GEO6 GEOMETRY AND MEASURESUREMENT Student Pages for Packet 6: Drawings and Constructions

Transcription

Name ___________________________
Period __________
Date ___________
GEO6
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STUDENT PAGES
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GEOMETRY AND MEASURESUREMENT
Student Pages for Packet 6: Drawings and Constructions
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GEO6.1 Geometric Drawings
 Review geometric notation and vocabulary.
 Use a compass and a ruler to make geometric drawings.
 Learn about and understand congruence.
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GEO6.2 Geometric Constructions
 Learn the SSS axiom for triangle congruence.
 Use a compass and straightedge to make classic Euclidean
constructions.
 Justify constructions by completing proofs.
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15
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GEO6.3 Vocabulary, Skill Builders, and Review
1
Geometry and Measurement Unit (Student Pages)
GEO6 – SP
Drawings and Constructions
WORD BANK (GEO6)
Definition
Example or Picture
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Word
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altitude
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angle bisector
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diagonal
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central angle
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midpoint
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perpendicular
bisector
Measurement and Geometry Unit (Student Pages)
GEO6 – SP0
6.1 Geometric Drawings
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GEOMETRIC DRAWINGS
Ready (Summary)
Set (Goals)
We will make geometric drawings using a
straight edge and a compass. We will
review geometric notation and vocabulary.
Go (Warmup)
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 Review geometric notation and
vocabulary.
 Use a compass and a ruler to make
geometric drawings.
 Learn about and understand
congruence.
These are obtuse angles.
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These are acute angles.
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Right angles are angles that measure exactly 90o. These are right angles.
2. Describe obtuse angles in your own
words.
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1. Describe acute angles in your own words.
Geometry and Measurement (Student Pages)
GEO6 – SP1
LABELING GEOMETRIC FIGURES
Object
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Name each object. Use the words: angle, line, line segment, ray or point.
Picture
1.
How to Label
A dot with a capital letter near it.
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P
Two points named on the line with a
“double arrow line” on top. Order does
not matter.
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A
2.
B
AB or BA
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The endpoint and another point on the
ray named with a “single arrow ray” on
top. Order does matter.
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3.
C
DC only
The endpoints named on the line
segment with a “no- arrow segment” on
top. Order does not matter.
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E
4.
F
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EF or FE
Three points. The middle point is
always the vertex of the angle.
5.
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K
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 GHK or  KHG
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6. Write two names for the ray that has
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N as an endpoint.
N
_____ and _____
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7. Write the two names for the segment that
has endpoints at M and L .
_____ and _____
8. Write two names for the angle at x.
M
x
L
J
Q
_____ and _____
GEO6 – SP2
LABELING GEOMETRIC FIGURES (continued)
Picture
How to Label
M
Three points in any order.
9.
L
Example:
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T
LMN
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Object
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Name each object. Use the words: congruent triangles or triangle.
Congruent and corresponding parts
are named in order.
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RST  UVW
Examples:
S
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10.
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ST  VW
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V
W
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Congruent sides are labeled with the
same number of “hash marks.”
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Congruent angles are labeled with
distinguishing marks as well.
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RST  UVW
The two triangles on the right are congruent because they have the same size and same
shape.
11.
Z
C
X
CB  __________
A
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13.
B
ZYX   __________
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12.
Y
XYZ   __________
14.
Draw the appropriate markings on all angles and all sides.
GEO6 – SP3
DRAWING AN ISOSCELES TRIANGLE
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An isosceles triangle is a triangle with two equal sides. Use a centimeter ruler and follow the
directions to draw an isosceles triangle. Use the words in the box to fill in the blanks.
1. Draw a 6-cm horizontal line segment from and extending to the right of point A. Label the
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other endpoint B.
2. Find the point on AB that divides the segment into two equal or congruent parts. This point
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is known as the ____________________ of AB . Label it point M.
3. Locate point M on AB and draw a 4-cm line segment, DM , that is perpendicular (  ) to
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AB at point M.
Label the other endpoint D. ( AB _____ MD )
4. Connect point A with point D and point B with point D to make triangle ABD (∆ABD).
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How long is AD ? _____ cm. How long is BD ? ______ cm.
