Download Geometric Figures

Geometric Figures
Chapter Focus
You will identify and describe basic geometric figures. Angle pairs such as
adjacent angles and vertical angles will be used to solve problems. You will
draw shapes that satisfy given conditions, and you will use transformations
to produce congruent figures on a coordinate plane.
('~~~'MON
Chapter at a Glance
:
Lesson
Standards for Mathematical Content
8-1
Building Blocks of Geometry
PREP FOR
PREP FOR
8-2
Classifying Angles
CC.7.G.5
8-3
Line and Angle Relationships
CC.7.G.5
8-4
Angles in Polygons
CC.7.G.2
8-5
Congruent Figures
PREVIEW Of
Problem Solving Connections
Performance Task
Assessment Readiness
CORE
"~""
.......... CC.7.G.5, CC.7.G.6 CC.S.G.2
Unpacking the Standards
Understanding the standards and the vocabulary terms in the
standards will help you know exactly what you are expected to learn
in this chapter.
What It Means to You
You will learn about supplementary, complementary, vertical, and
adjacent angles. You will solve simple equations to find the measure of
an unknown angle in a figure.
EXAMPLE
II line p. Find the measure of each angle.
Line n
p
n
Adjacent angles formed by two intersecting lines are supplementary.
mL5 + 55°
1800
mL5:= 1800
-
55°
Vertical angles are congruent.
mL6:= 55° mL7 = 1250 Alternate interior angles are congruent. mL3 == 55 0
Alternate exterior angles are congruent.
mL2
1250
Corresponding angles are congruent.
mL1
55 0
mL4 = 125 0
Draw (freehand, with ruler and protractor,
and with technology) geometric
shapes with given conditions. Focus
on constructing triangles from three
measures of angles or sides, noticing
when the conditions determine a unique
triangle, more than one triangle, or
no triangle.
Use geometry software to construct an obtuse triangle with angles that
measure 120°, 26°, and 34°.
protractor (transportador) A tool for
measuring angles.
Use the" dilate" tool to enlarge and reduce the triangle.
What can you conclude about constructing triangles with given angles
of 120°, 26°, and 34°?
You can draw several triangles with various side lengths with angles
that measure 120°, 26°, and 34°.
Key Vocabulary
adjacent angles (angulos adyacentes) Angles in the same plane that have a common vertex and a common side. complementary angles (angulos complementarios) Two angles whose measures add to 90°. congruent angles (angu/os congruentes) Angles that have the same measure. line (linea) A straight path that has no thickness and extends forever. line segment (segmento de recta) A part of a line made of two endpoints and all points between them. plane (plano) A flat surface that has no thickness and extends forever. point (punto) An exact location that has no size. ray (rayo) A part of a line that starts at one endpoint and extends forever in one direction. reflection (reflexi6n) A transformation of a figure that flips the figure across a line. rotation (rotaci6n) A transformation in which a figure is turned around a point. supplementary angles (angulos suplementarios) Two angles whose measures have a sum of 180°. translation (traslaci6n) A movement (slide) of a figure along a straight line. vertical angles (angulos opuestos por el vertice) A pair of opposite congruent angles formed by intersecting lines.
Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ _ii:!'<
Building Blocks of Geometry
Essential question: How do you identify and describe basic geometric figures?
A~ln! is an exact location in space with
no size. A fift~ is a straight path with no thickness that extends forever in
opposite directions. point P
~
LM
A D~~i~ is a flat surface with no thickness that extends forever.
plane ABC
Ati:, is a part of a line. It has one
-->
endpoint and extends forever in one
direction.
RS
A~~p~i~~ijl is a part of a line or ray
.p
M
~
s
~
H
GH
between two endpoints.
~
Drawing Geometric Figures
Use a ruler or straightedge to help you sketch each geometric figure in
the drawing box.
~
co
Q.
A
Draw line XY.
E
8
O'l
c:
~
::a::J
0­
1::
::J
~
co
:c
.S:
Draw ray XZ so that point Z does not lie on lineXY. The endpoint is always listed fIrst in the name of a ray, so the endpoint of ray XZ is point _ _ __ iE
~
c:
E
Draw segment YZ.
O'l
::J
o
:c
@
The endpoints of segment yz are
and _ _ __
points
Draw plane XYZ.
