1. Lesson 3-2 Properties of Parallel Lines (page 78)
Essential Question
How can you apply parallel lines (planes) to make deductions?

2. 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 a 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.

3. Postulate 10 || - lines ⇒ corr. ∠’s ≅
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
|| - lines ⇒ corr. ∠’s ≅ x 1 2 4 3 5 6 y 8 7

4. Theorem 3-2 || - lines ⇒ AIA ≅
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 k 3 1 n 2

5. Given k || n ∠1 ≅ ∠ 3 Vert. ∠’s R ≅ ∠3 ≅ ∠ 2 || - lines ⇒ Corr. ∠’s ≅
transversal t cuts k and n
Prove: ∠1 ≅ ∠2
Proof:
Statements Reasons
___________________________________ _____________________________________________ t k 3 1 n 2
See
page 78!
k || n
Given
∠1 ≅ ∠ 3
Vert. ∠’s R ≅
∠3 ≅ ∠ 2
|| - lines ⇒ Corr. ∠’s ≅
∠1 ≅ ∠ 2
Transitive Property

6. Theorem 3-3 || - lines ⇒ SSIA Supp.
If two parallel lines are cut by a transversal, then same side interior angles are supplementary.
|| - lines ⇒ SSIA Supp.
Given:
k || n
transversal t cuts k and n
Prove:
∠1 is supplementary to∠ 4 t k 1 n 4 2

7. k || n Given || - lines ⇒ AIA ≅ ∠1 ≅ ∠ 2 Def. of ≅ ∠’st
Given: k || n
transversal t cuts k and n
Prove: ∠1 is supplementary to ∠ 4
Proof: k 1 n 4 2
Statements Reasons
____________________________________ _____________________________________________
k || n
Given
∠1 ≅ ∠ 2
|| - lines ⇒ AIA ≅
m∠1 = m ∠2
Def. of ≅ ∠’s
m∠2 + m ∠4 = 180º
Angle Addition Postulate
m∠1 + m ∠ 4 = 180º
Substitution Prop.
Def. of Supp. ∠’s
∠1 is supplementary to ∠ 4

8. This proof is part of your assignment.
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
t ⊥ l ; l || n
Prove: t ⊥ n l 1 n 2 t
This proof is part of your assignment.
See page 81 #13 for help.

9. l || n m∠2 = 90º Given Def. of ⊥ lines Def. of ⊥ lines
Given: transversal t cuts l and n
t ⊥ l ; l || n
Prove: t ⊥ n
Proof: 1 l n 2 t
Statements Reasons
t ⊥ l _____________________________________________
m ∠ 1 = 90º _____________________________________________
_____________ __________ Given
∠1 ≅ ∠2 or m∠1 = m∠2 _____________________________________________
________________________ Substitution Property
t ⊥ n _____________________________________________
Given
Def. of ⊥ lines
l || n
|| - lines ⇒ Corr. ∠’s ≅
m∠2 = 90º
Def. of ⊥ lines

10. If two parallel lines are cut by a transversal, then …
corresponding angles
are congruent .
alternate interior angles
same-side interior angles
are supplementary .

11. Example #1. If m ∠1 = 120º, x || y, and m || n, then find all the other measures.
WHY?
120º
m ∠2 = ______
m ∠3 = ______
m ∠4 = ______
m ∠5 = ______
m ∠6 = ______ x
120º
120º
60º 6 4 5
120º
120º y
120º 2
120º 1 3
60º
120º
120º m n

12. Example #2. Find the values of x, y, and z.
105º
75º 35
x = ______
y = ______
z = ______
(3x)º
(5y)º
105º
z º 15
75º
x || y 75 x y
3 x = 105
5 y = 180
z = 75
WHY?
x = 35
WHY?
5 y = 75
WHY?
y = 15

13. Example #3. Find the values of x, y, and z.13
32º
x = ______
y = ______
z = ______
(4x)º
52º 8
52º
96º
32º 96
z º
(3y +8)º
4 x = 52
3 y + 8 = 32
z = 180
WHY?
WHY?
WHY?
x = 13
3 y = 24
z + 84 = 180
y = 8
z = 96

14. How can you apply parallel lines (planes) to make deductions?
Assignment
Written Exercises on pages 80 to 82
DO NOW: 1 to 5 odd numbers
GRADED: 7, 9, 11, 15, & 17
NOTE: #13 was completed in your notes.
Prepare for a quiz on
Lesson 3-2: Properties of Parallel Lines
How can you apply parallel lines (planes) to make deductions?