5. AD is ____________________ to BD ( AD _____ BD ).
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6. What kinds of angles are  AMD and  BMD? ____________________
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What kinds of angles are  MDA and  MDB? ____________________
What kinds of angles are  MAD and  MBD? ____________________
7. MD is called the height, or ____________________ of ∆ABD.
acute angles
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altitude
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congruent
midpoint
right angles
A
GEO6 – SP4
DRAWING A CIRCLE
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Use a centimeter ruler and a compass. Follow the given directions to draw a circle.
1. With a radius of 3 cm, draw a circle with center at point C. How many degrees is a full turn
of a circle? _____.
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2. Draw a horizontal diameter through C. Label the endpoints F and H. Name both radii.
_____ and _____ .
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3. Name the diameter. _____ How long is this diameter? _____ cm.
4. Write an equation that relates the length of the diameter (d) to the length of the radius (r) of
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any circle. ____________________ .
5. Find and label a point R on the circle. Draw  FHR. What kind of angle is  FHR?
____________________ .
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6. Find and label a point N on the circle so that  FCN is an obtuse angle.
We call this a central angle, because its vertex is at the center of the circle.
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Name another central angle.
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____________________.
7. Any segment that has both endpoints on a
circle is called a chord. Name both chords
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C
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in this diagram. _____ and _____ .
GEO6 – SP5
DRAWING A TRAPEZOID
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Use a centimeter ruler and follow the given directions to draw a trapezoid.
1. Starting at point W, draw a 4 cm horizontal line segment. Label the other endpoint X.
2. From point X, draw XY so that it is perpendicular to WX and is 3 cm.
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3. From point Y, draw YZ to the left of point Y so that it is parallel to WX and is 2 cm.
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4. Draw in ZW . What is the approximate length of this segment? _____ cm
5. Refer to trapezoid WXYZ.
Are there any right angles? _____
Name them if any exist __________________
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Are there any acute angles? _____ Name them if any exist __________________
Are there any obtuse angles? _____ Name them if any exist __________________
6. Draw a line segment from point Z to point X. This is called a diagonal. Draw another one in
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this figure. Name this segment _____
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7. Draw an altitude from point Z to WX at a point called V. What kinds of angles are  WVZ
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and  XVZ? __________
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Diagonal (a segment that
is not the side of a
polygon, but connects
two vertices)
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Parallel (two lines in the
same direction)
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GEO6 – SP6
YOU WRITE THE DIRECTIONS
square
rectangle
parallelogram
segment
midpoint
congruent
parallel
perpendicular
right
angle
height
altitude
obtuse
angle
acute
angle
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triangle
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Word Bank
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Make a geometric drawing here. Write detailed directions of your drawing on a separate piece
of paper. Give the directions to your partner and see if your partner can duplicate your drawing
using only your written directions. You must use at least six words from the word bank.
A Drawing From Your
Partner’s Directions
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Your Drawing
GEO6 – SP7
6.2 Geometric Constructions
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GEOMETRIC CONSTRUCTIONS
Ready (Summary)
Set (Goals)
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 Learn the SSS axiom for triangle
congruence.
 Use a compass and straightedge to
make classic Euclidean constructions.
 Justify constructions by completing
proofs.
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Go (Warmup)
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We will learn about the Side-Side-Side
axiom (SSS) for triangle congruence. We
will use SSS to complete proofs that justify
three classic Euclidean compassstraightedge constructions.
x
A
B
y
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60o
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Use the figure above to answer the questions.
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1. Starting at point A and tracing clockwise, the rotation around the full circle (from A back to
A) is _____ degrees.
2. Starting at point A and tracing clockwise, the rotation around half of the circle (from A to B)
is _____ degrees. The is referred to as a straight angle.
3. The small square in the diagram means that the angle is _____ degrees. This is referred to
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as a right angle.
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4.  x is _____ degrees. It is also a _____________________.
5.  y is _____ degrees.
6. From the diagram we see that two right angles form a ______________________.
GEO6 – SP8
CREATING A TRIANGLE
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Your teacher will give you three pieces of spaghetti in different lengths. Tape the three pieces
together to make a triangle.
1. Trace your triangle and your partner’s triangle below.
Your triangle
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Your partner’s triangle
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The Side-Side-Side axiom states that if two triangles have corresponding sides that are
congruent, then the triangles are congruent to one another.
