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Unit 7: Parallel and
Perpendicular Lines
You will:
•Learn theorems about parallel and
perpendicular lines
Resource: Prentice Hall
3.2: Properties of Parallel
Lines
Lines are parallel when
they are
• coplanar
• and they do not
intersect
A line that crosses two
parallel lines is called
a transversal.
Text Resource: Prentice Hall
The ANGLES created by the intersection of a
transversal with parallel lines have some
interesting properties:
There are several angle pairs created
by this intersection.
Corresponding Angles
Alternate Interior Angles
Same Side Interior Angles
4
3
7
2
1
5 6
Resource: Prentice Hall
8
Use the diagram above.
•Identify which angle forms a pair of same-side interior angles with 1.
•Identify which angle forms a pair of corresponding angles with 1.
Same-side interior angles are on the same side of transversal t between lines p
and q.
4, 8, and 5 are on the same side of the transversal as
but only 1 and 8 are interior.
So
1 and
8 are same-side interior angles.
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
1,
Properties or Parallel Lines
GEOMETRY LESSON 3-1
(continued)
Corresponding angles also lie on the same side of the transversal.
One angle must be an interior angle, and the other must be an exterior angle.
The angle corresponding to 1 must lie in the same position relative to
line q as 1 lies relative to line p. Because 1 is an interior angle, 1 and
are corresponding angles.
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
5
Properties or Parallel Lines
GEOMETRY LESSON 3-1
x
Compare 2 and the vertical angle of 1. Classify the angles
as alternate interior angles, same–side interior angles, or
corresponding angles.
The vertical angle of
1 is between the parallel runway segments.
2 is between the runway segments and on the opposite side of
the transversal runway.
Because alternate interior angles are not adjacent and lie between
the lines on opposite sides of the transversal, 2 and the vertical angle
of 1 are alternate interior angles.
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
Corresponding Angle
Postulate:
If a transversal intersects two parallel lines, then
corresponding angles are congruent.
a
2
1
b
Text Resource: Prentice Hall
Saint Agnes Academy
Use the Corresponding Angles Postulate to
prove 1  3
a
2
Given: a || b
Prove: 1  3
3
b
1
Statement
Reason
1. a || b
1. Given
2. 1  2
2. Corr. s Postulate
3. 3  2
4. 1  3
3. Vertical s Thm (2-1)
4. Transitive Property
(ch 2)
Text Resource: Prentice Hall
Saint Agnes Academy
t
You have just proved the Alternate
Interior Angles Theorem
If a transversal intersects two parallel lines, then
alternate interior angles are congruent.
a
3
1
b
Text Resource: Prentice Hall
Saint Agnes Academy
Use the Alternate Interior Angles Theorem to
prove m1 + m4 = 180
a
Given: a || b
Prove: m1 + m4 = 180
3 4
b
1
Statement
Reason
1. a || b
1. Given
2. 1  3
3. m1 = m 3
2. Alt Int s Thm
3. Def of congruent
4. m3 + m 4 = 180
4.  Add’n Postulate (1-8)
5. m1+ m 4 = 180
5. Substitution
t
You have just proved the Same Side
Interior Angles Theorem
If a transversal intersects two parallel lines, then
same-side interior angles are supplementary.
a
4
1
b
Text Resource: Prentice Hall
Saint Agnes Academy
Which theorem or postulate gives the reason that
m  3 + m  2 = 180?
3 and
m
3+m
2 are adjacent angles that form a straight angle.
2 = 180 because of the Angle Addition Postulate.
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
In the diagram above, l || m. Find m  1 and then m  2.
m  1 = 42
m  1 + m  2 = 180
Corr s Postulate
 Addition Postulate
42 + m  2 = 180
Substitution
m  2 = 138
Subtraction Property of
Equality.
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
In the diagram above,
|| m. Find the values of a, b, and c.
a = 65
c = 40
Alternate Int s Thm
Alternate Int s Thm
a + b + c = 180
Angle Add’n Postulate
65 + b + 40 = 180
Substitution
b + 105= 180
Additon
b = 75
Subtraction Prop of =
3-1
Homework 3.1
page 118
Due at the beginning of the next class.
Name
Section #
Page #
Remember the
honor code.
No Copying!
Show your work
here IN PENCIL
I pledge that I have neither
given nor received aid on
this assignment
Text Resource: Prentice Hall
Saint Agnes Academy
Check in ink!
GEOMETRY LESSON 3-1
Pages 118-121 Exercises
1. PQ and SR with
transversal SQ;
alt. int. s
2. PS and QR with
transversal SQ;
alt. int. s
3. PS and QR with
transversal PQ;
same-side int. s
4. PS and QR with
transversal SR;
corr. s
5.
