   6
4   5
1 2
3 4
5 6
7 8

30. If two parallel lines are crossed by a transversal, th
en alternate interior angles are congruent.
Alternate Interior Angles Theorem
t
1
4
5
l
m
Given: l  m
Prove: 4  5
Statements Reasons
1.l  m
2. 4   1
3. 1   5
4. 4   5
1. Given
2. Vertical
AnglesThm
3. Corres-
ponding
Angles Post.
4. Transitive
Property of
Congruence

31. Alternate Interior Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
82 
z 
t
82 
z = 82

32. Alternate Exterior Angles
• Two angles that lie outside the two lines o
n opposite sides of the transversal
t
2 and  7
1 and  8
1 2
3 4
5 6
7 8

33. Alternate Exterior Angles Theor
em
If two parallel lines are crossed by a transversal, th
en alternate exterior angles are congruent.
t
2   7
1   8
1 2
3 4
5 6
7 8

34. Alternate Exterior Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
120 
w
t
120 
w = 120

35. Consecutive Interior Angles
• Two angles that lie between the two lines
on the same sides of the transversal
t
m3 and m5
m4 and m6
1 2
3 4
5 6
7 8

36. • If two parallel lines are crossed by a transversal,
then consecutive interior angles are supplement
ary.
Consecutive Interior Angles Theore
m
t
m3 +m5 = 180º
m4 +m6 = 180º
1 2
3 4
5 6
7 8

37. Consecutive Interior Angles
• Find the measures of the missing angles,
assuming the black lines are parallel.
?
t
135 
45 
180º - 135º

38. Angles and Parallel Lines
If two parallel lines are crossed by a transversal, the
n the following pairs of angles are congruent.
• Corresponding angles
• Alternate interior angles
• Alternate exterior angles
If two parallel lines are crossed by a transversal, the
n the following pairs of angles are supplementar
y.
• Consecutive interior angles

39. Review Angles and Parallel Lines
Alternate interior angles
t
D
C
B
A 1 2
3
4
5 6
7
8
Alternate exterior angles
Corresponding angles
Consecutive interior angles
Consecutive exterior angles

40. Examples
t
16 15
14
13
12 11
10
9
8 7
6
5
3
4
2
1
s
D
C
B
A
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.
m 2=80º m 3=100º m 4=80º
m 5=100º m 6=80º m 7=100º m 8=80º
m 9=100º m10=80º m11=100º m12=80º
m13=100º m14=80º m15=100º m16=80º
Click for Answers

41. More Examples
t
16 15
14
13
12 11
10
9
8 7
6
5
3
4
2
1
s
D
C
B
A
1. The value of x, if m3 = (4x + 6)º and the m11 = 126º.
If line AB is parallel to line CD and s is parallel to t, find:
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.
ANSWERS:
1. 30 2. 35 3. 33
Click for Answers

42. Proving Lines Parallel
Recall: Corresponding Angles Postulate
If two lines cut by a transversal are parallel, t
hen corresponding angles are congruent.
So what can you say about the lines here?
Contrapositive:
If corresponding angles are NOT congruent,
then two lines cut by a transversal are NOT
parallel.
145
144 NOT PARAL
LEL!

43. So what can you say about the lines here?
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
144
144
PARALLEL!

44. If corresponding angles are congruent, then
two lines cut by a transversal are parallel.
Proving Lines Parallel
Converse of the Corresponding Angles Postulate
D
C
B
A
A
BC
D
two lines cut by a transversal are parallel

45. 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.
D
C
B
A
A
BC
D

46. If alternate interior angles are congruent, then two l
ines cut by a transversal are parallel.
Converse of the Alternate Interior A
ngles Theorem
t
1
4
5
l
m
Given: 5  4
Prove: l  m
Statements Reasons
1. 5   4
2. 4   1
3. 1   5
4.l  m
1. Given
2. Vertical
AnglesThm
3. Transitive
Property of
Congruence
4. Converse
of the Corr-
esponding
Angles Post/

47. 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.
D
C
B
A
A
BC
D

48. Proving Lines Parallel
Converse of the Consecutive Interior Angles Theorem
If consecutive interior angles are SUPPLEMENT
ARY,
then two lines cut by a transversal are parallel.
D
C
B
A
A
BC
D

49. Proving Lines Parallel
Examples
• Find the value of x which will make lines a
and lines b parallel.
1. 2.
3. 70

(x - 20)

b
a
60

3x

b
a
4.
60

3x

b
a
80

2x

b
a
ANSWERS: 20; 50; 45; 20

50. Ways to Prove Two Lines Parallel
• Show that corresponding angles are congr
uent.
• Show that alternative interior angles are co
ngruent.
• Show that alternate exterior angles are co
ngruent.
• Show that consecutive interior angles are
supplementary.
• In a plane, show that the lines are perpend
icular to the same line.