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∆CAT  ∆DOG
C
A
D
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2. Make hash marks on the triangles above
to show corresponding, congruent sides.
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a. CA  _____
b. AT  _____
c. TC  _____
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3. Mark corresponding, congruent angles on
the triangles above.
a.  D   _____
b.  O   _____
c.  G   _____
GEO6 – SP9
SOME GEOMETRY FACTS
Two triangles are
congruent if their
corresponding sides are
congruent to one
another and their
corresponding angles
are congruent to one
another.
reflexive
property of
congruence
In geometry, a figure is
congruent to itself.
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Congruent Parts of
Congruent Triangles
are Congruent
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CPCTC
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SSS
axiom
Two triangles are
congruent if their
corresponding sides are
congruent to one
another.
Symbols
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Picture
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definition of
congruent
triangles
Words
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Short Name
A compass opening
defines a length.
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radius
facts
All arcs drawn with the
same compass opening
have equal radius
lengths.
All radii of a given circle
have the same length.
GEO6 – SP10
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CONSTRUCTION 1: BISECT AN ANGLE
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Use a straightedge and a compass to divide this angle into two congruent parts (bisect).
What do we know?
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Directions
1. Draw an arc from V that intersects both
rays. Label the points of intersection A
and B.
VA  VB , because all ________________
from a given circle have the ____________.
drawn with the same compass opening have
3. Draw VX
VX  VX _______________ property
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2. With the same compass setting, draw an
arc from A and an arc from B into the
interior of the angle, long enough so they
intersect. Label the point of intersection
X. Draw AX and BX .
AX  BX , because all _______________
equal _____________length.
Use the diagram you just constructed to prove that AVX  BVX .
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Statement
AX  BX ; VX  VX
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VA  VB ;
Reason
Given above
∆AVX  ∆BVX
_____ axiom
AVX  BVX
___ ___ ___ ___ ___
GEO6 – SP11
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CONSTRUCTION 2: DRAW A PERPENDICULAR THROUGH
A POINT ON A LINE
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P
Use a straightedge and a compass to draw a perpendicular to P.
Directions
What do we know?
TP  WP , because all _______________
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1. From P, draw an arc that intersects the
line to the left and to the right. Label the
points of intersection T and W.
from a given circle have the ___________.
2. Draw an arc from T and an arc from W
with the same compass setting (but
longer than in step 1) so that they
intersect above P. Label the point of
intersection N. Draw TN and WN .
drawn with the same compass opening have
3. Draw PN .
PN  PN __________________ property
TN  WN , because all _______________
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equal _____________length.
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Use the diagram you just constructed to prove that NP  TW .
Statement
Reason
Given above
∆TPN  ∆WPN
_____ axiom
TPN  WPN
___ ___ ___ ___ ___
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TP  WP ; TN  WN ; PN  PN
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TPN and WPN must both be right
angles
NP  TW
The sum of TPN and WPN must be
______ (a straight angle), and if they are
congruent, they must each be _______.
Two lines that form right angles are
____________________ to one another.
GEO6 – SP12
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CONSTRUCTION 3: COPY AN ANGLE
Use a straightedge and a compass to copy  C with the vertex at F and the given ray.
What do we know?
CG  _____, because
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1. Draw an arc from C that intersects both
rays. On the horizontal, label the point of
intersection G. Label the other point of
intersection D.
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Directions
2. Using the same compass setting, draw an
arc from F on the given ray. On the
horizontal, label the point of intersection
H.
GD  _____, because
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3. Draw GD . Then place the sharp point
your compass on G and the pencil tip on
D to “measure” GD . Using this compass
setting, draw an arc from H that intersects
the arc above it. Label this point of
intersection K. Draw HK .
FH  _____, because
4. Draw KF .
KF  _____, because
Use the diagram you just constructed to prove that  F   ____ .
Statement
Given above.
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CG  _____
Reason
FH  _____
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GD  _____
∆______  ∆______
 F   ____
GEO6 – SP13
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CONSTRUCTION 4: DRAW A PERPENDICULAR SEGMENT
FROM A POINT OFF OF A LINE
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P
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Use a straightedge and a compass to draw a perpendicular from P to the line.
Directions
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1. From P, draw an arc that intersects the line to the left and to the right. Label the points of
intersection T and W.
