Download Chapter 3 Geometry Handouts

Geometry
-I Chapter 3 Day #1
TOpic:Parallel Lines
I.
voCabUlaryl
1. Parallel lines - lines that are in the same plane that do not intersect n.
I
2. Parallel Planes - planes that do not intersect n.
3. Skew Li~es - Lines not in the same plane.C.~o
Y\~
f f\ )
4. Transversal+ a line that intersects with 2 or more other lines in a
plane.
II.
Examples:
I
N
For exercises 1-3~refer to the figure at the right.
1. Name all planes that intersect plane OPT.
2. Name all segments that are parallel to NU.
s
3. Name alii segments that intersect MP.
I
For exercises 4-6~refer to the figure at the right.
N
4. Name all segments parallel to QR.
S. Name all segments skew to AG.
T
6. Name all segments parallel to QX.
G
{,
III.
Vocabulary
(see diagram to the right)
1. Same Sidle(Consecutive) Interior Angles
2. Alternate Interior Angles
3. AlternaJ
Exterior Angles
4. correspJnding
Angles
--
~
\
{
~r
2
IV.
I
Examples
A. Decide whetherleaCh pair of angles labeled are alternate interior angles,
same-side interior anqles, corresponding angles, OR alternate exterior
angles.
#1.
"
#2.
2
#3.
#4.
1
2
Use the figure in the example for the next TWO sections:
B.
Name the transversal that forms each pair of angles.
1. <9 & <13
2. <5 & <14
3. <4 & <6
q
p
.••
I.\~
0(
.
vx:
-,...
:Jl.. ~"".
::: L'~'
tt
•
e
I
C. Name the transversal that forms each pair of angles and identify each pair of
angles as alternate interior, alternate exterior, corresponding, or same side
(consecutive) interior angles.
#1. <1 & <5
#2. <6 & <14
#3. <2 & <8
#4. <3 & <11
#5. <12 & <3
#6. <4 & <6
#7. <6 & <16
#8. <11 & <14
#9. <10 & <16
V.
Refer to the drawing shown to answer the questions.
1. Find an example of parallel planes.
_----==--o~
2. Find an exarple
of parallel lines.
3
Homework ~orkSheet- Chapter 3 Day #1
I.
Short Answer
Name the correct Jeometric term that goes with each definition.
1.
2.
3.
4.
5.
II.
Word Bank:
Parallel lines
Skew lines
Parallel Planes
Alternate Exterior Angles
Alter~ate Interior Angles
Corresponding Angles
Transversal
Same Side Interior Angles
Two planes that DO NOT intersect.
Lines that are NOT coplanar and DO NOT intersect.
Two coplanar lines that DO NOT intersect.
A line that intersects two or more lines in a plane at different points.
A pair of angles determined by two lines and a transversal consisting of an
interior angle AND an exterior angle that have different vertices and that
lie on the sarne side of the transversal.
Short Answer
For exercises 1-2, refer to the following figure at the right.
1. Name all planes that intersect plane STX.
x
2. Name all segments that are parallel to XY.
y
III.
Short Answer
Name the transversal that forms each pair of angles AND identify each pair of
angles as alternate interior, alternate exterior, corresponding or same side
(consecutive) interior angles.
IV.
1.
<9 & <13
2.
3.
4.
5.
6.
7.
8.
<6 & <16
<3 & <10
<8 & <14
<4 & <8
<1 & <11
<4 & <9
<10 & <16
~12
413
n
rn
Short Answe~
Indicate whether each of the fol/owing are examples of intersecting, paral/el, or
I
skew lines.
#1. Railroad tracks
#2. Airline flight paths
#3. Streets in downtown NYC
#4. A picket fence
I.
4
v.
Short AnswJ
The map at the r;gh!t shows the downtown area of a city. Name two points of
locations that reprerent each type of angle.
1. Alternate
interiolr angles
2. Same-side interior
angles
Post Office
3. Alternate exteridr angles
4. Corresponding angles
VI.
Multiple Cho~e
Use the figure belo, to answer the next THREEquestions:
1. Which line segment is parallel to GE?
a. Segment DH
b. Segment FG
c. Segment KI
d. Segment HI
2. Which two line segments are skew?
a. DE & GE
b.
GK & DH
c.
EI & GK
GE
d.
KI
G/I
~I
E
I
I
I
I
I
I
)Ii - -11/
'/
d. HI & DF I
3. Which line segment is parallel to plane FGKJ?
a. FD
b. HI
c.
o
F~
J
K
I
Use the figure shown to answer the next FOURquestions:
4.
