Languages
Pages
Legal
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"
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,=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,=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 !!!
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!&&
Theorem& Picture/Rephrase&
& &
& &
& &
& &
& &
& &
& &
& &
& &
& &
& &
& &
&
Top Related