Write a set of directions for drawing at least 3 geometric figures like those
described above. Trade your directions with a partner, and draw the figures
he or she describes. Then check each oth~r's work.
Chapter 8
311
Lesson 1
•
•
•
•
Point: Use a capital letter.
Line: Use any two points on the line or a lowercase letter.
Plane: Use any three points in the plane that are not on the same line .
Ray: Use the endpoint and another point on the ray. List the endpoint first .
• Line segment: Use the endpoints.
Naming Geometric Figures
i~ A
List all possible names of the line shown.
You can name a line by using _ _ _ points.
The order in which you list the points does I does not matter.
Possible names for the line are fiB, ___
_____ ,and _____
.•~.. List all possible names of the ray with endpoint D shown.
You can name a ray by using ____ points, one ofwhich must be the endpoint. The endpoint of the ray must be listed first I second. Possible names for the ray are
and ______ List all possible names of each figure.
2a. the plane
2b. the segment with endpoints K and M
____ and ______
K
~M
2c. How is a line segment similar to a ray? How are they different?
m. What mistake
Error Analysis Christi named the line shown as
did she make? What is a correct name for the line?
Chapter 8
312 T
~
Lesson 1
Name ________________________
Identify the figures in the diagram.
1. three points __________________
2. one line _ _ _ _ _ _ _ _ _ __
3. a plane _______________
4. four rays ________________
5. three line segments ____________ Identify the figures in the diagram. 6. four points _ _ _ _ _ _ _ __
7. three lines ______________
8. a plane _____________
9. three rays _______________
10. four line segments __________________
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C
III
a.
E
o
u
en
c
~
Identify the figures in the diagram.
11. four points __________
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o
12. two lines ___________
~
III
I
.~
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C
o
1:
en
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o
13. a plane ________________
14. four rays ____________
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15. five line segments ______________
16. Identify the line segments that are congruent in the figure.
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A,---+f---"i'--__
F
Chapter 8
313
Practice and Problem Solving
Write the correct answer. The drawing shows a section of the Golden Gate Bridge in San Francisco. 1. Identify two lines that are suggested
by the bridge.
2. Identify a ray and a line segment that
are suggested by the bridge.
3. Identify two lines in the figure that are
in the same plane.
4. Identify a plane in the figure.
Choose the letter for the best answer. The drawing is an artist's sketch for an abstract painting. r-t--t=---t=-H-t----lF
7. Which line segment is congruent to
8. Which line segment is congruent to
EO?
OF?
A AC
C OF
F EO
H CJ
B CD
o
G HM
J FM
Chapter 8 GH
314
Practice and Problem Solving
Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _______ Date----mw
Classifying Angles
Essential question: How can you use angle pairs to solve problems?
Recall that two rays with a common endpoint form an angle. The
two rays form the sides of the angle I and the common endpoint
marks the vertex. You can name an angle several ways: by its
vertexl by a point on each ray and the vertexl or by a number.
B
c
Angle names: LABCI LCBA I LBI Ll It is useful to work with pairs of angles and to understand how pairs of angles relate to each other. Congruentangles are angles that have the same measure . • =rr:CI!'",
•• J%. ....
B
Measuring Angles
Using a rulerl draw a pair of intersecting lines. Label each angle from 1 to 4.
Use a protractor to help you complete the chart.
.,.>
M~aslJ..ef)f.AJlgle·
Angle
mL1
mL2
mL3
mL4
mL1 +mL2
mL2 + mL3
mL3 + mL4
mL4+ mL1
.•.
Conjedure Share your results with other students. Make a conjecture about pairs
ofangles that are next to each other.
Chapter 8
315
Lesson 2
~~i~~~~~i~~ are pairs of angles that share a vertex and one side but do not
overlap.
~~~==~~:~~~ are two angles whose measures have a sum of 90°.
~
~;~_l~are
two angles whose measures have a sum of 180°. You
have discovered in Explore 1 that adjacent angles formed by two intersecting lines
are supplementary.
t:••~~B;M:~I~]:"0\, Identifying Angles and Angle Pairs
Use the diagram below.
..14: Name a right angle. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
iii' Name a pair of adjacent angles. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Name a pair of complementary angles. _ _ _ _ _ _ _ _ _ _ _ _ _- - - - - - ­
Name an angle that is supplementary to LCFE. _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
~...