2. From T, draw an arc below the line (be sure that it is somewhere underneath P). From W,
draw an arc with the same compass opening that intersects the previous arc below the
line. Label the point of intersection N.
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3. Draw PN , which is perpendicular to the line. Label the point of intersection X.
GEO6 – SP14
6.3 Vocabulary, Skill Builders, and Review
FOCUS ON VOCABULARY (GEO6)
1
2
3
4
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7
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Fill in the crossword puzzle using the clues below.
9
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11
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12
13
14
Down
2. AC of this parallelogram
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Across
1. Bisector of a segment
3. Forms right angles.
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5. A _______ is 90 
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C
A
B
A
4. Line segment or ray in an angle that
divides it into two congruent angles.
6. Greater than 90  and less than 180 
11. Another word for altitude
7. A proven mathematical statement
12.A line with two end points
8. Squares A and B are _____.
13. The perpendicular distance from a vertex
to the opposite side of a plane figure.
14. Convincing argument to justify and
mathematical statement.
9. Lines that never cross and are always the
same distance apart.
A
B
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8. An angle that has the center of a circle as
its vertex.
10. Less than 90 
proof
diagonal
altitude
perpendicular
theorem
central angle
Word Bank
parallel
congruent
angle bisector
right angle
midpoint
height
acute angle
obtuse angle
segment
GEO6 – SP15
SKILL BUILDER 1
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Draw a picture, write an appropriate formula, and substitute to solve each problem.
2. The perimeter of a square ballroom is
49.6 cm. What is the area of the
ballroom?
a. Sketch the figure
c.
Substitute and solve
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c. Substitute and solve
b. Write an appropriate formula
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1. The height of a rectangle measures 62
cm and its length measures 53 cm. Find
the area.
d. Answer the question
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GEO6 – SP16
SKILL BUILDER 2
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2. A triangular lawn has an area of 59.2
square meters. The base of the lawn is
14.8 meters wide. Find the height of the
lawn.
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1. The area of a trapezoid is 40.25 sq.
inches and with bases that measures 7
inches and 4.5 inches. Find the height of
the trapezoid.
Sa
GEO6 – SP17
SKILL BUILDER 3
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2. The length of cereal box is 21 cm. The
width 6.5 cm. The height is 6 cm more
than the length of the box. Find the
volume.
e. Sketch the figure
f. Write an appropriate formula
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g. Substitute and solve
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1. Find the area of a circle whose diameter
is 24 cm.
h. Answer the question
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Express each ratio as a fraction in simplest form.
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3. 12 : 36 _________________
4. 18 out of 20 students are UCLA basketball fans. _______________
5. 25 of the 35 students received an A grade on the last quiz. ________________
GEO6 – SP18
SKILL BUILDER 4
Backyard
Office
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BED2
Dining Room
Kitchen
Kitchen
Living Room
Scale
2 cm = 9 ft
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BATH1
BED1
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The scale drawing below of a floor plan was created using a 2 cm = 9 ft scale.
Use the ratio strip to answer the questions.
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1. If the length of the living room on the scale drawing is 7 cm, what is the actual length of
the living room?
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2. Jo has an 8ft by 10ft area rug. Will this rug fit in the living room? Explain.
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3. What is the actual area of the dining room?
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4. Red built a round dining table that has a diameter of 1 m. Will the dining table fit in the
dining room?
2 cm
8 cm
4 cm
10 cm
2 cm = 9 ft
9 ft
27 ft
63 ft
GEO6 – SP19
SKILL BUILDER 5
Convert each measurement.
A map (scale drawing) of a city shows 1 in = 0.6 miles. On the map, the distance from
the library to the park is 4 inches. What is the actual distance from the library to the park?
2.
A half marathon is 13.1 miles. Convert this distance to kilometers and to meters.
(HINT: 1 kilometer  0.6 miles)
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1.
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3. How many cups are in 2 gallons? (HINT: 4 cups = 1 quart, 4 quarts = 1 gallon)
Determine the unit rate.
5. $25 in 2 hours
6. 6 cups of fruit for 4 milkshakes.
7. 7 books in 5 weeks
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4. 60 kilometer in 6 hours
GEO6 – SP20
SKILL BUILDER 6
Use a straightedge and follow the directions to draw each figure.
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1. Equilateral triangle
a. Draw a horizontal base of 8 cm and label it
PR .
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b. Find the midpoint and label it M.
