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DAY23
GOAL:
Use Definitions, Postulates, Properties and Theorems
to prove other shortcuts for measuring.
GEOMETRIC POSTULATES;
Basic rules for measuring
Ruler Postulate
Segment Addition Postulate
Protractor Postulate
Angle Addition Postulate
Distance = | a – b |
If B is between A and C then AB + BC = AC
Angle Measure = |a – b|
If B is in the interior of ∠ AOC, then m ∠ AOB + m ∠ BOC = m ∠ AOC
ALGEBRAIC PROPERTIES
CONGRUENCE PROPERTIES
Addition, Subtraction, Multiplication, Division, Reflexive, Symmetric,
Transitive, Distributive
Reflexive, Symmetric, Transitive
_______________________________________________________________________
DEFINITIONS
Straight Angle Definition
Linear Pair Definition
Perpendicular Pair Definition
Supplementary Angles Definition
Complementary Angles Definition
Vertical Angles Definition
USED IN PROOFS
An angle is a straight angle iff its measure is 180
Two angles form a linear pair iff non-shared sides form a
straight angle.
Two angles form a perpendicular pair iff non-shared sides
form a right angle.
The measure of two angles adds up to 180 iff the angles
are supplementary.
The measure of two angles adds up to 90 iff the angles
are complementary.
Two angles are vertical angles iff their sides form two
pairs of opposite rays.
THEOREMS:
Have to be proven before being used
TO
1.
2.
3.
PROVE THEOREMS
Start with the given information.
Use a Definition to explain the given information.
Reason back from what you would like to prove using Definitions, Properties, Postulates and
Theorems that have already been proven.
4. Conclude with what needs to be proven.
___________________________________________________________________________
LINEAR PAIR THEOREM:
If two angles form a linear pair, then the angles are
supplementary.
GIVEN: ∠1 and ∠2 form a linear pair
PROVE: ∠1 and ∠2 are supplementary
STATEMENT
REASONS
1. ∠1 and ∠2 form a linear pair.
Given
2. ∠TAM is a straight angle.
Linear Pair Definition
(Two angles form a linear pair iff their non-shared
sides form a straight angle)
!
3. m∠TAM = 180
Straight Angle Definition
(An angle is a straight angle iff its measure is
180)
4.
m∠1+ m∠2 = m∠TAM
5. m∠1+ m∠2 = 180
6.
Angle Addition Postulate
!
Substitution
∠1 and ∠2 are supplementary angles
Supplementary Angle Definition
(The measure of two angles is 180 iff the angles
are supplementary)
Given
QED
Definition of Supplementary
The measure of two angles sums to 180°
iff the angles are supplementary.
Substitution Property of Equality
If mÐ 1 + m Ð 2 = mÐ TAM and mÐTAM = 180°
Then mÐ1 + m Ð2 = 180°
Definition of Linear Pair
Two angles form a linear pair
f two angles sums
180°
re supplementary.
m ∠BAT = 180°
TO PROVE THEOREMS
f Straight Angle
a straight angle, 1. Start with the given information.
asure is 180°.
ition Postulate 2. Use a Definition to explain the given information.
es are adjacent,
f their individual 3. Reason back from what you would like to prove using
the measure of the
Theorems that have already been proven.
y their non-shared
4.Conclude with what needs to be proven
ides
.
Definitions, Properties, Postulates and
AT form linear
pair
______________________________________________________________________
If two angles form a linear pair, then the angles are
supplementary.
AT = 180°
_____________________________________________________________________
on Property of
uality
VERTICAL ANGLES THEOREM:
If two angles are vertical angles, then the angles have
c = d then a+b = d
equal measures.
Vertical Angles Theorem
form a linear pair
If two angles are vertical angles
∠2
m a linear pair
∠1 ∠3
Then
they
have
equal
measures,
m ∠ 3 = 180°
= 180°
GIVEN: ∠1 and ∠2 are vertical angles
air Theorem
∠ 1 have
and ∠equal
2 are vertical
Angles
∠1Given
and ∠2
measures
form a linearPROVE:
pair,
plementary.