Name an angle that is supplementary to LBFD. _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
iF Name an angle that is supplementary to LCFD. _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Name a pair of non-adjacent angles that are complementary. _ _ _ _ _ _ _ _ _ __
2a.
What is the measure of LDFE? Explain how you found the measure.
2b.
Are LCFB and LDFE adjacent angles? Why or why not?
Are LBFD and LAFE adjacent angles? Why or why not?
Chapter 8
316
Lesson 2
Wtf~~l!;le\
Finding Angle Measures
Find the measure of each angle.
~;
LBDC
c
A
LBDC and _ _ _ _ are _ _ _ _ _ _ _ _ _ _ angles. The sum of their measures is _ _ _ __ Write an equation to help you find the measure of LBDC. 75 +x
In the box, solve the equation for x.
mLBDC=
If· LEHF
• •
E
• •
G
LEHF and _ _ _ _ are _ _ _ _ _ _ _ _ _ angles. The sum of their measures is _ _ _ __ In the box, write and solve an equation to help you find mLEHF.
mLEHF=
3a.
Find the value of x, and mL/ML.
L
j
mL/ML=x=
Chapter 8
317
N
Lesson 2
PRACTICE
For 1-5, use the figure.
1. mLQUP+mLPUT= _ _ _ _ _ _ _ _ _ _ __
2. Name a pair of supplementary angles.
3. Name a pair of complimentary angles.
4. Name a pair of adjacent angles.
5. What is the measure of LQUN? Explain your answer.
Solve for the indicated angle measure or variable.
6. mLYLA= _ _ __
1. x =_______
S
L~
•
J
A
8. The railroad tracks meet the road as
shown. The town will allow a parking
lot at angle] if the measure of angle] is
greater than 38°. Can a parking lot be
built at angle]? Why or why not?
9.
A student states that when the sum of two angle measures equals 180°, the
two angles are complementary. Explain why the student is incorrect.
Error Analysis
Chapter 8
318 Lesson 2
Name _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ Date _ _ __
<:
Tell whether each angle is acute, right, obtuse, or straight.
1.
/
2.
3.
Use the diagram to tell whether the angles are complementary,
supplementary or neither.
4. LAQC and LGQC
5. LBQD and LDQE
6. LCQEand LEQF
~
[
E
o
7. LGQF and LFQE
u
Cl
.::
~
:0
;;;J
a.
t::
;;;J
o
8. LBQC and LDQC
u
:;;
:r:
.5
:to
~
.::
.s
..r::
Cl
;;;J
9. Angles Wand Xare supplementary. If mLWis 3r,
what is mLX?
o
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@
10. Angles S and Tare complementary. If mLS is 64°,
what is mLT?
11. Angles C and 0 are supplementary. If mLC is 83°,
what is mLD?
12. Angles U and V are complementary. If mLU is 41°,
what is mLV?
Chapter 8
319
Practice and Problem Solving
Write the correct answer.
The drawing shows a scene on a calendar.
1. L1 and L2 are complementary angles. If L1 measures 35°, what is the measure of L2? 2. L3 and L4 are supplementary angles. If L3 measures 50°, what is the measure of L4? 3. Which angle is an obtuse angle: L6 or L7? 4. Which angle labeled on the drawing
is a right angle?
Choose the letter for the correct answer. Use the diagram to complete Exercises 5 and 6. 5. Which of the following could be the measures of LTZU and LQZR? A mL TZU 55° and mL QZR = 55° B mL TZU 25° and mL QZR = 90° C mLTZU= BO° and mLQZR= 100° D mLTZU = 35° and mLQZR = BO° s
R
6. If LRZS measures 35°, what is the measure of LSZT? F 155° G 145° H 55 0
J 45°
7. LA and LB are complementary angles. The measure of LB is 4 times the measure of LA. What are the measures of the angles? A mLA 16° and mLB = 64°
B mLA 1Bo and mLB = 72°
C mLA:!:: 36° and mLB = 144°
D mLA 45° and mLB = 1350
Chapter 8 B. The hands of a clock form an acute
angle at 1:00. What type of angle do
they form at 4:00?
F acute
G right
H obtuse
J straight
320
Practice and Problem Solving
Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _______ Date----.iil
Line and Angle Relationships
Essential question: How do you use vertical angles to solve problems with
figures?