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c. Draw the perpendicular bisector to PR at
M.
d. Find point Q so that MQ has a length of 3
cm.
f. What kind of triangle is ∆PQR?
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e. What is the approximate length of PQ ?
_____ cm RQ ? _____ cm
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2. Parallelogram
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N
P
a. Draw a horizontal base of 6 cm. Label it GH
.
b. Find the midpoint of GH . Label it N.
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c. Draw NK 3 cm long so that it is
perpendicular to GH at N. Draw GK .
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d. Complete parallelogram HGKL by drawing
KL and HL . (Recall that opposite sides
must be congruent and parallel.)
e. What is the approximate length of GK ?
_____ cm
G
GEO6 – SP21
SKILL BUILDER 7
5
2
1
7
3
8
2.
1 1

6 4
4.
 1  3 
 - 4  - 8 
  
5.
10 
3 7
- 
8 12
3.
6.
3  5
1   -2 
4  8
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1
4
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1.
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Compute.
9
20
8.
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7.
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Write each fraction or mixed number as a decimal.
3
4
5
36
24
9.
-
12.
7 2  3  6
Simplify each expression.
42  7  3
11.
72  28
62
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10.
GEO6 – SP22
TEST PREPARATION (GEO6)
1. Which of the following shows CD ?
C
D.
D
C
2. Which of the following correctly labels this drawing?
EF
B.
C.
EF
E
B.
CM
A
C.
AB
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UW  TU
B.
TW  TU
C.
TUV  TVW
D.
TVU  TVW
B
M
D.
4. Refer to the figure. Which of the following statement is true?
A.
EF
C
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CA
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3. Which line segment is perpendicular to AB ?
A.
F
D.
EF
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A.
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C.
B.
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C
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A.
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Show your work on a separate sheet of paper and choose the best answer.
CB
T
U
U
V
W
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5. Two triangles are congruent if the corresponding sides are congruent to one another.
Sa
A.
C.
The statement is always true.
The statement is sometimes true.
B.
The statement is never true.
D.
The statement is true ONLY if the
angles are congruent.
GEO6 – SP23
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This page is intentionally left blank for notes.
GEO6 – SP24
KNOWLEDGE CHECK (GEO6)
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Show your work on a separate sheet of paper and write your answers on this page.
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1. Draw and label line segment BD.
C
D
N
A
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2. Find and label a point R on the circle so that ACR is an acute angle.
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3. Find and label the congruent angles of TUW .
4. Find and label the congruent sides of TUW .
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U
V
W
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5. What do you know about TVU and TVW ?
GEO6 – SP25
Home-School Connection (GEO6)
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Here are some questions to review with your young mathematician.
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Q
T
P
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M
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2. Draw a 2-cm line segment, PQ , that is perpendicular to MN .
Label the other endpoint Q. ( MN ____ PQ )
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1. Write two names for the ray that has Q as an endpoint.
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Parent (or Guardian) signature ____________________________
Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by
using appropriate tools (e.g. straightedge, ruler, compass, protractor, drawing software).
Draw quadrilaterals and triangles from given information about them (e.g. a quadrilateral having
equal sides but no right angles, a right isosceles triangle).
Identify and construct basic elements of geometric figures (e.g. altitudes, mid-points, diagonals,
angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles)
by using a compass and straightedge.
Demonstrate an understanding of conditions that indicate two geometrical figures are congruent
and what congruence means about the relationships between the sides and angles of the two
figures.
Formulate and justify mathematical conjectures based on a general description of the
mathematical question or problem posed.
Express the solution clearly and logically by using the appropriate mathematical notation and terms
and clear language; support solutions with evidence in both verbal and symbolic work.
Note the method of deriving the solution and demonstrate a conceptual understanding of the
derivation by solving similar problems
pl
e:
D
MG 5.2.1
o
Selected California Mathematics Content Standards
MG 6.2.3
MG 7.3.1
MG 7.3.4
m
MR 7.1.2
MR 7.2.6
Sa
MR 7.3.2
FIRST PRINTING
Measurement and Geometry Unit (Student Pages)
DO NOT DUPLICATE © 2009
GEO6 – SP26

GEO6 GEOMETRY AND MEASURESUREMENT Student Pages for Packet 6: Drawings and Constructions

Transcription

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