Prove that ∠ 1 and ∠ 2 have equal measures.
Given
LINEAR PAIR THEOREM:
∠BAT = m∠ CAT
180° - m ∠ 3
180° - m ∠ 3
STATEMENT
Statements
1. ∠1 and ∠2 are vertical angles
1 = m∠ 2
ve equal measures)
of Linear Pair
m a linear pair iff 2.
sides form a straight
are supplementary
e supplementary
operty of Equality
mber is subtracted3.
es of an equation,
uation is equivalent
vertical angles
f Supplementary
two angles sums to
4.
180°
e supplementary.
roperty of Equality
1) for (180° - m ∠ 3)
° - m ∠ 3)
∠1 and ∠3 form a linear pair
∠2 and ∠3 form a linear pair
REASONS
Reasons
Given
Linear Pair Definition
. form a linear pair iff their non(Two angles
shared sides form a straight angle)
∠1 and ∠3 are supplementary
∠2 and ∠3 are supplementary
m∠1+ m∠3 = 180!
m∠2 + m∠3 = 180
!
!
5. m∠1 = 180 − m∠3
Linear Pair Theorem
(If two angles form a linear pair then they are
supplementary)
Supplementary Angle Definition
(The measure of two angles is 180 iff the angles
are supplementary)
Subtraction Prop. of Equality
m∠2 = 180 − m∠3
!
6.
m∠1 = m∠2
Substitution
_______________________________________________________________________________
2.
4.
Proof:
Reasons
Statements
LINEAR PAIR THEOREM:
If two angles form a linear pair, then the angles are
l. Angle Addition Postulate
l. mll + m/-3 = 180;
supplementary
mL2*m/-3-180
2. Substitution Prop.
2. m/-l + mL3 - mL2
+ mL3
VERTICAL ANGLES THEOREM:
If two angles are vertical angles, they have equal measures
m/-3
3. Reflexive Prop.
m13 =
3.
_______________________________________________________
4. Subtraction Prop. of =
ON SMALL BOARD 4. mll
2 - mL2
3
GROUP
1-4
Example
IGOMILOG- 1.
?
'-!
2.
3
5-8
In the diagram,
/-4:
/-5.
Name two other angles congruent to 15'
J
J
Solution
L8: /-5
∠5
L4:congruent
L5, 17:
7 : other
L4 and
In the diagram, ∠4 ≅ ∠5 . Since
NameLtwo
angles
to L5.
35,and
and
wLUOY==35,
bisectsLSOa,
LSOa,wLUOY
OIbisects
diagram,OI
thediagram,
InInthe
angle.
ofeach
each
themeasure
measure
120.
Findthe
m/-YOW=
angle.
3.
Findofthe
value
of X
Find
Exs.
m/-YOW
11-14
=120.
Classroom Exercrses14.
14.wLZOW
wLZOW
13.wLZOY
wLZOY
13.
15.nIVOW
nIVOW
Find the measures
15.
16.m/-SOU
m/-SOU
and a supplement of LA.
of a complement
16.
17.wLTOU
wLTOU
17.
l.mLA=10
18.mLZOT
mLZOT
18.
2.mLA-75
3.mLA=89
4.
mLA:/
LM
value
Findthe
thevalue
x.x.
5.ofofName
Find
two right angles.
6. Name two adjacent complementary angles.
angles that are not adjacent.
7. Name two complementary
(4x+8)"
(4x+8)"
8. a. Name a supplement of /- MLQ.
, b. Name another pair of supplementary angles.
19.
19.
a.
b.
c.
d.
e.
m/-QIR =
m/-PIQ =
m/-VIT -
WIVIQ =
wLSIT:
?
?
?
?