~djat~:~~ have a common vertex and a common ray, but
do not have any interior points in common. Vertical angles are
opposite angles formed by two intersecting lines.
Angles Formed by Intersecting Lines
Use the figures below to determine relationships between pairs of angles
formed by intersecting lines.
A Use a protractor to measure the numbered angles in the figures to the nearest degree.
Record the measures in the table.
Figure 1 Figure 2
Figure 1
Figure 2
.11·· Add the measures of each pair of adjacent angles and record the sums in the table.
Figure 1
Figure 2
'(:.. Describe the pattern in the sums of the measures of the adjacent angles of each figure.
I)
In each figure, Ll and L3 are vertical angles, and Land L
are vertical angles. Describe the pattern in the measures of each pair of vertical angles. Chapter 8
321 Lesson 3
!
1a. Make a Conjecture Make a conjecture about any pair of adjacent angles formed
by a pair of intersecting lines.
I
i
,~
~. Make a Conjecture Make a conjecture about any pair of vertical angles.
PRACTICE
The figure is formed by a pair of intersecting lines.
Use the figure for each problem.
1. Use a protractor to fmd the measures of L1, L2, and L3.
mL1
=___---', mL2 =___----', and mL3 = ____
2. Explain how you could find the measure of L 1 without using a protractor. 3. Explain how you could find the measure of L2 without using a protractor.
@
::t:
oc: '"S:r ::J s: ;:s
::J
The figure shows the intersection of Pine Street, West Avenue, and
Shady Lane. Use the figure for each problem.
...::t:
8
c:
4. Write and solve an equation to find the measure of L 1.
"S!:c:
;:+
.,;'
:r
:;'
-----7~~---West
Ave.
'"on
3
al
5. What is the measure of L2? Justify your answer
without using a protractor.
6.
~
Shady Ln.
L2 and L3 are complementary angles. Write and solve an equation to find the measure of L3.
Chapter 8
322 Lesson 3
,
I
Name _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ __
~
'"--,
Find the measure of each angle.
2. L2 andL4
1. L5 and L6
~
.
­
6314
4. L6 and L7
3. L4 and L8
y
3 4
87" 8
6. L3 and L4
5. L1 and L2
x
Chapter 8
323
Practice and Problem Solving
Write the correct answer.
Use the diagram below for 1-3.
1. What is the measure of L1?
2. What is the measure of L2?
3. What is the measure of L3?
Choose the letter for the best answer.
The map shows the area around Falcon Park.
~~2~l
4. If 4 measures 112°, what is the measure of 2? Birch St.
S. Which two angles are vertical
angles? F L2 and L3
H L2 and L4 G L2 and L6
J L2 and LS
Orchard St.
6. Which two angles are adjacent angles? A L1 and L4
C L3 and LS B L6 and L8
D LS and L8 7. If L4 measures 112°, what is the measure of L1? Chapter 8 324
Practice and Problem Solving
Name~
_ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date--_',if
Angles in Polygons
Essential question: How can you draw shapes that satisfy given conditions?
Draw each triangle with the given conditions.
Triangle 1
Angles: 30° and 80°
Included side: 2 inches
Triangle 2
Angles: 55° and 50°
Included side: 1 inch
Use a ruler and a protractor to draw each triangle with the given angles
and included side length.
/(,. Draw Triangle 1.
Step 1: Use a ruler to draw a line that is 2 inches long. This will be the
included side.
Step 2: Place the center of the protractor on
the left end of the 2-in.1ine. Then
make a 30°-angle mark.
Step 3: Draw a line connecting the left side of
the 2-in.line and the 30°-angle mark.
This will be the 30° angle.
L
2 in.
Step 4: Repeat Step 2 on the right side of the
triangle to construct the 80° angle.
Et.
Step 5: The side of the 80° angle and the side
of the 30° angle will intersect. This is
Triangle 1 with angles of 30° and 80°
and an included side of 2 inches.
Draw Triangle 2.
Chapter 8
325
Lesson 4
1a.
Conjedure When you are given two angle measures and the length of the included side,
do you get a unique triangle?
:,~:W,0"'~!''';''':,;:,x""
Co;";,:"""
Two Sides and a Non-Included Angle
Use a ruler, protractor, and compass to construct a triangle with given
lengths of 2 inches and 1! inches and a non-included angle of 45°.
