Download Chapter 1 Points, Lines, Planes, and Angles page 1

Lesson 3-2
Properties of Parallel Lines
(page 78)
Essential Question
How can you apply parallel
lines (planes) to make
deductions?
Properties of Parallel Lines
• Refer to the results from exercises #18,
#19, and #20 on page 76 (and the results
from the top of page 78).
• We have choice of what to accept
without proof and make a postulate.
• This textbook uses #18 as a postulate.
• This is not the same in all geometry books.
Postulate 10
If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
|| - lines ⇒ corr. ∠’s 
1
x
4
5
y
8
3
6
7
2
Theorem 3-2
If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
|| - lines ⇒ AIA 
t
Given: k || n
transversal t cuts k & n
Prove: ∠1 ∠ 2
3
k
1
n
2
t
Given: k || n
transversal t cuts k and n
3
k
1
Prove: ∠1 ∠ 2
n
2
Proof:
Statements
See
page 78!
k || n
1.
___________________________________
2.
___________________________________
3.
___________________________________
4.
___________________________________
Reasons
Given
_____________________________________________
Vert. ∠’s R 
∠1 ∠ 3
_____________________________________________
∠3 ∠ 2
_____________________________________________
∠1 ∠ 2
_____________________________________________
|| - lines ⇒ Corr. ∠’s 
Transitive Property
Theorem 3-3
If two parallel lines are cut by a transversal,
then same side interior angles are supplementary.
|| - lines ⇒ SSIA Supp.
t
Given:
k || n
transversal t cuts k and n
Prove:
∠1 is supplementary to ∠ 4
k
1
n
4
2
t
Given: k || n
transversal t cuts k & n
Prove: ∠1 is suppl. to ∠ 4
k
1
n
4
2
Proof:
Statements
Reasons
Given
k || n
_____________________________________________
∠1 ∠ 2
_____________________________________________
1.
____________________________________
2.
____________________________________
3.
____________________________________
_____________________________________________
4.
____________________________________
m∠2 + m ∠ 4 = 180º
_____________________________________________
5.
____________________________________
_____________________________________________
6.
∠1
is supplementary to ∠ 4
____________________________________
_____________________________________________
m∠1 = m ∠ 2
m∠1 +m ∠ 
|| - lines ⇒ AIA 
Def. of  ∠’s
Angle Addition Postulate
Substitution Prop.
Def. of Supp. ∠’s
Theorem 3-4
If a transversal is perpendicular to one of two parallel
lines, then it is perpendicular to the other one also.
Given: transversal t cuts l & n
l
1
n
2
t ⊥ l ; l || n
Prove:
t⊥n
t
This proof is on page 81 #13.
Given: transversal t cuts l and n
t ⊥ l ; l || n
Prove:
t⊥n
Proof:
2.
1
n
2
t
Statements
1.
l
t⊥l
m ∠ 1 = 90º
l || n
3. ________________________
Reasons
Given
_____________________________________________
Def. of ⊥ lines
_____________________________________________
Given
|| - lines ⇒ Corr. ∠’s 
∠1  ∠2 or m∠1 = m∠2
_____________________________________________
5. ________________________
m∠2 = 90º
Substitution Property
4.
6.
t⊥n
Def. of ⊥ lines
_____________________________________________
If two parallel lines are cut by a transversal,
then …
1) corresponding angles
are congruent .
2) alternate interior angles
are congruent .
3) same-side interior angles
are supplementary .
Example #1.
If m ∠1 = 120º, x || y, and m || n,
then find all the other measures.
120º
m ∠2 = ______
x
120º
6
120º
4
5
120º
y
1
120º
m
120º
2
3
60º
n
60º
m ∠3 = ______
120º
m ∠4 = ______
120º
m ∠5 = ______
120º
m ∠6 = ______
Example #2. Find the values of x, y, and z.
105º
75º
(3x)º (5y)º
35
x = ______
105º
75º
zº
15
y = ______
x || y
x
3 x = 105
x = 35
WHY?
y
5 y + 105 = 180
5 y = 75
WHY?
y = 15
75
z = ______
z = 75
WHY?
Example #3. Find the values of x, y, and z.
(4x)º
13
x = ______
32º
52º
96º
zº
4 x = 52
x = 13
WHY?
32º (3y +8)º
3 y + 8 = 32
3 y = 24
WHY?
y=8
52º
8
y = ______
96
z = ______
z + 32 + 52 = 180
z + 84 = 180
z = 96
WHY?
Assignment
Written Exercises on pages 80 to 82
RECOMMENDED: 1, 3, 5, 19
REQUIRED: 7, 9, 11, 15, 17, 21, 23, 25
Prepare for a quiz on
Lesson 3-2: Properties of Parallel Lines
How can you apply parallel
lines (planes) to make
deductions?