/x'-n
Exs.S-E
ffix
O
B
9. In the diagram, assume that m/-CDB:90. Name:
/\
a. TWo congruent supplementary angles
an obtuse angle is ?
c. a right angle is
angles that are not congruent
_______________________________________________________________________________
Two
supplementary
b.
le is ?
aresupplements.
supplements.
22, /-l and /-2are
complementary angles
22 d. A straight
G angle
have a complement? 22, /-l and /-2 c. Two
G
arc
supplements
and
/-4
L3
supplements
/-4 arc
L3 and 10.
thitofofnLDFB
assume
diagram,
In the
- 90 and FE bi- CDEA
andL4.
L4.
/-2and
measures
the
findthe
m/-l=m13
/-2
measures
a.a.lflfm/-l
-27,find
=m13-27,
measure.
each ofofl-2
termsofofx.x.
L4ininterms
arrdL4
l-2arrd
measures
flrndFind
themeasures
m/-3IAFD.
mll ==sects
the
m/-3
b.b.IfIfmll
==x,x,flrnd
Exs. 9, l0
congruent?
c. mLBFE
be
b.
wLAFE
supplements
must
their
m/-AFD
a.
are
congruent,
angles
congruent?
c.
If
two
be
supplements
c. If two angles are congruent, must their
?
,/ l"
?
your conclusion.
Points, Lines, Planes, and Angles
the
measuresofofthe
themeasures
andthe
findthe
thevalue
valueofofr rand
supplementary,find
aresupplementary,
LAand
andLB
LBare
ItItLA
angles.
angles.
23.mLA-2x,mLB-x-15
BB 23.mLA-2x,mLB-x-15
nd a supplement
l.
mLB
mLB-- 2x2x-- 1616
16,mLB
m/-A==xx** 16,
24,m/-A
24,
the
themeasures
measuresofofthe
andthe
findthe
thevalue
valueofofyyand
complementary, find
arecomplementary,
andlD
lDare
LCand
of LB.lflfLC
- iz.s
angles.
angles.
4. m/B - 3x
25.mLC
mLC=3/
+5,mLD=2y
=2y
25.
=3/+5,mLD
+2
26.m/-C
m/-C=/
-3y+2
26.
=/ -8,mLD
-8,mLD-3y
d complementary. Find their measures.
findthe
the
equationtotofind
theequation
d supplementary. FindUse
Solvethe
equation.Solve
their
writeananequation.
measures.
giveninformation
informationtotowrite
Usethe
the
given
described.
two
angles
measures
of
the
measures of the two angles described.
ry angles.
entary angles.'
hat may or may not
P
angle'
theangle'
largeasasthe
twiceasaslarge
anangle
angleisistwice
27.AAsupplement
supplementofofan
27.
angle.
theangle.
largeasasthe
hvetimes
timesasaslarge
an,angleisishve
28.AAcomplement
complementofofan,angle
28.
twicethe
the
lessthan
thantwice
sixless
anglesisissix
complementaryangles
twocomplementary
oneofoftwo
29.The
Themeasure
measureofofone
29.
the
other.
of
measure
measure of the other.
42.
anglesisis42.
supplementaryangles
twosupplementary
themeasures
measuresofoftwo
betweenthe
30.
Thedifference
differencebetween
30. The
itssupplement.
supplement.
andits
itscomplement,
complement,and
theangle,
angle,its
measuresofofthe
Findthe
themeasures
Find
the
complementofofthe
largeasasaacomplement
timesasaslarge
sixtimes
angleisissix
anangle
supplementofofan
31.AAsupplement
31.
angle.
angle.
the
timesthe
eighttimes
angleisiseight
anangle
supplementofofan
themeasure
measureofofaasupplement
timesthe
32.Three
Threetimes
32.
angle.
theangle.
measure of a complementofofthe
/ 3l
l.3.i isin
/isinX;misiny;Xlly
Z;misinZ.
andmm
2.4.I tand.
doare
notcoplanar.
intersect.
(Def. of ll planes)
Given
l.3.Given
Def. ofplanes
coplanar
2.4.Parullel
do not intersect.
s.
tllm
(Def.