Which
a.
b.
c.
d.
.
is a pair of alternate interior angles?
<3 & <6
<2 & <7
<6 & <5
<4 & <6
I
5. Which angle CORIRESPONDS
to <7?
a. <1
b. <3
c. <4
d. <6
I
6. Which pair of angles are alternate exterior angles?
a. <1 & <5
b. <3 & <6
c. <5 & <8
d. <1&<8
7. Which pair of an~les are same-side interior angles?
a. <1 & <5
b. <3 & <6
c. <4 & <8
d. <3&<5
t
2
3
4
<,
5
6
7
u
8
Geometry
j
1
Chapter 3 Day #2
Topic: Angles & Parallel Lines
I
Important ~heorems & Postulates
I.
Choose the correct word to complete each sentence.
1. Alternate
Exterior Angles Theorem - If two parallel lines are cut by a
transversal, then alternate exterior angles are congruent
2. Alternate
Interior Angles Theorem - If two parallel lines are cut by a
transversal, then alternate interior angles are congruent
3. Corresponding Angles Postulate - If two parallel lines are cut by a transversal,
then corresponding
angles are congruent
4. Same-Side (Consecutive Interior) Angles Theorem - If two parallel lines are cut
by a transversal, then consecutive interior angles are
_
5. Perpendicular Trc.'lnsversalTheorem -- In a plane, if a line is perpendicular
,
one of two parallel lines, then it is
II.
to the other.
to
-t
Use the figulre for the exercises below:
1. Name four pairs of vertical angles.
2. Name all angles that form a linear pair with <7.
3. Name all angles that are congruent to <1.
4. Name all angles that are congruent to <4.
5. Name all angles that are supplementary
to <3.
6. Name all angles that are supplementary
to <2.
In the figure, m<3 = 102. Find the measure of each angle.
III.
p
q
1. <5
2.
3.
4.
5.
6.
<6
~~~~~~~~m
<11
<7
\..
n
<15
<14
In the figure, m<9 = 80 and m<5 = 68. Find the measure of each angle.
IV.
1. <12
2.
3.
4.
S.
6.
<1
<4
<3
p
<7
<16
q
2
V.
Find x and y in each figure:
#1.
fOr>
#2.
--------
~~ +1'0
VI.
Find m<l1. in each figure.
#2.
#1.
<,
~
VII.
'('ft
Word Problems
#1. Fencing
A diagonal brace strengthens the wire fence and prevents it from sagging. The brace
makes a 50 degree angle with the wire as shown. Find y.
#2. Zip lines
Ana made a zip line for her tree house. To do this, she attached a pulley to a cable. She
then strung the cabl~ at an angle between the tree house and another tree. She made
the drawing of the zip line at the right. The two trees are parallel.
a. What is the measure of <17
b. Are <1 and t~e given angle: same-side interior angles, alternate interior angles,
alternate exterior angles, or corresponding angles?
I.
~
"
.)
Geometry
I.
-I Chapter 3 Day #2 HOMEWORK
I
Short Answer
In the figure, m<3 =
bo and m<12 = 55.
I
Find the measure of each angle.
1.
2.
3.
4.
5.
<1
<6
<2
<10
<13
6. <15
II.
Short Answer
In the figure, m<9 = 75. Find the measure of each angle.
1. <3
2. <5
3. <6
te
it,m
4. <8
5. <11
-C
6. <12
III.
In
Problems
1. Find x and y in the figure.
2. Anthony is building a picnic table for his patio. He cut one of the legs at an angle
of 40 degrees. At what angle should he cut the other end to ensure that the top
of the table is parallel to the ground?
·=
,~'. >:.
R
,
3. Find x figure.
.,.:;
~.
4
4.
Find x and y ~nthe figure.
(4x- 5)°
5. What is the value of x and the measure of <1?
(5x - 25)°'
IV.
Multiple Choice
Use the picture below to answer the next FOUR questions:
1.
2.
Which angle is congruent to <17
a. <2
b.
c.
d.
Which
<5
<6
<7
angle is NOT supplementary
a. <2
b. <4
c.
I
5
to <67
b. corre~ponding angles postulate
c. samerside interior angles theorem
d. Alternate exterior angles theorem
If m<5 = 42, what is m<47
a. 42
b. 48
c. 128
d. 138
8
z
<5
d. <8
3. Which can be used to prove directly that
a. Alternate interior angles theorem
4.
f
2
3"'" 4
< 1 =< 87
1
Geometry - C~apter 3 Day #3
!
Topic: Proving Lines Parallel
I.
