parallel and perpendicular lines
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PARALLEL and PERPENDICULAR LINES
Geometry CH 3
Relationships Between Lines
Theorems about Perpendicular Lines
Angles formed by Transversals
Parallel Lines and Transversals
Showing lines are Parallel
Using and // Lines ┴
APPENDIX
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
In this section we are going to learn about lines that are…PARALLEL, PERPENDICULAR, INTERSECTING and
SKEW.
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Intersecting lines taped to the wall
Intersecting lines taped to the ceiling
Intersecting Lines:
Any lines that “touch” are intersecting
Even though you can’t see where these two lines touch, we know that eventually they will, so they are still intersecting
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Intersecting Lines:
Perpendicular lines are lines that make right angles.
And if 1 angle is a right angle, thenAll the angles are right angles sm
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
A special case of intersecting lines are Perpendicular lines:
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Parallel lines are always the same distance apart.
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
PARALLEL LINES are lines in the same plane that do NOT INTERSECT
We usually draw them like this
And include these “arrow tick marks” to indicate they are parallel.
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
m
np
s
Which 2 lines appear to be parallel? Lines m and p.
m // p.
1. a and c
2. a and b
Are the following lines parallel, intersecting or skew?
a
b
c
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
INTERSECTING
PARALLEL?
Remember, they LOOK parallel, but we don’t know for sure unless we measure or are told.
3. k and j
Are the following lines parallel, intersecting or skew?
k
j
f
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
PARALLEL
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Representations of PARALLEL LINES. This one has all the angles numbered
3-D
This is the room you are in right now
(teacher) Can you see 2 lines that don’t touch, but aren’t parallel?
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
These lines will not touch, because they are in different planes
Parallel lines must be in the same plane.
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
These are SKEW lines
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
SKEW LINES are lines that do not touch…
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
NOT INTERSECT…
NOT INTERSECT…
…not because they are parallel… …but because they are not in the same plane.SKEW LINES
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Sometimes lines are parallel, but you cannot see the plane they are onAre Lines M and L parallel?
They look like they might be, but they are not in the same plane.
The rule about being in the same plane includes …
creating a plane
L and M are parallel because they don’t intersect, and are in the same plane, even though we couldn’t see the plane.
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Just like lines, there are PARALLEL PLANES
PARALLEL PLANESAre flat surfaces that go on forever and never touch.
Like the ceiling and floor in a room that goes on forever,
Relationships Between Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Perpendicular line theorems:All right angles are congruent.
Anytime you see 2 right angles, they are the same shape
If 2 lines are perpendicular, then they form 4 right angles.When you see 1 right angle, you know there are 4 right angles
A BBA
Theorems About Perpendicular Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
For each of the following problems, find X.
9012 xx90122 x1212 1022 x
22 51x
1. 2. 3. 4.
Theorems About Perpendicular Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
12xx 107 x
x 153 y ZZ
Z
For each of the following problems, find X.
51x
90107 xx90108 x808 x
10x
1. 2. 3. 4.
Theorems About Perpendicular Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
12xx 107 x
x 153 y ZZ
Z
For each of the following problems, find X.
51x
10x
90153 y753 y25y
1. 2. 3. 4.
Theorems About Perpendicular Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
12xx 107 x
x 153 y ZZ
Z
For each of the following problems, find X.
51x
10x25y
90 zzz903 z30z
1. 2. 3. 4.
Theorems About Perpendicular Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
12xx 107 x
x 153 y ZZ
Z
When one line crosses two or more lines we call it a TRANSVERSAL
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
These two lines crossed by a TRANSVERSAL create 8 angles.
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
123 4
5 67 8The order of the angles is not important.
What is important is the relationship between the angles
The angles BETWEEN the two lines are called INTERIOR
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
123 4
5 67 8
Which angles are INTERIOR ANGLES?3, 4, 5, 6
The angles OUTSIDE the two lines are called EXTERIOR
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
123 4
5 67 8
Which angles are EXTERIOR ANGLES?1,2,7,8
These are INTERIOR angles
These are EXTERIOR angles
These are EXTERIOR angles
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
TRANSVERSAL
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
This line is called the…
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Which line is the transversal?#1 #2
#3 #4
A
BC
K
L M
X W
HE
R
S
When we are talking about angles on the same side of the transversal, we call them…
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
123 4
5 67 8
SAME-SIDE
When we are talking about angles on alternate sides of the transversal, we call them…
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
123 4
5 67 8
ALTERNATE
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Are they on the SAME SIDE or ALTERNATE sides of the transversal?
