unit 3 workbook - leman's...

Geometry – Unit 3 Targets & Info Name: This Unit’s theme – Parallel Lines and Transversals Approximately Sept 27 – Oct 15 Use this sheet as a guide throughout the chapter to see if you are getting the right information in reaching each target listed. By the end of Unit 3, you should know how to…

Target found in…

Did I reach the target?

DIAGRAMS & EXAMPLES!

Identify and use correct vocabulary: Corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles, vertical angles, linear pair, transversal, parallel, perpendicular, slope, y-intercept

Chapter 3

Use angle relationships to find the measures of angles in a diagram

Chapter 3 Section 2, pages 89-95

State if lines are parallel and justify your statement with a postulate or theorem


Find the slope of a line given a graph, two points, or the equation of a line


Write the equation of a line given: a) two points b) a point on the line and the slope c) a point on the line and the equation

of a parallel or perpendicular line

Chapter 3 Sections 5 & 6

Complete a two column proof by providing reasons that justify each given statement

Chapter 3 Section 4 pages 106-112

Complete a blank two column proof using given information and a diagram.

*** You will be allowed to use a sheet with all theorems/postulates from the unit on the test. You do not need to memorize the theorems. ***

Lesson&1:&&Lines&and&Angles&!parallel&lines:!!!lines!that!are!coplanar!and!do!not!intersect!

skew&lines:!!lines!that!are!not!coplanar!

parallel&planes:!!planes!that!do!not!intersect!

!Parallel&Postulate!

! If!there!is!a!line!and!a!point!not!on!the!line,!then!there!is!exactly!one!line!through!the!point!parallel!to!the!given!line.!!

!!Perpendicular&Postulate&!

! If!there!is!a!line!and!a!point!not!on!the!line,!then!there!is!exactly!one!line!through!the!point!perpendicular!to!the!given!line.!

&&transversal:!!a!line!that!intersects!two!or!more!coplanar!lines!at!different!points!!! ! !!!!!!!!!!!!!!corresponding!angles!! ∠1!and!∠5!! ∠2!and!∠6!! ∠3!and!∠7!! ∠4!and!∠8!

alternate!interior!angles!! ∠3!and!∠6!! ∠4!and!∠5!

alternate!exterior!angles!! ∠1!and!∠8!! ∠2!and!∠7!

consecutive!(sameDside)!interior!angles!! ∠3!and!∠5!! ∠4!and!∠6!

l"m"

p"

a"b"

c"

q"

r"s"

1! 2!3! 4!

5! 6!7! 8!

!Name!a!pair!of!corresponding!angles.!!Name!a!pair!of!alternate!interior!angles.!!Name!a!pair!of!consecutive!interior!angles.!!Name!a!pair!of!alternate!exterior!angles.!!!!Tell!which!kind!of!angles!each!of!the!following!are.!!∠1!and!∠3!

∠1!and!∠2!

∠1!and!∠6!

∠1!and!∠8!

∠3!and!∠11!

∠2!and!∠6!

∠2!and!∠7!

∠5!and!∠11!!!!!

Postulate!

! If!two!parallel!lines!are!cut!by!a!transversal,!then!the!corresponding!angles!are!congruent.!

!!!

!!

!

!

!

1! 2!3!4!5!6!7! 8!

7! 8!

1!2!

3!4! 5!

6!9! 10!11!

1! 2!

l" m"

l!||!m"

Given:!!l!||!m"

Prove:!!∠2!≅!∠3!!!!!

!!If!two!parallel!lines!are!cut!by!a!transversal,!then!the!alternate!interior!angles!are!congruent.!!!!!!!!!!!!!!!!!

!

Given:!!l!||!m"

Prove:!!∠1!≅!∠3!!!

!!If!two!parallel!lines!are!cut!by!a!transversal,!then!the!alternate!exterior!angles!are!congruent.!!!!!!!!!!!!!!!

l" m"

1!2! 3!

l" m"

1!2!

3!

l" m"

1! 2! 3!

!!

Given:!!l!||!m"

Prove:!!∠2!and!∠3!are!supplementary!!!!

!!If!two!parallel!lines!are!cut!by!a!transversal,!then!the!consecutive!interior!angles!are!supplementary.!

!

!

!

!

!

!

!

!

!

!

!

Given:!!! l!||!m!

! ! t!⊥!l"

Prove:!!t!⊥!m!!!!If!a!transversal!is!perpendicular!to!one!of!two!parallel!lines,!then!it!is!perpendicular!to!the!other!!!