(See steps 2 and,
5.
Def.
of
ll
Hnes
of
ll
planes)
4.)
Corresponding Angles, Alternate Interior Angles, Alternate
Exterior
Angles,
3. /isin Z;misinZ.
3.
Given
Same Side Interior Angles, Same Side Exterior Angles
4. t and m are coplanar.
Def. of coplanar
The following terms, which are needed for 4.
future theorems about parallel
s.lines,
tllm
(See steps 2 and, 4.)
5.
Def.coplanar
of ll Hnes lines.
onlyby
to acoplanar
lines. line and two
Angles apply
formed
transversal
A transversal is a line that intersects two or more coplanar lines in different
points.
[n the next
diagram,
is a needed
transversal
of n aia
*. Theabout
angles
formed
The following
terms,
whichI are
for future
theorems
parallel
have
special
names.
lines, apply only
to coplanar lines.
Interior angles: angles 3, 4, 5,6
angles: angles
7, 8
A transversal
is a line that intersects twoExterior
or more coplanar lines l,in2,different
points.
[n the
Alternate
next angles
diagram,
interior
(alt.I is
a transversal
int.
A-) are two of
nonadjacent
n aia *. The
in- angles formed
have
special
terior
anglesnames.
on opposite sides of the transversal.
Interior
angles:
L3 and
/-6 angles 3, 4, 5,6L4 and L5
Exterior angles: angles l, 2, 7, 8
Alternate
interior
(alt.(s-s.
angles
Same-side
interior
int.int.
angles
A-)A-)
are are
twotwo
nonadjacent
ininterior angles
terior
angles
on the
opposite
sameonside
of the sides
transversal.
of the transversal.
L3
L3and
and,/-6
L5
L4
L4and
andL516
Same-side
Corresponding
interior
(s-s.d")
angles
(corr.
int.areA-)two
angles
areangles
two interior
angles
in correspondoning
thepositions
same side
of thetotransversal.
relative
the twb lines.
L3
1l and,
andL5
15
L2
and L4
/-6 and 16L3 and, /-7
/-4 and /-8
Corresponding angles (corr. d") are two angles in corresponding positions
relative Angles
to the twb ∠1
lines.
∠2 and ∠7
Alternate
Exterior
and ∠8
Classroom Exercises
1l and
15 Exterior
and /-6
L2Angles
Same
Side
L3 ∠7
and, /-7∠2 and /-4
∠1 and
∠8and /-8
1. The blue line is a transversal.
a. Name four pairs of corresponding angles.
b.
Name Exercises
two pairs of alternate interior angles.
Classroom
c. Name two pairs of same-side interior angles.
1. The blue line is a transversal.
56 / Chapter 2
a. Name four pairs of corresponding angles.
b. Name two pairs of alternate interior angles.
c. Name two pairs of same-side interior angles.
56
2
Chapter each
/ Classify
pair of angles as alternate interior angles, same'side
interior angles, corresponding angles, or none of these.
2. 12 and L4
4. 17 and Ll5
6. L7 and LIO
8. Lll and 114
3. L6 and /-10
5. 17 and /-12
7. Ll4 and Ll5
9. 1l and Lll
10.
Name alternate
exterior
alternate exterior angles, yorr
10. Although
we have not
definedangles.
two pairs of them.
whatside
theyexterior
are. Name
guess
11. can
Name
same
angles.
11. Name two pairs of angles we would call same-side exterior
angles.
t2. Suppose one pair of alternate interior angles are congruent
(say, /-2: /-7). Explain why the other pair of alternate
13.
interior angles must also be congruent.
Suppose a pair of same-side interior angles (say, /-2 arrd
L3) are supplementary. What must be true of any pair of
corresponding angles?
eac!-pair of lines as intersectrlQ parallel, or ske[_
c. AB and ID
b. AB and FK
a. AE and Ei
e
<+€
<+e
f. CN and FG
e.