123 4
567 8
Alternate Interior angles:Alternate Exterior angles:
Same side Interior angles:
Same side Exterior angles:
3&5 4&6 1&7 2&8
3&6 4&5 2&7 1&8
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
123
45
6
7
8
CORRESPONDING ANGLES are at the same spot at the other intersection
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
The northwest corner of this intersection
The northwest corner of this intersection
These are CORRESPONDING
ANGLES
123
45
6
7
8
Corresponding angles:
2 and
CORRESPONDING ANGLES are at the same spot at the other intersection
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
3 and 7 and
1 and 5864
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
What is the relationship between…What is the relationship between…What is the relationship between…What is the relationship between…What is the relationship between…
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
a bd c
e fh g
1. a&h2. d&f3. c&f4. b&h5. c&g
Same side exteriorAlternate InteriorSame side InteriorAlternate ExteriorCorresponding
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
State the relationship between the angles given.
6. b and e ??
a bd c
e
wy
xz j k
mp
7. f & b
fgh
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
What if we get a more complicated picture like this?
CORRESPONDING
a bd c
e
wy
xz j k
mp
8. c & j
fgh
Angles Formed by Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
If more than 3 lines end up highlighted, the angles have NO
RELATIONSHIP,
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
In this next section, we are going to continue focusing on this picture
But, we are going to make a small change:
We are going to see what happens when these lines are
parallel.
Activities: Measurements Using Lined Paper, Eratosthenes’ Measurement
Parallel Lines and Transversals
A slash through the parallel symbol || indicates the lines are not parallel.
AB || CDA
D
B
C
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
SIDE NOTE:
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Watch what happens to corresponding angles when the lines are parallel.
Click on the picture to see.
POSTULATE
POSTULATE Corresponding Angles Postulate
1
2
1 2
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
h
a600 bc d
e fg
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
So if we start with just this information…And we know the measure of 1 angle
1200
1200 600
By Vertical Angles
By CORRESPONDING ANGLES, each of the angles at the other intersection
have the same measure of their matching angles above
1200
600 1200
600These 2 are supplemen
ts
Then by vertical angles, we know this
one is…
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ha
75b
c de fg
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
So, whenever we have this picture…
10575105
10575105
75
If we know 1 angle…
We know them all !!☻
One shortcut people use to figure this out is:
All the OBTUSE angles have 1 measure,
all the ACUTE angles have the other.
a bc d
e fg h
All the small angles are congruent
All the big angles are congruent
A big one plus a little one
adds to 180!
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Find the X, Y and Z
45
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
X ZY135 13545
20
Find all the missing angles
160160 20
2020 160
160
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Find all the missing angles
1100700
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
1100 700
700 1100
7001100 1100
700 1100
700
7001100700 1100
THEOREMS ABOUT PARALLEL LINES
THEOREM Alternate Interior Angles Theorem
3
4
3 4
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
THEOREMS ABOUT PARALLEL LINES
THEOREM S.S.-Int AnglesTheorem
5
6
m 5 + m 6 = 180°
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
THEOREMS ABOUT PARALLEL LINES
THEOREM Alternate Exterior Angles Theorem
7
8
7 8
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
THEOREMS ABOUT PARALLEL LINES
THEOREM Perpendicular Transversal
j k
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Prove the Alternate Interior Angles Theorem:SOLUTION
GIVEN p || q
p || q Given
Statements Reasons
1
2
3
4
PROVE 1 2
1 3 Corresponding Angles Postulate
3 2 Vertical Angles Theorem
1 2 Transitive property of Congruence
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Use properties of parallel lines to find the value of x.
SOLUTION
Corresponding Angles Postulatem 4 = 125°
Linear Pair Postulatem 4 + (x + 15)° = 180°
Substitute.125° + (x + 15)° = 180°
Subtract.x = 40°
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
Over 2000 years ago Eratosthenes estimated Earth’s circumference by using the fact that the Sun’s rays are parallel.
When the Sun shone exactly down a vertical well in Syene, he measured the angle the Sun’s rays made with a vertical stick in Alexandria. He discovered that
m 2 1
50 of a circle
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ACTIVITY: Eratosthenes’s Measurement
m 2 1
50 of a circle
Using properties of parallel lines, he knew that m 1 = m 2
He reasoned that
m 1 1
50 of a circle
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ACTIVITY: Eratosthenes’s Measurement
The distance from Syene to Alexandria was believed to be 575 miles
m 1 1
50 of a circle
Earth’s circumference
150 of a circle 575 miles
Earth’s circumference 50(575 miles)
Use cross product property
29,000 miles
How did Eratosthenes know that m 1 = m 2 ?
