parallel and perpendicular lines

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


Geometry CH 3 Parallel & Perpendicular Lines MENU

In this section we are going to learn about lines that are…PARALLEL, PERPENDICULAR, INTERSECTING and

SKEW.



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



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



A special case of intersecting lines are Perpendicular lines:



Parallel lines are always the same distance apart.



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.



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



INTERSECTING

PARALLEL?

Remember, they LOOK parallel, but we don’t know for sure unless we measure or are told.

3. k and j


k

j

f



PARALLEL



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?



These lines will not touch, because they are in different planes

Parallel lines must be in the same plane.



These are SKEW lines



SKEW LINES are lines that do not touch…



NOT INTERSECT…

NOT INTERSECT…

…not because they are parallel… …but because they are not in the same plane.SKEW LINES



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.



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,

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


For each of the following problems, find X.

9012 xx90122 x1212 1022 x

22 51x

1. 2. 3. 4.



12xx 107 x

x 153 y ZZ

Z


51x

90107 xx90108 x808 x

10x

1. 2. 3. 4.



12xx 107 x

x 153 y ZZ

Z


51x

10x

90153 y753 y25y

1. 2. 3. 4.



12xx 107 x

x 153 y ZZ

Z


51x

10x25y

90 zzz903 z30z

1. 2. 3. 4.



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


These two lines crossed by a TRANSVERSAL create 8 angles.



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



123 4

5 67 8

Which angles are INTERIOR ANGLES?3, 4, 5, 6

The angles OUTSIDE the two lines are called EXTERIOR



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



TRANSVERSAL



This line is called the…



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…



123 4

5 67 8

SAME-SIDE

When we are talking about angles on alternate sides of the transversal, we call them…



123 4

5 67 8

ALTERNATE



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



123

45

6

7

8

CORRESPONDING ANGLES are at the same spot at the other intersection



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



3 and 7 and

1 and 5864



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…

a bd c

e fh g

1. a&h2. d&f3. c&f4. b&h5. c&g

Same side exteriorAlternate InteriorSame side InteriorAlternate ExteriorCorresponding



State the relationship between the angles given.

6. b and e ??

a bd c

e

wy

xz j k

mp

7. f & b

fgh



What if we get a more complicated picture like this?

CORRESPONDING

a bd c

e

wy

xz j k

mp

8. c & j

fgh



If more than 3 lines end up highlighted, the angles have NO

RELATIONSHIP,



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


A slash through the parallel symbol || indicates the lines are not parallel.

AB || CDA

D

B

C



SIDE NOTE:



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.



h

a600 bc d

e fg



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…

ha

75b

c de fg



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!



Find the X, Y and Z

45



X ZY135 13545

20

Find all the missing angles

160160 20

2020 160

160



Find all the missing angles

1100700



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.




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.




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.




THEOREM Perpendicular Transversal

j k

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.



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



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°



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



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




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 ?




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

Actually, they are only very “close” to parallel.

You need:a sheet of lined papera rulera protractora pencil



ACTIVITY: Measurements Using Lined paper

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




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?




Showing Lines are Parallel


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:



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

http://rds.yahoo.com/_ylt=A0WTefURkjRM3D8ABt.jzbkF/SIG=12bkg0uro/EXP=1278600081/**http:/www.lafilastrocca.it/contents/media/converse.jpg



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.



A BI-CONDITIONAL statement is one in which both the original statement and the converse are true.

STATEMENT: IF______________, THEN ______________.

CONVERSE: IF ______________, THEN _____________.


All of these rules are BI-CONDITIONAL


IF the lines are parallel, Then you get this pattern

6060

6060

120120

120120



6060

6060

120120

120120

IF you get this pattern, Then the 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:



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:



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:



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



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



Are the lines parallel?

62

YES

62

How do you know?Because alt. int. are congruent


m

n

m || n by alt.int converse.


100

100


m

n

m || n by Alternate Exterior converse

By what theorem are these lines parallel?



m

n

m || n by Corresponding angles converse




m

n

m || n by S.S. Interior converse


100

80



m

n

m || n by S.S. Exterior converse


100

80


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.



Using ┴ and ‖ Lines


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



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



If two lines are // to the same line, then they are // to each other.

k

m

n

t


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



APPENDIX


OPENERSASSIGNMENTSEXTRA PROBLEMS

APPENDIX

Problems


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


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


PARALLEL

Intersecting

SKEW

PARALLEL

SKEW

APPENDIX

A

BC

D

Problems 3.1


6. Are lines AB and CD parallel, perpendicular or

skew?SKEW

APPENDIX

Problems 3.1


7. Name the plane parallel to plane CDE

PLANE ABG (or any combination of ABGH)

APPENDIX

Problems 3.1


8. What is the intersection of planes FCG and CDE?

Line CF

APPENDIX

Problems 3.1


9. Which lines are parallel to FC?

Line ED

Line HA

Line BG

APPENDIX

Problems 3.1


10. Which segments intersect FE?

Line FG

Line FC

Line EH

Line ED

APPENDIX

Problems 3.1


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


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


36°?°

Problems 3.2


16. Find the missing angle:

9036 x54x

APPENDIX

3x°2x°

Problems 3.2


17. Solve for X:

9032 xx905 x18x

APPENDIX

2x + 5x + 25

Problems 3.2


18. Solve for X:905225 xx90303 x603 x20x

APPENDIX

?° 168°

Problems 3.2


19. Find the missing angle:

180168 x12x

APPENDIX

2x + 10 3x + 20

Problems 3.2


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


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


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


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


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


138°

42°138°

42° 138°

42°138°

APPENDIX

<

<

6. Find the value of the variable.

<<

70°

g

Problems 3.4


converseangleingcorrespondbyg 70

APPENDIX

a

b

cx + 20

36°

7. Find x.x + 20 = 36 - 20 - 20 x = 16

Problems 3.4


<<

130°

2w – 10

Problems 3.4


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


<

<

(3w – 9)

(2w + 4)

<

<

Problems 3.4


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


<

<

y

14 + 3*40

134

Problems 3.4


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


b =40 °

d =75 °

APPENDIX

14. Challenge Problem: Solve for the variables

4x 6x + y x + 5y

> >

Problems 3.4


18046 xyx18010 yx

yxyx 56

xy 10180

yyx 55 yx 45

)10180(45 xx xx 407205

72045 x72045 x16x20y

APPENDIX

Problems 3.5


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


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


Can you prove that lines a and b are parallel? why or why not?

Yes; Alternate Exterior Angles Converse.

ANSWER

Problems 3.5


Yes; Corresponding Angles Converse.

ANSWER

Problems 3.5



APPENDIX

No; Alt. Int. angles must be congruent, not supplementary

ANSWER

m 1 + m 2 = 180°

Problems 3.5



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


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


parallel and perpendicular lines

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