EF
and
NM
d. EF and IH
14. Classify
--.4,
-.+__!
nt.
CORRESPONDING
ANGLES POSTULATE
Exs. 1-11
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL
Lines
o 15. THEN CORRESPONDING ANGLES ARE CONGRUENT
andangles?
a protractor in the last
rsversals,
numbered
Given: Parallel
cut by a Transversal
Conclude: Alternate Interior Angles are Congruent
that corresponding
angles Lines
are conhe did
not
inHowever,
our previous
postulates
and
theo∠1
≅
∠2
If k || l, then
(Corresponding Angles Postulate)
m. we Lines
will accept it as a postulate.
llel
allel
corresversals, and a protractor in the last
d that corresponding angles are conin ourthen
previous
postulatesangres
and theorsal,
corresponding
are
em. we will accept it as a postulate.
________________________________________________________________________________
ALTERNATE INTERIOR ANGLE THEOREM
ersal, then
IFcorresponding
TWOtheorems.
PARALLEL
LINES ARE CUT BY A TRANSVERSAL
angres are
ove the following
n
THEN ALTERNATE INTERIOR ANGLES ARE CONGRUENT
parallel lines are cut by a transverhen alt. int. A- are :.
Given: Parallel Lines cut by a Transversal
Conclude: Alternate Interior Angles are Congruent
arc:.
al,Athen
alternate interior angles are
If
k || l,theorems.
then ∠1 ≅ ∠2
(Alternate Interior Angle Theorem)
rove
following
sitive the
Prop.
of
=z
m∠1 = m∠3(VAT )
m∠3 = m∠2(CAP)
al, then alternate interior angles are
.
m∠1 = m∠2(substitution)
ALTERNATE EXTERIOR ANGLE THEOREM
/-1.Reasons
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL
/-1. l. Given
THEN ALTERNATE INTERIOR ANGLES ARE CONGRUENT
2. Yert. A- arc:3.Given:
If two parallel
are cut by a transReasons
Parallellines
Lines
cut by a Transversal
Conclude: Alternate Exterior Angles are Congruent
versal, then corr. A- arc
o.
4.
Transitive
Property
l.________________________________________________________________________________
Given
2. Yert. A- arc:SAME SIDE INTERIOR ANGLE THEOREM
3. If two parallel lines are cut by a transIF TWO PARALLEL LINES
ARE CUT BY A TRANSVERSAL
versal, then
corr.
A- arc
l, then same-side
interior
angles
areo.
SAME SIDE INTERIOR ANGLES ARE SUPPLEMENTARY
4.THEN
Transitive Property
Given: Parallel Lines cut by a Transversal
al, then same-side interior angles are
If k || l, then m∠1+ m∠4 = 180
Conclude: Same Side Interior Angles are
Supplementary
(Same Side Interior Angle Theorem)
m∠1 = m∠2(AIAT )
m∠4 = m∠2(LinearPT / sup.∠def )
m∠1 = m∠2(substitution)
SAME SIDE EXTERIOR ANGLE THEOREM
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL
THEN SAME SIDE INTERIOR ANGLES ARE SUPPLEMENTRY
Given: Parallel Lines cut by a Transversal
Conclude: Same Side Exterior Angles are
Supplementary
PROVING PARALLEL LINES THEOREMS
If parallel lines are cut by transversal, then …..
Corresponding Angles Postulate (CAP)
If two parallel lines are cut by a transversal, then
10/16/16, 1'48 PM
corresponding angles are congruent.
Alternate Interior Angle Theorem (AIAT)
If two parallel lines are cut by a transversal, then
Alternate interior angles are congruent.
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Alternate Exterior Angle Theorem (AEAT)
Page 3 of 10
If two parallel lines are cut by a transversal, then
Alternate exterior angles are congruent.
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Page 4 of 10
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Same Side Interior Angle Theorem (SSIAT)
10/16/16, 1'48 PM
If two parallel lines are cut by a transversal, then
Same side interior angles are supplementary.