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ACTIVITY: Eratosthenes’s Measurement
How did Eratosthenes know that m 1 = m 2 ?
SOLUTION
Angles 1 and 2 are alternate interior angles, so
1 2
By the definition of congruent angles,
m 1 = m 2
Because the Sun’s rays are parallel,
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ACTIVITY: Eratosthenes’s Measurement
parallel
Actually, they are only very “close” to parallel.
You need:a sheet of lined papera rulera protractora pencil
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ACTIVITY: Measurements Using Lined paper
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
On your paper, draw two lines that are not parallel.
Next draw a line that touches both of them (called a transversal)
Measure the angle at the top right of each intersection (called corresponding angles)
50
47
From those 2 angles find the measures of every other angle in the picture.
133133
47
50130
130
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ACTIVITY: Measurements Using Lined paper
Next follow the exact same instructions, except:
For the first two lines you draw, trace two of the blue lines on your paper
1. What is different about your results?
2. Describe the pattern you see?
3. What is it about the way you drew your lines that makes this happen?
4. If you drew in a different transversal, how would you picture (and angle measures be different?)
5. How would they be the same?
Parallel Lines and Transversals
Geometry CH 3 Parallel & Perpendicular Lines MENU
ACTIVITY: Measurements Using Lined paper
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
So far, we’ve studied basic conditional (if-then) statements:
IF it rains, THEN you need an umbrella.IF she has small feet THEN she will need small shoes.IF the angle is more than 900, THEN it is obtuse.
Next, we will study converse and bi-conditional statements:
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
The CONVERSE of an if-then statement is the statement formed by switching the hypothesis and conclusionSTATEMENT: IF it is raining, THEN you need an umbrella.CONVERSE: IF you need an umbrella, THEN it is raining
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
A BI-CONDITIONAL statement is one in which both the original statement and the converse are true.
STATEMENT: IF today is Tuesday, THEN yesterday was
Monday.CONVERSE:
IF yesterday was Monday, THEN today is Tuesday.
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
A BI-CONDITIONAL statement is one in which both the original statement and the converse are true.
STATEMENT: IF______________, THEN ______________.
CONVERSE: IF ______________, THEN _____________.
Geometry CH 3 Parallel & Perpendicular Lines MENU
All of these rules are BI-CONDITIONAL
Showing Lines are Parallel
IF the lines are parallel, Then you get this pattern
6060
6060
120120
120120
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
6060
6060
120120
120120
IF you get this pattern, Then the lines are parallel.
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
IF the lines are parallel, THEN corresponding angles are congruent
Corresponding Angle Postulate:
IF corresponding angles are congruent, THEN the lines are parallel.
Corresponding Angle CONVERSE:
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
IF the lines are parallel, THEN alt-int. angles are congruentAlt-Int Angle Postulate:
IF alt-int. angles are congruent, THEN the lines are parallel.Alt Int Angle CONVERSE:
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
IF the lines are parallel, THEN alt-ext. angles are congruentAlt-Ext Angle Postulate:
IF alt-ext. angles are congruent, THEN the lines are parallel.Alt Ext CONVERSE:
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
IF the lines are parallel, THEN ss-int. angles are supplementsSS-Int Angle Postulate:
IF ss-int. angles are supplements, THEN the lines are parallel.SS Int CONVERSE:
IF The lines are parallel . . . THEN Alt. Int. are congruent
bad c
e fgh
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
Alt. Ext. are congruentCorr. Are congruentS.S. Int. are supplementsS.S. Ext. are supplements
eandclike handblike gandclike
fandclike gandblike
THEN The lines are parallel.
IF ANY of these are true…Alt. Int. are congruentAlt. Ext. are congruentCorr. Are congruentS.S. Int. are supplementsS.S. Ext. are supplements
bad c
e fgh
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
Are the lines parallel?
62
YES
62
How do you know?Because alt. int. are congruent
Geometry CH 3 Parallel & Perpendicular Lines MENU
m
n
m || n by alt.int converse.
Showing Lines are Parallel
100
100
Geometry CH 3 Parallel & Perpendicular Lines MENU
m
n
m || n by Alternate Exterior converse
By what theorem are these lines parallel?