!!

!!

1!

2!l"

m"

t"

Lesson&1&Practice:&&Lines&and&Angles&!

!Complete!the!following!proof:!!! 1.! Given:!a!!||!!b"

! ! ! ! l"!||!!m!!! ! Prove:!!∠1!≅!∠3!!!! Statements! Reasons!!! 1.! a!!||!!b" 1."

" " l"!||!!m! !!! 2.! ∠1!≅!∠2! 2.!!! 3.! ∠2!≅!∠3! 3.!!! 4.! ∠1!≅!∠3! 4.!!!!! 2.! Given:!r!!||!!s"!! ! Prove:!!∠1!and!∠3!are!supplementary!!!! Statements! Reasons!!! 1.! r!!||!!s" 1.!!! 2.! ∠2!≅!∠3! 2.!! !! 3.! ∠1!and!∠2!are!a!linear!pair! 3.!!! 4.! ∠1!and!∠2!are!supplementary! 4.!!! 5.! m∠1!+!m∠2!=!180°! 5.!!! 6.! m∠2!=!m∠3! 6.!!! 7.! m∠1!+!m∠3!=!180°! 7.!!! 8.! ∠1!and!∠3!are!supplementary! 8.!! !!&

a"

b"

l" m"

1!

2!3!

1! 2!

3!r"

s"

Page&153,&#7=10,&12=20,&22=39&

y40°75°

x

x40°

z

y

70°(2y+10)

12x5z

120°

50°

yx

70°

60°

(3x+2y)(x+4y)

110°

120°

(3y+8)°

x70°

Lesson&2:&&Using&Parallel&Theorems&!

Solve!for!each!variable.!!! 1.! ! 2.!!!!!!!!! ! x!=!__________!!!!y!=!__________! ! x!=!__________!!!!y!=!__________!!!!!!!! 3.! ! 4.!!!!!!!!! ! x!=!__________!!!!y!=!__________! ! x!=!__________!!!!y!=!__________!!! ! z!=!__________! ! z!=!__________!!!!!!!! 5.! ! 6.! !!!!!!!!!! ! x!=!__________!!!!y!=!__________! ! x!=!__________!!!!y!=!__________!!!!!!!!

x30°

40°

x150°

130°

5y2z

x

50°

z

yx

56°C D

B

A

Ey

x 120°

110°

y

x

82°42°

!!! 7.! ! 8.!!!!!!!!! ! Hint:!!Draw!a!third!parallel!line!!! ! x!=!__________! ! x!=!__________!!!!!! 9.! ! 10.! !!!!!!!!! ! x!=!__________!!!!y!=!__________! ! x!=!__________!!!!y!=!__________!!! ! ! ! z!=!__________!!!!!!! 11.! ! 12.! !!!!!!!!!! ! BE!bisects!∠ABD! ! x!=!__________!!!!y!=!__________!!! ! x!=!__________!!!!y!=!__________! ! !!! ! z!=!__________!!

321

A

C

D F

B

E

4

3

21

A

R T

B

S

!Given:! AS!||!BT!

! ∠1!≅!∠2!!Prove:! ∠3!≅!∠4!!!!!!!!!!!Given:! BC!||!DF!

! BC!bisects!∠ABE!!Prove:! ∠1!and!∠3!are!supplements!!!! !

432

1E

A

C D

B

32

1K C

A B

D

Lesson&2&Practice:&&Using&Parallel&Theorems&!

!! 1.! Given:!!! BE!||!CD!

! ! ! ∠2!≅!∠3!!! ! Prove:!!! ∠1!≅!∠4!!!!!! ! Statements! Reasons!!!! 1.! BE!||!CD!! 1.!!! 2.! ∠1!and!∠2!are!supplementary! 2.!!! 3.! ∠3!and!∠4!are!a!linear!pair! 3.!!! 4.! ∠3!and!∠4!are!supplementary! 4.!!! 5.! ∠2!≅!∠3!! 5.!!! 6.! ∠1!≅!∠4!! 6.!!!!!! 2.! Given:!!! DC!||!AB!

! ! ! AK!bisects!∠DAB!!! ! Prove:!! ∠1!≅!∠2!!!! ! Statements! Reasons!!!! !! !!!!!!!!!!