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Page 5 of 10
Same Side Exterior Angle Theorem (SSEAT)
If two parallel lines are cut by a transversal, then
Same side exterior angles are supplementary.
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Page 6 of 10
WAYS TO PROVE TWO LINES ARE PARALLEL
1. Show that a pair of Corresponding Angles are congruent.
(Converse of Corresponding Angles Postulate; Converse of CAP)
2. Show that pair of Alternate Interior Angles are congruent.
(Converse of Alternate Interior Angle Theorem: Converse of AIAT)
3. Show that pair of Alternate Exterior Angles are congruent.
(Converse of Alternate Exterior Angle Theorem: Converse of AEAT)
correasure:
4. Show that a pair of Same-Side Interior Angles are supplementary.
Exam.ple
Which segments are parallel?
(Converse of Same-Side Interior Angles Theorem: Converse
ofI SSIAT)
Solution (ll HI and 7N are parallel sinc
sponding angles have the same m
Exam.ple
I Which
5. Show
thatsegments
a pairare
ofparallel?
Same-Side Exterior Angles are supplementary.
mlHIL-23+61 -84
(ll HI of
Solution
(Converse
Exterior
Angles Theorem: Converse of SSEAT)
andSame-Side
7N are parallel
since correm/TNI - 22 + 62 :84
sponding angles have the same measure:
(2) W and .lN are not paralle
mlHIL-23+61 -84
6t # 62.
m/TNI - 22 + 62 :84
CLASS DISCUSSION
(2) W and .lN are not parallel since
Exampl,e 2 Find the values of x and 7 that
6t # 62.
ACll DF and trll ar.
1. Which segments are parallel?
2. Find the values of x and y that make
Solution
,qCll opitmLCBF = mlBFE. (w
Exampl,e 2 Find the values of x and 7 that make
3*+20-n+50
ACll DF and trll ar.
Solution
,qCll
opitmLCBF = mlBFE.
3*+20-n+50
x-15
AEll BF
ake
if /AEF and /-F are supplemenrary. (Why?)
(2y-5)+(x+50)=180
(2y-s)1(15+s0):180
2y
=
/-
The following theorems can be proved using pr
rems. we state the theorems without proof, howev
120
work.
The following theorems can be proved using previous postulates and theorems. we state the theorems without proof, however, for you to use in future
work.
emenrary. (Why?)
120
0
Theorem 2-9
Through a point outside a line, there is exactly on
Through a point outside a line, there is exactly one line parallel to the given line.
given line.
Theorem 2-9
vious postulates
Through
and
theo- outside a line, there is exactly one line perpendicular
a point
r, for yougiven
to useline.
in future
P
a
Then line m exists
and is unique.
parallel to the given line.
Given this:
line perpendicular
to the
Theorem 2-8
Through a point outside a line, there is exactly one li
Theorem 2-8
Given this:
if /AEF and /-F are sup
(2y-5)+(x+50)=
(2y-s)1(15+s0):
2y
/-60
y?)
80
80
x-15
AEll BF
2x-30
'
since
2x-30
'
(why?)
to the
Given this:
P
a
P
Given this:
PC
Then line ft exists
Th
an
Th
an
CONVERSE OF AIAT:
If two lines are cut by a transversal to that Alternate Interior
Angles are congruent, then the lines are parallel.
CONVERSE OF AEAT:
If two lines are cut by a transversal to that Alternate Exterior
Angles are congruent, then the lines are parallel.
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CONVERSE OF SSIAT:
Page 7 of 10
If two lines are cut by a transversal to that Same-Side Interior
Angles are supplementary, then the lines are parallel.
CONVERSE OF SSEAT: If two lines are cut by a transversal to that Same-Side Interior
Angles are supplementary, then the lines are parallel
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HW#25
Worksheet / Fill Up the Blank Proofs
Page 9 of 10
GROUP WORK
Classroom Exercises
'ru
State which segments (if any) are parallel. State the postulate or theorem that
justilies your answer.
