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 Chapter 2 Section 3, pages 98-104 Find the slope of a line given a graph, two points, or the equation of a line Chapter 3 Section 5, pages 113-119 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. ***
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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
Chapter 2 Section 3, pages 98-104
Find the slope of a line given a graph, two points, or the equation of a line
Chapter 3 Section 5, pages 113-119
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"
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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.!
!!!
!!
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!
!
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,=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
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21
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!Given:! AS!||!BT!
! ∠1!≅!∠2!!Prove:! ∠3!≅!∠4!!!!!!!!!!!Given:! BC!||!DF!
! BC!bisects!∠ABE!!Prove:! ∠1!and!∠3!are!supplements!!!! !
432
1E
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1K C
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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!!!! !! !!!!!!!!!!
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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
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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,=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¶llel,&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 !!!
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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!=!__________!
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!!
! ! 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!!!!!!!!!!!!!
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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!
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C!
D!
F!
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! 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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