6

5

1

2 34

60°

105°

(3x+11)°(3y+1)°(4x+5)°

x

y

80°44°

(13y-10)°(9x+12)°6y°

x

110° 30°

yx

y

z

40°

!Solve!for!each!variable:!!! !!! ! ! ! !!!!!!! 3.! x!=!__________!!y!=!__________! ! 4.! x!=!__________!!y!=!__________!!! ! ! ! ! ! z!=!__________!!!!!!!!! !!! ! ! ! !!!!!!!!! 5.! m∠1!=!_________!!m∠2!=!_________! ! 6.! x!=!__________!!!!! ! m∠3!=!_________!!m∠4!=!_________! ! ! y!=!__________!!! ! m∠5!=!_________!!m∠6!=!_________!!!!!! ! ! ! ! !!!!!!!! 7.! x!=!_________!!y!=!_________! ! 8.! x!=!__________!!y!=!__________!!!!

145°

110°

x

z

45

y

x80°

35°

32°

35°

x

dc

ba

125°

80°

(3x+8)°130°

3y°

75°

(3x+4y)°

120°

130°

(5x+2y)°

!! ! !!! ! ! ! ! !!!!!!!!! 9.! x!=!_________!!! ! 10.!x!=!__________!!y!=!__________!!! ! ! ! ! ! z!=!__________!! !!!!!!!!!!!! !! 11.! a!=!_________!!b!=!__________! ! 12.!x!=!__________!!!!! ! c!=!_________!!d!=!__________!!!!!! ! ! ! !!!!!!!!!!! 13.! x!=!__________!!y!=!__________! ! 14.!x!=!__________!!y!=!__________!

Lesson&3:&&Proving&Lines&are&Parallel&!

If!two!parallel!lines!are!cut!by!a!transversal,!then!the!corresponding!angles!are!congruent.!!!!State!the!converse.!!

!

!

!

!! ***Also!a!Postulate***!!!! Given!the!following!information,!what!can!you!conclude?!!!!!!!!!!Given:!!∠2!≅!∠3!!"

Prove:!!l!||!m"""""""

If!two!lines!are!cut!by!a!transversal!so!that!the!alternate!interior!angles!are!congruent,!then!the!lines!are!parallel.!

""""!!!

!!!!!

!

!

1! 2!

l" m" ∠1!≅!∠!2!

l" m"

1!

2!

3!

j

k

l 3

21

!If!two!lines!are!cut!by!a!transversal!so!that!the!alternate!exterior!angles!are!congruent,!then!the!lines!are!parallel.!

!

! Given:!!∠1!≅!∠3!!!

! ! What!can!you!prove?!

!!!!

!!!If!two!lines!are!cut!by!a!transversal!so!that!the!consecutive!interior!angles!are!supplementary,!then!the!lines!are!parallel.!!

!! ! Given:!!∠2!and!∠3!are!supplementary!!!! ! What!can!you!prove?!!!!!!!!!!! Given:! j!||!k!! ! k!||!l!!! Prove:!!! j!||!l!!!!!

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

l" m"

1!

2! 3!

l" m"

1! 2! 3!

s

t

u

q r

1514131211

1098

7654321

w

!!In!a!plane!if!two!lines!are!perpendicular!to!the!same!line,!then!they!are!parallel!to!each!other.!!!Given:!m!⊥!p!

n!⊥!p!!

What!can!you!prove?!!

!&SUMMARY&!Name!6!ways!to!prove!lines!are!parallel.!!! 1.!!!! 2.!!!! 3.!!!! 4.!!!! 5.!!!! 6.!!!!Which!lines,!if!any,!can!be!proved!parallel!from!the!given!information?!!(TEST!QUESTION)!!! 1.! ∠1!≅!∠9!

! 2.! ∠5!≅!∠10!

! 3.! ∠7!≅!∠11!

! 4.! ∠12!≅!∠14!

! 5.! ∠6!≅!∠9!

! 6.! s!||!t!and!s!||!u!

! 7.! ∠2!≅!∠12!

! 8.! m∠13!+!m∠14!=!180°!

mn

p

432

1A

B C

D

s

t

u

q r

1514131211

1098

7654321

w

! 9.! s!⊥!w!and!u!⊥!w!

! 10.! ∠2!≅!∠4!

! 11.! ∠2!≅!∠3!

! 12.! ∠3!≅!∠14!

! 13.! m∠5!+!m∠6!+!m∠8!=!180°!

! 14.! ∠3!≅!∠12!

! 15.! ∠7!and!∠11!are!supplementary!

!

!

!

!

!

!

! Given:! ∠1!≅!∠2!! ! ∠3!≅!∠4!!! Prove:!!! AB!||!CD!!