1.
W
300
Use the given information to name the segments that
must be parallel. If there are no such segments, say so.
4.
m/-l : mL8
5.
L5: L3
8.mL5+mL6=m13+mL4
6.
9. mLAPL + mLPAR =
L2:
'l
A
P
L7
7. mL5 =m/-4
4
I
L
R
180
10. Reword Theorem 2-8 as two statements, one describing existence and the
other describing uniqueness.
"
How many lines can be drawn through P parallel ,o E
12. How many lines can be drawn through Q parallel to PR?
13. H ow ma ny lines can be drawn through P perpendicular to
11.
€
a. QR?
b.
oR:?
\
\
--
\
\-\
a
Parallel Lines and Planes
R
/
67
b. In a plane two lines perpendicular to a third line must be parallel.
17.
In a plane,
why i
tll/
and
klln.
Use the diagram to explain
ll n.
More Practice
Written Exercises
HW#26
Use the given information to name the segments that
must be parallel. If there are no such segments, write
nonc.
A
t. /_t:
2. m12
/_4
3. m15 = m17
5. m16 : m/-9 - 90
7, m/-7 - m/-10 - m/-l
9.
12:15
4.
L5:
- m1l0
L8
:
mL3
-
90
m/-2 = m15
-
m18
6, m16
s.AUtOr,Nrtor
10.
11. Write the reasons.
Given: Tiansversal / cuts lines / and n;
1l -
/_2
Prove: / ll n
Proof:
Statements
l. Tiansversal / cuts /
4.
11/-2:
/-2:
5.
tll n
2.
J.
Reasons
l.
and n.
/-3
/-1
13
2.
J.
4.
5
12. Restate Theorem 2-9 as two statements, one describing existence and the
othe.r
uniqueness.
pairs of parallel
lines in each figure. Which congruent or supplemenName
,nro describing
figure. Which congruent or supplemenName ,nro pairs of parallel lines in each
tary angles did you use to determine the parallel lines?
parallel
lines? or supplementheWhich
you use
toindetermine
angles of
didparallel
congruent
each figure.
lines
Name
,nro pairs
68
Chapter
/ tary
or supplemencongruent
parallel
in
each
figure.
Which
pairs
lines
Name
,nro Name
F 2of,nro
14.
or supplemencongruent
13.
parallel
in
each
figure.
Which
pairs of
lines
the parallel lines? 14.H
tary13.
angles did you use to determinethe
parallel
lines?
you
determine
use
to
tary 12.
anglesF
did
H
parallel
lines?
13.
the
tary angles did you use to determine
14. H
13. F
14. H or supplemen13. F,nro pairs
14. H
13.ofFparallel lines in each figure. Which congruent
Name
tary angles did you use to determine the parallel lines?
14. H
13. F
BB
B
Find the values of x and y that make the red lines parallel and the blue lines
parallel.
Find
the values of x and y that make the red lines parallel and the blue lines
paralleland
andthe
theblue
bluelines
lines
Find
thatmake
makethe
thered
redlines
linesparallel
thevalues
valuesofofx xand
andyythat
Find
parallel.
the
and the blue lines
lines parallel
15. Find the values of x and y that make the red 16.
Bparallel.
parallel.
parallel.
parallel and the
y
Find14.
16.blue lines
15. values of x and that make the red lines 15.
B the
15.
parallel.
15.
B
16.
16.
15.
16.
16.
15.
Civen: Ll - /-2: 14: /-5
abou/-5
fQ and R-S? Be preWhat
can
you /-2:
prove
17.
14:
Civen:
Ll/-2:
17.Civen:
14:
/-5
Civen:
Ll
/-2:
17.
14:
/-5
Ll
17.
14:
Civen:
Ll
/-2:
your
class, if asked.
to
reasons in /-5
17. Civen:pared
14:
Llcan
/-2:give
-/-5abou
- can
abou
fQ
andBe
R-S?
