can you
pattern
equal
cal
into his notebook.
Name_____________________________________________
Period____________ Score_________
ing of Marcos’s diagram is
Angle Relationships Exploration
ou identify
any lines that
ght
and provided
onmust
the Lesson
with Page.
the sameThis
number
of arrows
urce
type
of pattern is
Materials
Needed:
des of a parallelogram areTracing Paper
called(from
a tiling.
this] tiling, a
gram
Lesson In
1.3.2).
I. This and
diagram
shows a tilingto
of parallelograms.
am is copied
translated
fillIn addition,
an tracing paper can be used to
show that that the opposite angles of a
withoutUse
gaps
orpaper
overlaps.
parallelogram
have the
measure.
tracing
to trace angle A. Slide
it around
thesame
figure
and
mark the angles that are congruent.
Place tracing paper on top of the “root”
parallelogram and then translate it up
A
and to the right as shown at right. Then
rotate the vertical angles 180° about the
pairs
that you can identify in
common vertex to show that the opposite angles in a parallelogram must
have equal measure.
each parallelogram
is a translation of another, what
a. Name at least two types of angle
e stated about
the angles in the rest of Marcos’
the figure.
? Use a dynamic geometry
tool One
or way
tracing
paper
toask for volunteers to share what they
Suggested Lesson
to start today’s
lesson is to
Activity:
learned in Lesson 2.1.1 or you can use Reciprocal Teaching to review the
Lesson 2.1.1. Then focus the conversation on the
mine which angles must have thevocabulary
samefrommeasure.
relationships that the students studied: both geometric relationships (such
II. the
Anasenlarged
view
of one
theangle
parallelogram
vertical
pairs,
straight
angleportion
pairs, andof
right
pairs) and angletiling is shown
all angles that must be equal
same
color.
measure
relationships
(such
as
same,
that
is,
they
have
at the left, with some points and angles labeled. equal measure,
supplementary,]and complementary). Explain that this lesson will
J
end result only requires
two
colors.
J
continue to look for relationships between angle pairs.
b
a
c d
L
?
M
b.
surement
.
long
le
ram
a. A line
that crosses
twowarm-up,
or morewhich
other
is individually
called a transversal.
In
Problem
2-13 is a quick
canlines
be done
or in
teams
to
allow
you
to
assess
whether
or
not
students
can
recognize
the
Problem
continues
on
next
page
→
the diagram, which line is the transversal?
N
w x
y z
P
K
relationships they learned of in Lesson 2.1.1. You could use a modified
Hot Potato where each team member explains a different part, making
sure they
justify
answers.
Which
lines
are their
parallel?
Then move into problem 2-14. If you have access to a computer
with a projector, then show the PowerPoint presentation after
Trace ∠x
on tracing
paper
andstatement
shade and
its part
interior.
Then
translate
students
have read the
problem
(a). If you
do not
have
access
to
technology,
use
a
document
camera
or
overhead
transparencies
sliding the tracing paper along the transversal until it lies on top
to demonstrate the tiling of the parallelogram as described in the
anotherTechnology
angle and
matches
Notes
section. it exactly.
Much of the angle relationship development will depend on students
recognizing
that∠x?
an angle and its translated image must have equal
c. Which angle in the diagram corresponds
with
measure. Therefore, emphasize that the diagram is formed by a
translated parallelogram and encourage students to trace a parallelogram
on tracing paper and physically translate it on the Lesson 2.1.2 Resource
angles
because
they
are
d. In this diagram, ∠x and ∠b are called
corresponding angles because they are in the same
Page to verify that corresponding angles must have equal measure.
179
∠x by
of
nding
position at
ns of the transversal.
What
is
the
two different intersections of the transversal.
There
3 other
pairs
corresponding
Encourage students
to useare
tracing
paper to
verifyofthat
opposite angles ofangles. Name
nd b? Must one
be
greater
than
a
parallelogram
have
equal
measures.
They
can
trace
a parallelogram
in
them all. (Repeat the steps you used above to find each corresponding angle
pair if necessary.)
their text and then rotate the tracing paper about the center of the
you know. [ They are equal,
parallelogram to see that the opposite angles must have equal measure.
between this and what they
∠a and ______
∠b andExpect
∠x students to make
∠c connections
and ______
∠d did
andas ______
e of the original
they studied rotation symmetry in Lesson 1.2.6.
nslations are rigid
Move students on to problems 2-15 and 2-16. Note that these problems
e. How
of each topair
of corresponding
angles
in this
figure?
h and angle.
] would you describe the relationship
use transformations
preview
the angle relationships
students
will be
studying in Lessons 2.1.2 and 2.1.3: corresponding angles, alternate
interior angles, and same-side interior angles. Only corresponding angles
will be discussed using their formal name in this lesson. Note that the
Chapter 2: Angles and Measurement
175
f. Suppose b = 60º. Use what you know about vertical, supplementary, and corresponding angle
relationships to find the measures of all the other angles in the diagram.