4321A

O

J K

N

p

q4

3

2

1

Lesson&3&Practice:&&Proving&Lines&are&Parallel&!!

! 1.! Given:!!! JO!||!KN!

! ! ! ∠1!≅!∠2!

! ! ! ∠3!≅!∠4!!! ! Prove:! KO!||!AN!!! ! ! Statements! ! ! Reasons!!! 1.! JO!||!KN! ! 1.!!! 2,! ∠1!≅!∠3!! 2.!!! 3,! ∠1!≅!∠2!! 3.!!! 4.! ∠2!≅!∠3!! 4.!!! 5.! ∠3!≅!∠4!! 5.!!! 6.! ∠2!≅!∠4!! 6.!!! 7.! KO!||!AN!7.!!!!!!! 2.! Given:!!! ∠1!≅!∠2!!! ! Prove:! ∠3!≅!∠4!!!!!! ! ! Statements! ! Reasons!!!!!!!!!!!!!

z

y

x

65°

105°x

44°

36°

!!!!!!!!!!! 3.! x!=!__________!!y!=!__________! ! ! ! 4.! x!=!__________!!! ! z!=!__________!& &&&Page&160=163,&#7=10,&12=29,&32,&34,&54=57&

Lesson&4:&&Parallel&and&Perpendicular&Lines&and&Slope&(Algebra&Review)&!Slope:&!!!Find!the!slope!of!the!line!passing!through!points!(3,!5)!and!(D2,!1).!!!!!!Find!the!slope!of!the!given!line.!!!!!!!!!!!!!!!!!!!Slope=Intercept&Form:&&&&&Find&the&slope&of&the&following&lines:&!

1)!! y = 3x + 2 ! ! ! ! 2)!! y = − 25x − 7 ! ! ! 3)!! 3x − 2y = −6 !

!!!!!4)!! y = −5 ! ! ! ! 5)!! x = 3 !!!!!

!Parallel&Lines:&&&Perpendicular&Lines:&&&&&Are&the&following&lines&parallel,&perpendicular,&or&neither?&!1)!! y = 3x + 2 ! ! y = 3x − 6 !!!

2)!! y = 12x − 5 ! ! y = 2x + 3 !

!!3)!! y = 2 ! ! x = 9 !!!4)!!the!line!through!(D2,!6)!and!(8,!1)!!!!!!!!the!line!through!(4,!3)!and!(6,!2)!!!!!Find&the&equation&of&the&given&lines.&!1)!!m!=!2,!through!the!point!(D2,!5)!!!!!2)!!vertical!line!through!(0,!9)!!!!!3)!!passes!through!(D2,!7)!and!(3,!D3)!!!!!4)!!passes!through!(5,!2)!and!is!parallel!to! y = 2x +1 !!!!!!5)!!passes!through!(D1,!3)!and!is!perpendicular!to!2x + 3y = 1 !!!

4

2

#2

#4

#5 5

4

2

#2

#4

#5 5

4

2

#2

#4

#5 5

4

2

#2

#4

#5 5

Lesson&4&Practice:&&Parallel&and&Perpendicular&Lines&and&Slopes!!

1.!!Find!the!slope!of!each!of!the!following!lines:!!!!!!!!!!!!!! a.! slope!=!__________! b.! slope!=!__________!!!!!!!!!!

!

!

! c.! slope!=!__________! d.! slope!=!__________!!! !!2.! Find!the!slope!of!the!line!through!the!following!points:!

! ! a)! (0,!4)!and!(2,!D3)! b)! (5,!2)!and!(1,!2)!!!!! ! c)! (D4,!3)!and!(2,!D1)! d)! (3,!1)!and!(3,!D2)!

! !! ! 3.! Find!the!slope!of!the!following!lines:!

!! ! a)! y!=!5x!–!1! b)! 5x!–!2y!=!6!!! ! ! slope!=!__________! ! slope!=!__________!!

! ! c)! y!=!3! d)! 5 3y -x 21

= !

!! ! ! slope!=!__________! ! slope!=!__________!

4

2

#2

#4

#5 5

4

2

#2

#4

#5 5

!!

! ! 4.! Use!the!slopes!of!the!following!lines!to!determine!if!the!following!lines!are!parallel,!perpendicular,!or!!

! ! ! neither.!!EXPLAIN&WHY.!

! ! a)! y!=!4x!D!1! ! 2 x 41

y += !

!

! ! b)! 3x!–!2y!=!8!21

x 23

y −= !