Concepts
J
v" If two Ii es in
plane are cut by a transversal and certain conditions are
met, theln the lines must be parallel.
v"
(.
The fOll9wing onditions must occur in order for the two lines to be
parallel.
1. Correspdnding angles are congruent.
0-. ..:
2. AlternaJ
.•
i-_
exterior angles are congruent.
I
3. Same Side (Consecutive] interior angles are supplementary
4. Alternate
intenior angles are congruent
b~
5. Two linJ are perpendicular to the same line.
II.
Problems
I
Find x, so that line 'I is parallel to line m.
#1.
~
)
#2.
~
~)(-S
~<~------~\---------
f{I
#5.
#4.
M
~~
1
tt\
5
b
~
2
III.
Short Answer
Given the fOllowinJ information, determine which lines, if any, are parallel. State the
postulate or theorem that justifies your answer.
1. <3 = <7
2. <9 = <11
(;
.••.
... '/_....
_Vi""t
.••
e
3. <2=<16
•••.•..
"'(IV
I
1._
it•
i
c
.•••m
4. <4 = <13
5. <5 + <121=180
IV.
Short Answer
Given the folia wind information determine which lines, if any, are parallel. State the
postulate or theor1m that justifies your answer.
1.
m<BCGit m<FGC= 180
2. <CBF
(;..
= <GFH
o
~
:: ::~: =J:BCF
b
cJ
E
5. <ACD tGF
0
v.
Word Problem
Mrs. Jensen made quilt for her nephew. The pattern for one block is shown at the
right. If m<1 = 60 Jnd m<2 = 115, are the two gray strips of fabric parallel? Explain.
b
I.
3
Homework - .hapter 3 Day #3
J.
i
Short Answ .r
A. Given the folio
I
ing information, determine which lines, if any, are parallel. State
the postulate or theorem that justifies your answer.
1. <16 = <3
-I (
2. <4
= <13
3. <1~ + <10
= 11.80
I?
fV'
4. <1 - <7
B. Given the folio
the postulate
1.
~ t.
3 711 e 7'
/''1
'!' 12 15
I
f'
I
4l.
IJ/,
1#
Ill~
r" .-~11
ing information, determine which Jines, if any, are parallel. State
olil theorem
that justifies your answer.
fl
= <8
<2
7}
-~~
L
I
,
t.i ~
6 '7
2. <9 = <16
3. <2
= <10
4. <6 = <15
II.
#1.
(f\
"
l~
/10
,~'"
It, .I IS'
Problems: Findx so that each pair of lines are parallel.
#4.
,~
,,~
~1-"'> 3~"'11
~
VI.
MULTIPLEC~OICE:
Choose the word or qhra.se that best completes each sentence.
l.
/1'1
If two coplanar Ijnes are cut by a transversal, so that corresponding angles are
congruent, then
he lines are
.
a. Parallel
b. Perpendicular
c. Skew
d. Intersecti
g
2. In a plane, if two lines are perpendicular to the same line, then they are
a. Perpendi ular
I
b. Parallel
c. Skew
d. Intersectilng
_
):.
4
3.
For a line and a p,oint not on the line, there exists
that is parallel tithe
given line.
::::~:;I~:I:
5~fIlt.";
c. at most dne
4.
line through the point
&,
If two coplanar ~Inesare cut by a transversal so that oe'iLxatioe interior angles are
_____
-''jhen the lines are parallel.
a. (ample
lentary
b. Supplementary
c. congrue~t
5. If two coplanar +es are cut by a transversal so that alternate interior angles are
congruent, then fhe lines are
_
a. Perpendicular
b. Parallel
c. Skew
6. For what value of x is dlle?
o
»; (2x - 3)"
a. 20
b. 25
c.
35
d. 37
Use the figure showp to answer the next THREE questions:
7.
If l 1/ m, what is mc l ?
a. 22
b. 58
c. 122
\f
d. 130
8. For what value of x is I
a.
22
b. 54
c.
58
d. 122
9. If
III m, what
a.
22
b.
58
c.
122
d. 130
is ~<2?
1/
m?
..•.m
1
lp----------
Geometry - C~apter 3 Day #4 Proofs
Examples
1.
2.
:::::- r=b~:~1d
Given:
c
rl> 1\ q; c 1\ d
C&
;l~ ~
10
3 7
(I
Prove: t7 = <10
d
0
t(
"
3.
Given: {::15 + <8
Prove:
=
180
If, .g'
-
t II d
&
Given:
<4 + <5
Prove:
Be
=
P
c.
\0
~
e
f
\
4.
/
1
;}
·cl
,.a.