∠a = ______
= ______Geometry
Core∠w
Connections
∠b = 60˚
∠c = ______
∠d = ______
∠x = ______
∠y = ______
∠z = ______
corresponding angle relationships to find the measures of all the other angles in
corresponding
corresponding
angle
anglec relationships
to find 60°;
the
to find
measures
thew,measures
of all
theofother
all the
angles
other
angles in
Julia’s
diagram.
[ x, y,relationships
and
all measure
a, d,
and
z measure
120°.
] in
Julia’s diagram. [ x, y, and c all measure 60°; a, d, w, and z measure 120°. ]
Julia’s diagram.
Julia’s diagram.
[ x, y, and
[ x,cy,all
and
measure
c all measure
60°; a, d,
60°;
w, a,
and
d, zw,measure
and z measure
120°. ] 120°. ]
corresponding
angles
always
havehave
equal
measure?
2-17. Exploration:
Frank wondersDo
whether
corresponding
angles
always
equal
measure. For parts
2-17.
Frank wonders whether corresponding angles always have equal measure. For parts
2-17. III.
2-17.
Frank
wonders
Frank
wonders
whether
whether
corresponding
angles
always
angles
have
always
equal
havemeasure.
equal have
measure.
Forthe
parts
For parts
(a)
through
(d) through
below,
use
tracingcorresponding
paper
to
decide
if corresponding
angles
For
parts
(a)
(a) through(d)
(d)below,
below,use
usetracing
tracingpaper
papertotodecide
decideififcorresponding
correspondingangles
angleshave
havethe
thesame
(a)
through
(a)
through
(d)
below,
(d)
use
below,
tracing
use
paper
tracing
to
paper
decide
to
if
decide
corresponding
if
corresponding
angles
have
angles
the
have
the
same
measure.
Then
determine
if
you
have
enough
information
to
find
the
measures
measure. Then determine
if youThen
have determine
enough information
find theinformation
measures oftoxfind
andthe
y. Ifmeasures
you do, find
same measure.
if you havetoenough
same
measure.
same
measure.
Then
determine
Then
determine
if
you
have
if
you
enough
have
enough
information
information
to
find
the
to
find
measures
the
measures
of
x
and
y.
If
you
do,
find
the
angle
measures
and
state
the
relationship.
the angle measures
of x and
and state
y. If the
yourelationship.
do, find the angle measures and state the relationship.
of x andofy.x If
and
you
y. do,
If you
finddo,
thefind
angle
themeasures
angle measures
and stateand
thestate
relationship.
the relationship.
b.
a. a.
b.
Notice that the
a.
b.
a.
a.
b.
b.
x
x
transversal is horizontal
xy
x
x
y
y
y
y
yx
x
x
in examples a and b.
In both, the 135˚ angle
and ∠x are a pair of
[ x = 135° (corresponding angles),
[ y = 45° (straight angle pair), not
[ x = 135° (corresponding angles),
[ y = 45° (straight angle pair),corresponding
not
angles.
y
y
= 135°
[(vertical
x (corresponding
= 135°angles)
(corresponding
[ y = 45°
[ y(straight
= 45° (straight
angle
pair), not
y [=x135°
] angles),angles), enough
information
for xpair),
]anglenot
y = 135° (vertical angles) ]
enough information for x ]
y = c.
135°y (vertical
= 135° (vertical
angles)angles)
]
]
enoughenough
information
for x ] for x ]
d. information
c.
d.
c.
xd.
x
c.
c.
d.
d.
x
x
yx
x
y
y
x
y
x
y
y
y
y
[ x = 135° (corresponding angles),
[ x = 45° (straight angle pair),
[ x = 135° (corresponding angles),
[ x = 45° (straight angle pair),
y [=x45°
(straight
angle
pair)angles),
]
not
enough
information
for
y ] pair),
= 135°
[ x (corresponding
=y135°
(corresponding
angles),
[
x
=
45°
[
x
(straight
=
45°enough
(straight
angleinformation
pair),
angle
= 45° (straight angle pair) ]
not
for y ]
y =corresponding
45° y(straight
= 45° (straight
angle
angle] pair)
not enough
not
enough
information
information
for y measures
] for y ] equal?
e. Do
anglespair)
always
have ]equal measure?
If not,
when
are their
e.
Answer Frank’s question: Do corresponding angles always have equal measure?
e.
Answer Frank’s question: Do corresponding angles always have equal measure?
not,
are their
measures
equal?
[ No, corresponding
angles
have
equal
e. IfAnswer
e. when
Answer
Frank’s
Frank’s
question:
corresponding
Do corresponding
angles[ always
angles
have
always
equal
havemeasure?
equal measure?
If not,
when question:
areDo
their
measures
equal?
No,
corresponding
angles
have equal
measure
when
lines
intersected
by the
transversal
are parallel.
] have equal
If not, when
Ifonly
not,
are
when
theirthe
are
measures
their
measures
equal?
equal?
[ No,
corresponding
[ No,
corresponding
angles
have
angles
equal
measure
only
when
the
lines
intersected
by
the
transversal
are
parallel.