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
m
n
m || n by Corresponding angles converse
By what theorem are these lines parallel?
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
m
n
m || n by S.S. Interior converse
By what theorem are these lines parallel?
100
80
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
m
n
m || n by S.S. Exterior converse
By what theorem are these lines parallel?
100
80
Showing Lines are Parallel
IF any of these things are true, THEN they are ALL true.Alternate Interior angles are congruent
Alternate Exterior angles are congruent
Corresponding angles are congruent
Same side interior angles are supplementary
Same side exterior angles are supplementary
The lines are parallel.
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
Using ┴ and ‖ Lines
Geometry CH 3 Parallel & Perpendicular Lines MENU
Which lines must be parallel in this picture?If you can’t tell right away, trace the angles to find their relationship.
These are Corresponding anglesM||n by corresponding angle converse
APPENDIX
Proof:Given: 3 6
Prove: n // m
12
3 4
5 6
7 8
Statements
1. 3 6
2. 3 1
3. 1 6
4. n // m
Reasons1.Given
2. Vertical s are
3. Substitution
4. corresponding converse
nn
m
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
Proof:Given: m3 + m5 = 180
Prove: n // m
3
57
Statements
1. m3 + m5 = 180
2. m5 + m7 = 180
3. m3 + m5 = m5 + m7
4. m3 = m7
5. 3 7
6. n // m
n
m
Reason1. Given
2. Add. Post.
3. Subs.
4. Subtraction.
5. Def. of
6. Corresponding angle converse
Geometry CH 3 Parallel & Perpendicular Lines MENU
Showing Lines are Parallel
If two lines are // to the same line, then they are // to each other.
k
m
n
t
Using ┴ and ‖ Lines
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
k // m
n // m
Then these 2 are parallel
In a plane, if 2 lines are perpendicular to the same line, then they are // to each other.
r
s
t
Using ┴ and ‖ Lines
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
Using ┴ and ‖ Lines
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
APPENDIX
Geometry CH 3 Parallel & Perpendicular Lines MENU
OPENERSASSIGNMENTSEXTRA PROBLEMS
APPENDIX
Problems
Geometry CH 3 Parallel & Perpendicular Lines MENU
3.1 Relationships between lines3.2 Theorems about perpendicular lines3.3 Angles formed by transversals3.4 Parallel lines and transversals3.5 Showing lines are parallel3.6 Using parallel and perpendicular lines“Tough” angle problems
APPENDIX
Are the following lines parallel, intersecting or skew?
a b k J c d
e f
g h
1. ab and Jk
2. bk and hk
3. ec and gh
4. eg and fh
5. kh and af
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENU
PARALLEL
Intersecting
SKEW
PARALLEL
SKEW
APPENDIX
A
BC
D
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENU
6. Are lines AB and CD parallel, perpendicular or
skew?SKEW
APPENDIX
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENU
7. Name the plane parallel to plane CDE
PLANE ABG (or any combination of ABGH)
APPENDIX
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENU
8. What is the intersection of planes FCG and CDE?
Line CF
APPENDIX
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENU
9. Which lines are parallel to FC?
Line ED
Line HA
Line BG
APPENDIX
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENU
10. Which segments intersect FE?
Line FG
Line FC
Line EH
Line ED
APPENDIX
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENU
11. Which segments are skew to BC?
Line HA
Line ED
Line HG
Line EF
APPENDIX
M N
R
ST
OPWhat kind of lines are12. MN and PO?
13. RS and TS
14. RS and PO?
parallel
intersecting
skew
Problems 3.1
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
15. If angle 1 is 90,
21
4 3
What are the measures of
Angle 2?
Angle 3?
Angle 4?
90
90
90
Problems 3.2
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
36°?°
Problems 3.2
Geometry CH 3 Parallel & Perpendicular Lines MENU
16. Find the missing angle:
9036 x54x
APPENDIX
3x°2x°
Problems 3.2
Geometry CH 3 Parallel & Perpendicular Lines MENU
17. Solve for X:
9032 xx905 x18x
APPENDIX
2x + 5x + 25
Problems 3.2
Geometry CH 3 Parallel & Perpendicular Lines MENU
18. Solve for X:905225 xx90303 x603 x20x
APPENDIX
?° 168°
Problems 3.2
Geometry CH 3 Parallel & Perpendicular Lines MENU
19. Find the missing angle:
180168 x12x
APPENDIX
2x + 10 3x + 20
Problems 3.2
Geometry CH 3 Parallel & Perpendicular Lines MENU
20. Solve for X:
180203102 xx80305 x1505 x30x
APPENDIX
Identifying Angles -
t
kj
12
34
56
78
Corresponding angles are on the corresponding side of the two lines and on the same side of the transversal.Name all the pairs of Corresponding angles.