180
1\ AD
C(
5.
Given: t
II d;
<13= 2x + 5, <16 = 4x - 7
1;>
C'
Prove: Ix = 6
d
-
IS
I"e,
q
ql\VaJ\OJd
AJelUawalddns
aJe P>
"8
1>' :uaJ\!9
:JoJ JOOJd MOl e alPM
t.
"9
3
Chapter 3. Day!#4 Proofs HOMEWORK
1.
Given: 8" q; c " d
C. ~
Prove: ~9 = <5
~_~_-~~-¥
' .•..
? ...,
lS'f/b
2.
Given: d II d; p II q
Prove:
= <13
3.
Given:
k4
i3 + <5 = 180
Prove: dJ " e
=t>. :>
d tC-_
o
e
4.
Given: ~ 1\ d;
<l = 4x
Prove:
*=
S'
.--~,".,
.'"
~
I?
I
•
e'<...
...--I-
5; <8 = 7x - 1
16
Ct!!
diG
5.
Given: t8 + <5 = 180
Prove: EF 1\ GH
e
I
I
t ...
b
5
10
•••
F
H
p
p
!ueluawalddns
)
II J
:aJ\oJd
z
aJe E> pue
:N3/\19
:JoJ JOOJdMOH e al!JM
°9
1
Geometry - C apter 3 Day #5
Topic: Slopes of Lines art I
Today, we will look at rope of lines & segments and we will learn how to ~alculate it. We will
also look at the relationship
between slope and parallel/perpendicular lines.
I
I.
Slope
1. The slope of a line is the ratio of its vertical rise to its horizontal run.
2. In the coordinat~ plane, the slope of a line is the ratio of the change along the y-axis to
the change alon~ the x-axis.
3. There are four tJpes of slope: NEGATIVE, POSITIVE,ZERO, & NO SLOPE(sometimes
called (undefine~' slope.)
4. The slope of a li~e or a segment depends on its steepness, or the rate at which it rises or
falls. (Think: (ra~e of change').
II.
Ways to think of 'slope'
The following expressiohs can be used to represent slope of the line containing the points
(Xl> Yl) & (X2, Y2)'
AssLme that no denominator
YZ-Yl
XZ-Xl
III.
is zero
change in Y
change in x
~
vertical rise
horizontal run
\
"-
#1. AB
#3.
EM
IV.
A(~2, ~2
I
Theorems regarding the slope of PARALLELand PERPENDICULARlines
1. Two non-verticaillines are parallel if and only iftheir slopes are equal.
2. Two non-vertical lines are perpendicular
V.
if and only ifthe product oftheir
Short Answer
Find the slope of each lire.
1. A line parallel ti a line with slope %
2. A line perpendicular to the x-axis.
3. A line perpendiJular to a line with slope 5.
4. A line parallel to the x-axis.
5. y-axis
,
t----->
I
-----.
TW
3. A line parallel to
.J
V
Vp
,
1\
1. NP
2.
y
I I~
Find the slope of each line
....
v
r\
1/
,/
TW
4. A line perpendiJular to
V" 1\
NP
B(0,4)
...•.•..
~
C(-2,2)1\.
<----+
CD
"-
I\.
...•.•..
1/
Review of findi~g Slope
#2.
Y'.~
I"-
!/
V
N
,.0
x
T II\.
,W
,
I\.
IA:
~
II II\.
'f,.; H
......
M(4,2)
,
~
x
0
0(0,-2
I\.
......
1'1
~I(-1,-4)
..•....
~
3
E(4 - 2)
1'1
1\
"
slopes is -1.
2
Determine whether lints AB and MN are parallel, perpendicular,
or neither.
1. A(O, 3), 8(5, -7), M(-6, 7), N(-2, -1)
2. A(-1, 4), 8(2, -5" M(.J3, 2), N(3, 0)
Graphing
VI.
A. Graph the line that satisfies each condition.
1. Slope
= -1/
2.
=
Slope
; contains U(2, -2).
4/3; contains P(-3, -3).
3. Contains B( 4,2)' parallel to line FG with F(O,-3) and G(4, -2).
4. Contains Z(- ,0), perpendicular to line EKwith E(-2, 4) and K(2, -2).
5.
Contains Y(3, 0), parallel to line DJ with D (-3, 1) and J (3, 3).
6. Contains T(~, -2, perpendicular to line CX with C(O,3, and X(2, -1).
~-
I
I
G)
0---
y
Y
I
6)
~
0
x
0
®
I Y
I
I x
I
Y
I
I
I
q
x
0
I
I
~
Y
x
Y
I
~
10
I
I
x
I
I
I
I
I
I
!