]
f. Conjectures
aremeasure
often
written
in lines
the the
form,
“If…,
then…”.
A statement
in if-then
measure
only when
onlythe
when
intersected
lines
intersected
by the transversal
by
the transversal
are
parallel.
areform
parallel.
] is called
] a
statement.
Make
a conjecture
about“If…,
corresponding
by completing
f.conditional
Conjectures
are often
written
in the form,
then…”. Aangles
statement
in if-then this statement:
f.
Conjectures are often written in the form, “If…, then…”. A statement in if-then
form
is
called
a
conditional
statement.
Make
a
conjecture
about
corresponding
f.
Conjectures
f.
Conjectures
are often
arewritten
often
written
in the form,
in the
“If…,
form,then…”.
“If…,
then…”.
Aa statement
A statement
inabout
if-then
in if-then
form
is called
a conditional
statement.
Make
conjecture
corresponding
angles
byform
completing
this
conditional
statement:
“If
…, then
corresponding
“If
_______________________________________________________________________,
form
is
called
is
a
called
conditional
a
conditional
statement.
statement.
Make
a
Make
conjecture
a
conjecture
about
corresponding
about
corresponding
angles by completing this conditional statement: “If …, then corresponding
angles
have
equal
[this
If lines
are parallel,
then
corresponding
angles
by
angles
completing
by measure.”
completing
this
conditional
conditional
statement:
statement:
“Ifare
…,parallel,
then
“If
…,
corresponding
then
then
corresponding
angles
have
equal
measure.”
angles
have
equal
measure.”
[ If lines
thencorresponding
corresponding
angles
have
equal
measure.
] [ If lines[ If
angles
have
angles
equal
have
measure.”
equal
measure.”
are
lines
parallel,
are
parallel,
then
corresponding
then
corresponding
angles
have
equal
measure.
]
2-18.
For 2-18.
each diagram
below,
find
the
value
of
x
,
if
possible.
If
it
is
not
possible,
For each diagram below, find the value of x , if possible. If it isexplain
not possible, explain
angles
have
angles
equal
have
measure.
equalfind
measure.
]theyou
] of Be
2-18.how you
Forknow.
each
diagram
below,
value
x ,prepared
ifyou
possible.
If it
is every
not possible,
explain
State
the
relationships
use.
to
justify
how
you
know.
State
the
relationships
use.
Be
prepared
to
justify
g.
Prove that your conjecture in part (f) is always true. That is, explain why this every
g.you
Prove
that
your
conjecture
(f) is
always
true.
is,every
explain
whyyou
this know.
howdiagram
you
know.
State
the
relationships
use.
prepared
to
justify
IV.measurement
For each
below,
find
the
value
of your
x in
,you
ifpart
possible.
If
it isteam.
not
possible,
explain
how
find
to
other
of
team.
[Be
are
multiple
measurement
toare
other
members
ofThere
your
[That
There
multiple
isyour
a theorem.
[members
Iffind
lines
then
corresponding
are
g. conjecture
Prove
g. that
Prove
that
conjecture
youryou
conjecture
in
part
(f)
in [parallel,
part
isIfalways
(f) isare
true.
always
That
true.is,then
That
explain
is,angles
explain
whyare
this
whyangles
this are
conjecture
is
a
theorem.
lines
parallel,
corresponding
measurement
finduse.
to other
members
ofstudents
your
team.
[ There
multiple
Statecongruent
the
relationships
you
Bestudents
prepared
to justify
every
measurement
you
find.]
methods
for
each,
soyou
make
sure
justify
their
responses.
] are
methods
for
make
sure
justify
their
responses.
because
rigid
transformation
that
carries
one
angle
conjecture
conjecture
is a theorem.
isthere
aeach,
theorem.
[isIfasolines
[ If
are
lines
parallel,
parallel,
then
corresponding
then
corresponding
angles
are
because
there
is
a are
rigid
transformation
that]angles
carriesare
one angle
methods forcongruent
each, so make
sure
students
justify
their responses.
the congruent
other
(translation).
]a rigid
congruent
because
because
there
there
is atransformation
rigid
that carries
that c.carries
one angle
one angle
a. a. onto
b.
c.
the
otheris(translation).
] transformation
a. onto b.
b.
c.
onto
the
onto
other
the
(translation).
other
(translation).
]
]
Chapter 2: Angles anda.Measurement
181
b.
c.
Chapter 2: Angles xand Measurement
181
x
Chapter 2:
Chapter
Angles2:and
Angles
Measurement
and Measurement
x
115°
53°
53°
181
181
53°
115°
115°
x
x
x
10°
[ x = 65° ]
[ 4x !]25° = 3x[ +4x
! 25° = 3x + 10°
[ x = 65°[ Not
] enough information
[ Not enough] information
because
the enough
lines
areinformation
not
4x ]! 25°
+ 10°
[ x = 65° ]
[ Not
] sonotx = [35°
= 35°
because
the lines are
so =x 3x
]
parallel.
]
x
=
35°
because
the
lines
are
not
so
]
parallel. ]
parallel. ]