1 5,3 7,2 6,4 8
andandand orand
Problems 3.3
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
ba
e
dc
fg h
What is the relationship between angles
1. a & h?2. d & f?3. c & f?4. b & h?5. b & g?6. a & f?7. a & g?8. d & h?
Alt extAlt intSS intSS extalt extcorrSS extcorr
Problems 3.3
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
l || m
l
m
t108° 72°
108°
108°
108°
72°
72°
72°
What is the measure of each of the other 7 angles (follow the dancing arrow)?
? ???
? ??
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
In the figure a || b.
1. Name the angles congruent to 3.
2. Name all the angles supplementary to 6.
3. If m1 = 105° what is m3?
4. If m5 = 120° what is m2?
1, 5, 7
1, 3, 5, 7
105°
60°
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
5. Lines q and m are parallel. q||m Find the missing angles.
42°
q
m
b°
d°
f°
a °
c°
e°
g°
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENU
138°
42°138°
42° 138°
42°138°
APPENDIX
<
<
6. Find the value of the variable.
<<
70°
g
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENU
converseangleingcorrespondbyg 70
APPENDIX
a
b
cx + 20
36°
7. Find x.x + 20 = 36 - 20 - 20 x = 16
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
<<
130°
2w – 10
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENU
8. Find the value of the variable.130102 w1402 w70w
APPENDIX
9. Find x
62
7x - 8
7x – 8 + 62 = 180
7x + 54 = 180
7x = 126
x = 18
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
<
<
(3w – 9)
(2w + 4)
<
<
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENU
10. Find the value of the variable.1809342 ww18055 w1855 w37w
APPENDIX
11. Solve for x and y.
14 + 3x
5x - 66
n
m
14 + 3x = 5x -66
-3x -3x
14 = 2x – 66
+66 +66
80 = 2x
2 2
40 = x
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
<
<
y
14 + 3*40
134
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENU
203 x
102 x
180102203 xx180305 x1505 x30x
APPENDIX
12. Find x.
13. Find the missing angles:
70 °b°
70 °
d ° 65 °
Hint: The 3 angles in a triangle sum to 180°.
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENU
b =40 °
d =75 °
APPENDIX
14. Challenge Problem: Solve for the variables
4x 6x + y x + 5y
> >
Problems 3.4
Geometry CH 3 Parallel & Perpendicular Lines MENU
18046 xyx18010 yx
yxyx 56
xy 10180
yyx 55 yx 45
)10180(45 xx xx 407205
72045 x72045 x16x20y
APPENDIX
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
A) Are the lines parallel?B) If yes… what theorem says so?
YES Alt. Ext. CONVERSEYES SS-Int CONVERSE
NO
No
Find the value of x that makes m n.
SOLUTIONLines m and n are parallel if the corresponding angles are congruent.
3x = 60
x = 20
The lines m and n are parallel when x = 20.
(3x + 5)o = 65o
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
Is there enough information in the diagram to conclude that m // n?
ANSWERYes. m // n because the S.S. Ext. converse says that if same side exterior angles are supplementary, then the lines are parallel.
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
Can you prove that lines a and b are parallel? why or why not?
Yes; Alternate Exterior Angles Converse.
ANSWER
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
Yes; Corresponding Angles Converse.
ANSWER
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENU
Can you prove that lines a and b are parallel? why or why not?
APPENDIX
No; Alt. Int. angles must be congruent, not supplementary
ANSWER
m 1 + m 2 = 180°
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENU
Can you prove that lines a and b are parallel? why or why not?
APPENDIX
SOLUTION
GIVEN : 4 5
PROVE : g h
Prove that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
1. Given
g h4.
1. 4 5
2. 1 4
3. 1 5
2. Vertical Angles Congruence Theorem
3. Transitive Property of Congruence
4. Corresponding Angles Converse
STATEMENTS REASONS
Problems 3.5
Geometry CH 3 Parallel & Perpendicular Lines MENUAPPENDIX
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