I
I
0
x
I
3
HOMEWORK-"l Chapter 3 Day #5
I.
Short Answer
Indicate if the slope of
each line is POSITIVE, NEGATlVE, ZERO or UNDEFINED.
1. The line rises to the right.
2. The line is parallfl to the x-axis.
3. The rise of the line is zero.
4. The line falls to the right.
5. The line is perpehdicular to the x-axis.
tfe
I,
I
II.
Matching
Match the description of a type of line from the list below.
A negative number
Zero
Undefined
Type of Line
1. A horizontalline
.
A positive number
l
2. A line that rises irom left to right
3. A vertical line ~
4. A line that falls f~om left to right
III.
'
Finding slope ofllines
Determine the slope of each line.
t---+
1. LM
~
2. GR
3.
Ps
lC
y
IV.
IV
Finding slopes J1f segments
Determine the slope of each segment.
-
1. AB
•
I
14
2. HN
•
3. AC
~t
I I.
II
s
't-+
•
I I
,
I
•
c
V.
Short Answer
Find the slope of each line.
+---->
1. AB
VI
L
1\
t---+
3. LM
jQ
I 'A
+--+
v
II
4. EF
5. A line parallel td
4
y
I
2. Pi}
II
ill.
PQ.
to EF.
\
/
L
plj{
~rt
E
x
0
B
6. A line perpendidular to
7. A line perpendiJular
1<--+
8. A line parallel to AB.
('\
1/
M
1/
I I
\j
I
F
I
r~
I
'"
"
•
~
""X
4
Determine "1hethellines PQ and UV are parallel, perpendicular,
1. P (-3, -2), Q (9, 1 , U(3,
V(S, -2)
or neither.
VI.
F)'
2.
P (-4, 0), Q (0, 3), U(-4, j3), V(8, 6)
3. P(-lO, 7), Q(2, 1) U(4, 0), V(6, 1)
VII.
Multiple Ch~ice
For the following exercises; choose the CORRECTletter.
1. If two lines INTEBSECT,they
a. Have one common point
b. Are paral el
c. Are the sfme line
d. Are congrent./
2. What is the slopeI ofthe line passing through the points (2, 7) and (-I, 3)?
a. 2/7
b. %
c. 4/3
d. 1/3
I
,
3. Which pair of Slollpescould represent perpendicular
a. 1/7, 7
b. Yl, %
lines?
I
\1
I'
Neither 9arallel nor perpendicular
Both parallel and perpendicular
,
I'
,
\
r-
1
J
f-l b
-8
, -4
1
1
when
slopes is -1
slopes is greater than 0
slope
x
~
~
L
,1
4
2
-2 01\
\
IJ,
v
,
~
I'
r:
\
1\1
1
I.-
5. Two lines are perpendicular
a. The prod~ct oftheir
b. The prod~ct of their
c. They have the same
6.
1\
11
1 ~
I
~
I
y
0
-w
\
4. The lines in the fr'gUre at the right are
a. Parallel
b. Perpendicular
c.
d.
h
1'1
~
c. - %,4/3
d. 1/3,1/3
,
\
,
~
-L
~\
,
1\
"
'I
\
r-
v
~
~
1\
Their siofes arJ undefined.
If AB CD, what
a. 20°
b. 30°
c. 50°
IS the
I
value of In' as shown in the figure?
I
I
A
In
B ••••
....}
d. 70°
c
o
NAME
~
_
2
B. Graph the line that satisfies each condition.
I.
Graphing
1. Slope =2, contains A(0,2)
2. Slope = -2/3, contains R(-2, 1)
A. Graph the line that satisfies each condition.
1. Slope = -1/3; contains U(2, -1)
3. Contains Y(2, 0), parallel to line DJwith D (-2, 1) and J (3, 2)
2. Slope = 4/5; contains P(-4, -2)
4. Contains T(O, -1), perpendicular
to line CX with
qo,
4, and X(I, -1)
3. Contains B(-3, 1), parallel to line FGwith F(I, -3) and G(3, -2)
4. Contains Z(-4, 0), perpendicular
to line EKwith E(-2,3) and K(1, -1).
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Geometry - Chapter
3 Day #6
I
Topic: EquQtions of Lines
FORMULAS:
Slope formula
Slope Intercept Form
Point-Slope Form
Standard Form
1. Find the slope AND v-i'ntercept of each line.
.