! ! !!! ! c)! x!=!3! y!=!D2!!!!! ! d)! the!line!through!(2,!5)!and!(D1,!D1)!

! ! ! the!line!through!(1,!D3)!and!(3,!D4)!

!

!! ! 5.! Find!the!equation!of!the!line!following!lines.!

!

! a)! slope!=!32 ,!through!the!point!(3,!D5)! b)! !vertical!line!through!(4,!D1)!

!!!! c)! through!the!points!(D1,!4)!and!(1,!7)! d)! slope!=!0!and!the!yDintercept!=!5!!!!!!! e)! through!the!point!(3,!D2)!and!parallel!to!4x!–!y!=!6!!!!!! f)! through!the!point!(D1,!5)!and!perpendicular!to!y!=!3x!–!2!!!!! g)! ! ! ! ! ! ! ! h)! !

Chapter&3&Test&Review&Complete&the&following&proofs.&!! 1.! Given:!x!!||!!y"

! ! ! ! q"!||!!r!!! ! Prove:!!∠1!≅!∠4!!!!!!!! Statements! Reasons!!!!!!!!!!!!!!!!!!!!2.! ! Given:!m!!||!!n"!! ! Prove:!!∠1!and!∠4!are!supplementary!!! Statements! Reasons!!!!!!!!!!!!!

x"

y"

q" r"

1! 2!

3! 4!

1! !2!!3!

m"

n"!4!

!!!!!!!3.! ! Given:!m!!||!!n,!∠ ≅ ∠1 2 !!! ! Prove:!!n!||!p!!!!!!! Statements! Reasons!!!!!!!!!!!!!!!!!!!4.! ! Given:!∠ 1!and!∠ 5!are!supplementary.!!∠ ≅ ∠3 5 !!! ! Prove:!!n!||!p!!!!!! Statements! Reasons!!!!!!!!!!!!!!

! !1!!!

m"

n"!! p"2!

1! !2!!3!

m"

n"!4! p"5!

!!!!!5.!!Given:!!∠ ≅ ∠6 9 !!! Prove:!!∠ ∠3 4 and are supplements !!!!!! ! Statements! ! ! Reasons!!!!!!!!!!!!!!!!!!!! !6.! ! Given:!!!! JO KN || ,!!∠1!≅!∠2,!∠3!≅!∠4!!! ! Prove:!KO AN || ! !!! ! Statements! ! ! Reasons!!!!!!!!!!!!!!

E!

1!

2!3!

4!

5!6!7!

8! 9! 10!

A!

B!

C!

D!

F!

2!1!K! A!3! 4!

J!

N!O!

!!!!!7.! ! Given:!!∠3!≅!∠4!!! ! Prove:! ∠1!≅!∠2!!! ! ! Statements! ! Reasons!!!!!!!!!!!!!!!!8.! ! Given:! ∠1!≅!∠2!!! ! Prove:!!! ∠3!≅!∠4!!!!! !!! ! Statements! ! ! Reasons!!!!!!!!!!!!!

!

p!

q! 4!

3!

2!

1!

1!

2!

m"n"

3!

4!!

Extra&Practice&Proofs&!!!!!!!Given:!!∠ ≅ ∠ ∠ ≅ ∠1 2 3 4, !!! Prove:!!n"||"p!!!!!!!!!!!!!!!!!!!!!!!!!!Given:!!∠5 ≅ ∠10 !!! Prove:!!∠2 ≅ ∠4 !!!!!! !!

1!2!3!

4!5!

m"

p"

n"

k"

E!

1!

2!3!

4!

5!6!7!

8! 9! 10!

A!

B!

C!

D!

F!

321

A

C

D F

B

E

4

3

21

A

R T

B

S

! Given:!!m n|| ! !!! Prove:!!∠ ∠1 2 and are supplementary. !!!!!!!!!!!!! Given:!!a b c d|| || and ! !!! Prove:!!∠ ≅ ∠1 2 !!!!!!!!!!!!!!Given:! AS!||!BT!! ∠1!≅!∠2!!! Prove:! ∠3!≅!∠4!

!!!!!Write!a!paragraph!proof!

!!!!!Given:! BC!||!DF!!! BC!bisects!∠ABE!!! Prove:! ∠1!and!∠3!are!supplements!!!

1!3!2!

m! n!

1!3!

2!

a" b"

c"

d"

THEOREMS:&&Proven&true!&&

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unit 3 workbook - leman's...

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