6.
7.
1 and 2: corr. s
3 and 4: alt. int.
5 and 6: corr. s
8. alt. int.
s
1 and 2: same-side
int. s
3 and 4: corr. s
5 and 6: corr. s
1 and 2: corr. s
3 and 4: same-side
int. s
5 and 6: alt. int.
s
9. a. 2
b. 1
c. corr.
10. a. Def. of
b. Def. of right
c. Corr. s of i lines are
d. Subst.
e. Def. of right
f. Def. of
s
Text Resource: Prentice Hall
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Saint Agnes
.
GEOMETRY LESSON 3-1
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11. m 1 = 75 because
14. 70; 70, 110
corr. s of || lines are ;
m 2 = 105 because
15. 25; 65
same-side int. s of ||
lines are suppl.
16. 20; 100, 80
12. m 1 = 120 because
17. m 1 = m 3 = m 6 =
corr. s of || lines are ;
m 8 = m 9 = m 11
m 2 = 60 because
= m 13 = m 15 = 52;
same-side int. s of ||
m 2=m 4=m 5=
lines are suppl.
m 7 = m 10 = m 12
= m 14 = 128
13. m 1 = 100 because
same-side int. s of ||
lines are suppl.; m 2 =
70 because alt. int. s of
|| lines have = measure.
18. You must find the
measure of one . All s
that are vert., corr., or
alt. int. to that will have
that measure. All other s
will be the suppl. of that
measure.
19. two
20. four
21. two
22. four
23. 32
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
GEOMETRY LESSON 3-1
24. x = 76, y = 37,
v = 42, w = 25
25. x = 135, y = 45
26. The s labeled are
corr. s and should be
. If you solve
2x – 60 = 60 – 2x, you
get x = 30. This would
be impossible since
2x – 60 and 60 – 2x
would equal 0.
27. Trans means across or
over. A transversal cuts
across other lines.
Check in ink!
28. Answers may vary.
Sample: E illustrates corr. s ( 1 and 3,
2 and 4) and same-side int. s ( 1
and 2, 3 and 4);
I illustrates alt. int. s ( 1 and 4,
2 and 3) and same-side int. s
( 1 and 3, 2 and 4).
29. a. alt. int.
s
b. He knew that alt. int.
s
of || lines are
30. a. 57
b. same-side int.
s
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
.
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GEOMETRY LESSON 3-1
31. a. If two lines are || and cut by a transversal,
then same-side ext. s are suppl.
b. Given: a || b
Prove: 4 and 5 are suppl.
1. a || b (Given)
2. m 5 + m 6 = 180 ( Add. Post.)
3. 4
6 (Corr. s are )
4. m 5 + m 4 = 180 (Subst.)
5.
4 and
5 are suppl. (Def. of suppl.)
32. 1. a || b (Given)
2. 1
2 (Vert.
3. 2
3 (Corr.
4.
1
33. Never; the two planes do
not intersect.
34. Sometimes; if they are ||.
35. Sometimes; they may be
skew.
36. Sometimes; they may be ||.
37. D
38. G
are
s are
s
.)
.)
3 (Trans. Prop.)
39. D
40. I
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
GEOMETRY LESSON 3-1
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41. [2] a. First show that 1
7. Then show
that 7
5. Finally, show that
1
5 (OR other valid solution plan).
b.
1
7 because vert. s are .
7
5 because corr. s of || lines are
. Finally, by the Trans. Prop. of ,
1
5.
42. 121
43. 59
44. 29.5
45. (0.5, 7)
46. (–0.5, 3.5)
[1] incorrect sequence of steps OR incorrect
logical argument
47. (3, 3)
48. add 4; 20, 24
49. multiply by –2; 16, –32
50. subtract 7; –5, –12
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes
GEOMETRY LESSON 3-1
Extra Practice
In the diagram below, m || n. Use the diagram for Exercises 1–5.
1. Complete:
2. Complete:
6
and
and
4 are alternate interior angles.
8 are corresponding angles.
4
3. Suppose that m
3 = 37. Find m
4. Suppose that m
1 = x + 12 and m
6. 143
5 = 3x – 36. Find x. 24
5. If a transversal intersects two parallel lines, then same-side
exterior angles are supplementary. Write a Plan for Proof.
Given: m || n Prove: 2 and 7 are supplementary.
Show that m 2 = m 6. Then show that m 6 + m 7 = 180,
and substitute m 2 for m 6.
Text Resource: Prentice Hall
3-1 Academy
Saint Agnes