I
a. V=2x-7
b. x + V = 8.5
c. V-7::: x + 12
I
d. V + 5 = -2(x + 6)
,
2. Write an equation in s,lope-intercept form of the line having the given slope and vintercept.
a. m: 2; v-intercept: -3
b. m: -1/2; v-intercept: 4
c. m:}{; v-intercJpt: 5
d. m: 0; v-intercelpt:
-2
3. Write an equation in silope intercept form for the following:
a. x-intercept
is -5 and v-intercept
b. x-intercept
is 3/and v-intercept
is 3
is 9.
4. Write the point slope form of the equation
a. m
b. m
c. m
= }'2; (3,
-1)
(4, -2)
= }{; (-3, 2)
d. m = -5/2 ; (a, -~)
e. rn= 0; (-2, 5)
= -2;
5. Write the point-slope
rorm of the equation
for each line.
a. A line with sl0ge -1/2 containing (-2,5)
b. A line containing (-4.5, -6.5) and parallel to a line with slope 0.5.
6. Write an equation in ~Iope-intercept form for each line.
)I
1/
1/
c. t
1)1' r
The line perpendicular to line s that contains
Ii
..-i,..--Ir
.•• i,..--
x
,
•....
u
1\
.,
I
(a, 0)
\.
!/ \
V
u
,
~ /
a
/
e. The line parall~1 to line r that contains (1, -1)
f.
t
1\ 1/
a. r
b. s
d.
Y[;( 14
\
\.
_t""
's~
2
"'IIf",-
Y
cl'"-- •••.
7. Write an equation in slope-intercept form for each line.
-... r-...
I'"--t-.,
~
a. b
b. c
I);,
c. The line parallell to line b that contains (3, -2).
d. The line perpendicular to line c that contains (-2, -4)
'"61" -,
a
r-,
x
"
"
~
Free Response Question
#1. Rail Trail: A community ~ecently converted an old railroad corridor into a recreational
trail. The graph at the right shows a map of the trail on a coordinate grid. They plan to
construct a path to connect the trail to a parking lot. The new path will be perpendicular to
the recreational trail.
a. What is the slope of the line representing the existing path?
b. What is the slope of t~e line representing the NEW path?
c. Write an equation of 1!lheline representing the new path, and then graph it.
d. What are the coordin~tes of the point at which the path will meet the recreational
trail?
I
e. If each grid space is
yards by
yards, how long is the path to the nearest yard?
21
25
#2. Donna offers computer services to small companies in her city. She charges $55 per
month for maintaining a website and $45 per hour for each service call.
a. Write an equation to 1epresent the total monthly cost C for maintaining a web site
and for h hours of seryice calls.
I
b. Donna may charge her costs to represent them by the equation C = 25h + 125,
where $125 is the fixe~ monthly fee for a web site and the cost per hours is $25.
Compare her new pla~ to the old one if a company has 5 X hours of service calls.
Under which plan would Donna earn more?
3
Homework Worksheet - Chapter 3 Day #6
I.
Problems
/1
1. Find the slope AND v-intercept
a. y = 3x + 9
b. x - y = 9.2
c. 3.2 y - y = 6.6
,
of each line.
d. y-9=x+17
~
e. y + 7 = - 2(x + 4
2. Write an equation in ,lope-intercept
intercept.
a. m: 3; v-Intercept: -4
b. m: -1/3; v-intercept: 7
form of the line having the given slope and y-
c.
d.
3. Write
a.
b.
4. Write
m: 1/8;. y-inte~cePt: 6
m: 0; y-Intercept: -9
an equation in sllope intercept form for the following:
x-intercept is -4!and v-intercept is 6
x-intercept is 61and v-intercept is 7.
the point slope fprm of the equation
a. m = X; (4, -2)
b. m = -3; (-8, 10)
c. m:
m-
d.
5. Write
a.
b.
6. Write
-~/3 ; (0, -41 )1
0, (-3, 15)
1
the point-slope ~orm of the equation for each line.
A line with slope -1/2 containing (-3, 9)
A line containi~g (-5, -8) and parallel to a line with slope X .
an equation in sl19pe-intercept form for each line.
a. k
II
r....."-.'j(
b. I
c.
d.
e.
m
/1
/
V
n
perpendicular tlo line I, contains (-1, 6)
r'
I
1/
1/ \
i/
~t-
l- I-
I
I
!
-
b.
y
= -5/3
= -3/5
x+4
lve
x + 4 l¥S
I
d. y=3/Sx+4
1/15
c. y
I
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I I '/
I I)
I 1/
!
