Transcript 3.3 Parallel Lines and Transversals

PARALLEL LINES AND
TRANSVERSALS
3.3
GOAL
1
PROPERTIES OF PARALLEL LINES
This section will require you to think about and use
parallel lines. Although some of the theorems and
ideas may seem obvious, you must always be ready to
provide justification for any statements made.
POSTULATE
Corresponding Angles Postulate
lines  corr.  s 
THEOREMS
Alternate Interior Angles
lines  alt. int.  s 
Consecutive Interior Angles
lines  cons. int.  s su pp.
Alternate Exterior Angles
lines  alt. ext.  s 
Perpendicular Transversal
T ra n sv e rsa l  o n e o f 2
lin e s 
o th e r lin e
EXAMPLE 1
Extra Example 1
3
Given: p q
5
Prove: m  1  m  2  1 8 0 
7
4
Statements
1
6
8
2
p
q
Reasons
1. p q
1. Given
2.  1   4
2. A lt. E xt.  s T h m .
3. m  1  m  4
3. D e f.   s
4.  2 and  4 are supp.
4. Linear Pair Post.
5. m  2  m  4  1 8 0 
5. D ef. supp.  s
6. m  1  m  2  1 8 0 
6. Substitution
EXAMPLE 2
Extra Example 2
3
Given that m  2  1 1 0 ,
find each measure. Give the
postulate or theorem used.
5
7
4
1
6
8
2
a. m  8
70°; Linear Pair Postulate
b. m  1
70°; Corresponding Angles Postulate  w ith  8 
c. m  3
110°; Alternate Exterior Angles Thm.  w ith  2 
or Linear Pair Postulate  w ith  1
d. m  4
70°; Alternate Exterior Angles Thm.  w ith  1
or Linear Pair Postulate  w ith  2 
or Vertical Angles Theorem  w ith  8 
Checkpoint
A 2
B
100°
F in d m  1, m  2, a n d m  3 .
3
C
1
D
m  1  8 0 ; m  2  1 0 0 , m  3  8 0 
EXAMPLE 3
Extra Example 3
In the diagram above, how many angles have a measure of
100°?
eight
PARALLEL LINES AND
TRANSVERSALS
3.3
GOAL
2
PROPERTIES OF SPECIAL PAIRS OF
ANGLES
EXAMPLE 4
Extra Example 4
Use properties of parallel lines
to find the value of x.
(x – 8 )°
72°
1
Since m  1  7 2  by the Vertical Angles Theorem, and
m 1  ( x  8 )  180  by the Consecutive Interior Angles
Theorem, solve the equation:
72  ( x  8 )  180
x  116
EXAMPLE 5
Extra Example 5
Refer to Example 5. We now know that the diameter of
Earth is about 7973 mi. Recalculate the distance between
Syene and Alexandria using this figure.
Using the equation from Example 5:
1
50
d
o f a circle =
E a rth 's circu m fe re n ce
 1 
 (7 9 7 3 m ile s) 
d
 50 
5 0 1 m ile s  d
Checkpoint
Use properties of parallel lines to find the value of x.
x°
(x – 20)°
70°
n
m
x = 65
QUESTION:
If a transversal is perpendicular to one of two
parallel lines, what is the measure of all the
angles formed?
ANSWER:
90°