What
you
prove
pre-Be prefQabou
and
R-S?
What
youprove
prove
abou
fQ
and
R-S?
What
can
you
Be
pre-Be
fQ
and
R-S?
What
can
you
prove
fQ
andpreR-S? Be preWhat
canabou
you
prove
18.
L3
L6
Given:
pared
give
your
in
class,
if
asked.
to
reasons
pared
give
your
in
class,
if
asked.
to
reasons
paredtotogive
give
your
inabout
class,
ifinasked.
reasons
pared
your
in
class,
if asked.
reasons
pared
give
your
class,
if asked.
reasons
Be preother
angles?
What
can to
you
prove
18.
L3
L6
Given:
18.
L3
L6
Given:
18. Given:
L3
L6
18.
L3
L6
Given:
in
class,
if
asked.
reasons
to giveL3your
pared
18. Given:
L6
-can
-about
Beprepre-Be preabout
otherangles?
angles?
What
can
you
prove
Be preabout
otherother
angles?
What
can
you
prove
about
other
angles?
What
you
prove
Be
What
can
you
prove
Exs.17,lE
a
about
other
angles? Be preWhat
can
you
prove
in
class,
if
asked.
reasons
to
give
your
pared
in
class,
if
asked.
reasons
to
give
your
pared
in
class,
if
asked.
reasons
to
give
your
in
class,
if
asked.
reasons
to give
your
paredpared
pared to give your reasons in class, if asked.
Exs.17,lE Exs.17,lE
a
Exs.17,lE
aaExs.17,lE
aa
in
a proof Exs.17,lE
19. Copy what is shown for Theorem 2-6 on page 65. Then write
17.
14. H
13. F
Find the values of x and
y that make the red lines parallel and the blue lines
parallel.
e-
B
Find the values of x and
y
16.
15.
that make the red lines parallel and the blue lines
parallel.
e-
B
16.
15.
a
5. Then
Exs.17,lE
17.
write
17. a proof in
Civen: Ll - /-2: 14: /-5
write a proof in
What can you prove abou fQ and R-S? Be prepared to give your reasons in class, if asked.
18. Given: L3 - L6
What can you prove about other angles? Be prepared to give your reasons in class, if asked.
17.
5. Then
C
18.
Civen: Ll - /-2: 14: /-5
What can you prove abou fQ and R-S? Be pre18.
pared to give your reasons in class, if asked.
Given: L3 - L6
What can you prove about other angles? Be prepared to give your reasons in class, if asked.
a
Exs.17
>\*
I .Zls-\
I \
Df
E
19. Copy what is shown for Theorem 2-6 on page 65. Then write a proof in
form.
atwo-columnExs.17,lE
20. Copy what is shown for Theorem 2-7 on page 65. Then write a proof in
A
two-column
in
a proof form.
19.
Copy what is shown for Theorem 2-6 on page 65. Then write
Exs.2l,22
C
two-column form.
parallel
RX and
gh S 20.
a proof
is shown for Theorem 2-7 on page 65. Then
Copytowhat
21.write
nn inOl; CO
Given:
two-column form.
Prove:
t O,q.
t
>\*
/-2
I .Zls-\
C
22. Given: :C: 1l: nf t O,4
f
I \ A
D
E
>\* Prove: COtO,q'
Exs.2l,22
I .Zls-\
\ A
I the measure
Df FindE
of LRST. (Hint: Draw a lin'e through S parallel to RX and
Ll
t Ol; CO t O,q.
- /-2
:
50"\ 1l: nf t O,4
. 22. tGiven: :C:
21. Given: nn
Prove: Ll
YT
Prove:
COtO,q'
Exs.2l,22
rv.)
Parallel Lines and Planes / 69
23.RX24.XR
Find the measure of LRST. (Hint: Draw a lin'e through S parallel to RX and
2
rv.)
:
23.RX24.XR
:
. t
YT
50"\
Parallel Lines and Planes
. t
YT
50"\
Parallel Lines and
/
69
2