I~~vl
,J1 I
If I I
Multiple Choice
II
1. Which is the equation for the line PERPENDICULAR to y = -5/3 x + 111/3
a. y - 2 = -3/5 (x B)
J
!,(ie
1/1
\
f. parallel to line
contains (7, 0)
g. parallel to line m, contains (0, 0)
h. perpendicular Jo line rn, contains (-3, -3)
II.
y
'-4 mi
V
I
I
-+!- . In
I I
I
I
I
I
x
1
I
I
I
II
I I i
4
r-
2. What is the correct equation of the line
shown at the right?
a. V=3/2x+3
b. V = 2/3 x + 3
c. y = -3/2x - 3
d. V=-2/3x-3
I
,.
I.
-J.
1-
-
V
a.
y =
-5/2 x - 5
x
a
iL
1
3. The x-intercept of a linr is -5 and the v-Intercept
of the line is -2. What is
the equation of the line?
I
-I
Y
I
V
I
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~
-u
b. V = -5/2x - 2
c. V = 2/5x + 2
d. V = -2/5 x - 2
4. What is the slope inter1cept form of the equation:
V- 7 = -5/2 (x + 4)?
a. V - 2 = -5/2 (x 2)
J,
b.
V = -4/7 x + 2
y + 7 = -x + 5/2
c.
d. V = -5/2 x - 3
5.
Which
a.
b.
c.
d.
one
The
The
The
The
of the fol19wing correctly
v-coordinate of the point
x-coordinate of the point
v-coordinate of the point
I
x-coordinate of the point
describes the v-intercept of a line?
where the line intersects the x-axis.
where the line intersects the v-axis.
where the line crosses the v-axis.
where the line crosses the x-axis.
e. The ratio of thJ change in v-coordinates to the change in x-coordinates.
III.
Free Response
I
Jerri's current satellite televisipn service charges a flat rate of $34.95 per month for the basic
channels and an additional $10 per month for each premium channel. A competing satellite
television service charges a flat rate of $39.99 per month for the basic channels and an
additional $8 per month for e!ch premium channel.
1. Write an equation in slope intercept form that models the total month IV cost for each
satellite service, wherJ p is the number of premium channels.
2. If Jerri wants to includ~ three premium channels in her package, which service would be
less, her current service for the competing service?
3. A third satellite cornpenv charges a flat rate of $69 for all channels, including the
premium channels. If Jerri wants to add a fourth premium channel, which service would
be least expensive?
1
Geometry Chapter 3 Test Review
I
Parallel Lines
Parallel Planes i
IMPORTANT VOCABULARY
I s.kew
1
Corresponding
Angles
Intersect
Complementary
Angles
sUPPIementa1
Angles
Positive Slope
Negative Slope
Alternate
Interior
Angles
Theorem
Slope
I Slope-
I
"If slopes of 2 lines are equal, then
the lines are parallel."
I
Transversal
Same-Side
Interior Angles
Alternate
Exterior
Angles
Theorem
Undefined
slope
Corresponding
Angles
Postulate
Lmes
Intercept
Form
Point Slope
form
Vertical Lines
I Standard
Form
Alternate
Interior
Alternate
Exterior
Angles
Same-Side
Interior
Angles
Theorem
Horizontal
lines
Angles
Perpendicular
Transversal
Theorem
Perpendicular
lines
I Intercept
Plane
"If slopes of 2 lines are
opposite reciprocals,
then lines are
perpendicular ."
STANDARDS/GOALS:
• A.l.c.: I can write linear equations in standard form and slope-intercept form when given
two points, a point a~d the slope, or the graph of the equation.
• A.l.d.: I can recognizk the concept of slope as a rate of change, and can determine the
slope when given the tequation of a line in standard form or slope-intercept form, the
graph of a line, two points or a verbal description.
I
• A.1.e.: I can graph a linear equation using a table of values, x & y intercepts, or slopeintercept form
• C.l.d.: I can use various methods to prove that two lines are parallel or perpendicular by
using coordinates or angle measures.
• D.l.c.: I can identify ~orresponding, same-side interior, same-side exterior, alternate
interior, and alternate exterior angle pairs formed by a pair of parallel lines and a
transversal and use these special angle pairs to solve problems.
•
D.l.f.: I can apply properties and theorems of parallel and perpendicular lines to solve
problems.
• G.l.a.: I can use slopeI to distinguish between and write equations for parallel and
perpendicular lines.
• G.GPE.5.: I can prove/that two lines are parallel or perpendicular based on the slope of the
lines.
I.
Key Ideas
1. What is the slope of a vertical line?
I
I
2.
3.
4.
What is the slope of a horizontal line?
When two lines are parallel, their slopes are
?
Are the slopes of 2 pe1rpendicular lines equal to one another?
5.
What types of angles are located between the lines cut by a transversal?
Answers:
#1: undefined; #2. Zero) #3. Equal; #4. No, thevare opposite reciprocals; #5. Interior
II.
Short Answer: For questions 1 - 4, refer to the figure.
._~
1.
Name a segment sker to segment WY.
2.
3.
Name a plane that is parallel to plane ZYW.
Name a plane parallel to plane VSX.
4.
Name the intersection
vfflJ
S
T
;
x;+-}W
of planes VUY and ZYW.
,/
.
,/
z
y
2
For questions 5-7, name the 'transversal AND identify each pair of angles as alternate interior,
alternate exterior, corresporlding, or same side interior angles.
A
5. <2 and <12
.b
~n/
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t:l
6. <3 and <5
I
7. <7 and <15
5
t
c-<E
#8. Use the figure above to complete the following:
Given m nand m<8 = 86, find m<13.
II
I
#9. What concept that we have studied
III.
can
be used to prove that a...Lb in the figure shown?
I
Short Answer
For questions (1-3J, given the fol/awing information, determine which lines, if any are paral/el.
State the postulate or theorem that justifies your answer.
1.
<1 = <2
2.
<DAB
= <EBC
3. <ADE+ <BED= 180
IV.
Problems:
#1. Find x so that p
II q
#2. Find x so that a
II ~
~
• ,,-,
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~
"1..,;\\
~
/\'6
r .)
II
.
l :'2::>
b
bl-LI
#3. If a b, find x in the problem below:
<5 = X2 + 7x, <1 = lOx + 4 I
e,
( 1~
FS
~
<f~
,I
~
vb
~
~c.
#5. Determine whether line
and line KPare parallel, perpendicular, or neither.
C (1, -12), S (5, 4), K (1, 9), P (6, -6)
3
Word Problems
V.
1.
Mrs. Smith writes computer
manuals. She charges $125 to review writing specifications
plus
$50 plus per hour, h to wrIte the manual. Which equation represents the total fee, F that
Mrs. Smith earns for writing each computer manual?
2.
Printer's Ink charges $1.18 per page, p to copy a report plus $12 to bind it. Write an
equation that represents the total cost C, to copy and bind a report. What would be the cost
to copy and bind a 50-pag1e report?
I
3.
You and a friend are drivi7g go-karts on TWO different tracks. As you drive on a straight
section heading east you~ friend passes above you on a straight section heading south. Are
these sections of the two tracks parallel, skew, or neither? Explain.
4.
Cheryl is making a picture frame out of scraps of wood. list three pairs of values for x and y
for which the sides of the frame will be parallel. Explain how you determined these values.
~<\
:;:;1
~
~
yO
5.
yO
A student has attempted
to graph an equation that contains the point (2, -5) and has a slope
I
ofK
a.
What is the correct equation in slope-intercept
b.
What is the 1tudent's error on the graph?
form?
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-~
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6.
A real-estate developer iS~:~anning to build a new gated community in his city. The map of
the complex is given belo .1 Assume that all of the streets lie in a plane.
a. If Maple and , ain Streets are to be parallel, what must be true of <2 & <3?
b.
VI.
What streets will be parallel is <5
= <6?
Slope & Equations ~ Lines
#1. Write an equation in slope intercept form for the line that satisfies the given conditions:
-r)
a.
Contains the point (0,
b.
X intercept is -2 and v-intercept
c.
m = 0 and v-intercept
JI
and has slope -3/8.
c lel'\lleVl +>f-t-\lC':-.
is 8.
-7.
"It.\\P/
#2. What is an equation of ~ line perpendicular to: y = ~ x
#3. What is an equation of
+ 77
'7.{--~---
7-t-
._.__-::;c, tJ.~~
V / \lit-
I
>+.
a line parallel to -3x + By = 167
y
#4. Use the graph to answer ~he following:
a.
1\
Find the slope of line p
'\
\
b.
Find the slope of line q
c.
Find the slope of a llnejparallel to line q
d_ Find the slope of a linejperpendicular
#1.
1/
1
= 18~'
GIVEN: <4
PROVE:
= <14
x II
y
'rn\7 -
L..
X-c
GIVEN: Mil DE; <5 =l: <7
PROVE: <8
'/..
V
\;~
~
. ,..
•••
v
#3.
\
~
(VlL
f\
#2.
V
/
\
to line p.
Proofs
GIVEN: mil n
PROVE: <6 + <3
/. ii'r
./ /\
V
VII.
I'
'(
A
= <9
D
<
~
c
8
''\
f