A non-included angle is the angle not between the two given sides.
Step 1: Use a ruler to draw a straight line. This will be part of the triangle, but
does not have to measure a specific length.
Step 2: As in
D, place the center of the protractor on the left end of the
line. Then make a mark at the correct 45-degree point. Use your ruler
to make this side of the triangle 2 inches long.
Step 3: Make your compass the width of I! inches. Place the sharp point
on the end of the 2-inch side that you just drew in Step 2. Rotate
the compass until it intersects, or meets, the bottom line twice
(see figure).
Step 4: The point where the compass crosses the bottom line shows
where a line can be drawn that is exactly I! inches long. Use your ruler to verify the length and draw the line. 2a.
LL...
Is there another triangle that can be drawn with the given conditions?
When you are given two side lengths and the measure of a non-included
angle, do you get a unique triangle? Explain.
Chapter 8
326
Lesson 4
".I~'--'.. . "'"~~.,,,. L:;
a
Drawing Three Sides
Use geometry software to draw a triangle whose sides
have the following lengths: 2 units, 3 units, and 4 units.
Step 1: Draw three line segments of 2, 3, and 4 units
E
of length.
F
c=
D
C
=3
B
A
a=
Step 2: Let
be the base of the triangle. Place
endpoint C on top of endpoint Band
endpoint E on top of endpoint A. These will
become two of the vertices of the triangle.
D
B
E
A a =2 C
Step 3: Using the endpoints C and E as fixed
vertices, rotate endpoints F and D to see
if they will meet in a single point.
The line segments of 2, 3, and 4 units do / do not
form a triangle.
zJ
~4
b=3
E
A a=2 C
~&"-,_~,,,,,.,,_,
':1; _,
""'-4'<"'-~""<'''''',
c~
Of
B
,
i~!!!~!~!~~:l;j)l
Repeat Steps 2 and 3, but start with a different base length. Do the line segments make
the exact same triangle as the original?
Use geometry software to draw a triangle with given sides of 2, 3, and 6 units. Do these
line segments form a triangle?
Conjecture When you are given three side lengths that form a triangle,
do you get a unique triangle or more than one triangle?
Chapter 8
327
Lesson 4
PRACTICE
1. On a separate piece of paper, draw a triangle that has side lengths of 3 cm
and 6 cm with an included angle of 120°. Determine if the given information
makes a unique triangle, more than one triangle, or no triangle.
2. Use geometry software to determine if the given side lengths can be used to
form one unique triangle, more than one triangle, or no triangle.
3. On a separate piece of paper, draw a triangle that has degrees of 30°, 60°, and
90°. Measure the side lengths.
a. Can you draw another triangle with the same angles but different side lengths?
b. If you are given 3 angles in one triangle, will the triangle be unique?
4. Draw a freehand sketch of a triangle with three angles
that have the same measure. Explain how you made
your drawing.
Chapter 8
328 Lesson 4
Name _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _
-
IJOL'C _ _ __
Use a ruler, protractor, and compass to construct each figure.
1. Draw a triangle that has side lengths of 3 em and 4 em with
an included angle of 90°. Determine if the given information
makes a unique triangle, more than one triangle, or no triangle.
2. Draw a triangle that has angles that measure 45°,45°,
and 90°. Determine if the given information makes a unique
triangle, more than one triangle, or no triangle.
Chapter 8 329
Practice and Problem Solving
Use a ruler, protractor, and compass to construct each figure.
1. Draw a triangle that has sides that measure 5 em, 5 em, and 11 em. Determine if the given information makes a unique triangle, more than one triangle, or no triangle. 2. Draw a triangle that has two angles that measure 30° and 40°
with an included side length of 5 em. Determine if the given
information makes a unique triangle, more than one triangle,
or no triangle.
Chapter 8 330
Practice and Problem Solving
Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _______ Date ____ ;P1
Congruent Figures
Essential question: How can you produce congruent figures on the coordinate
plane?
A transformation is a change in the position, shape, or size
of a figure. A translation is a transformation that slides a
geometric figure. A translation moves each point of a figure
the same distance in the same direction. Translation
[F
' ", '
~;.•",",,",".,,''.,', ,:'/ '/ , '' ,', ., '
'EX P LOR E\ Translating a Figure
Translate !:lASe on the coordinate plane as described below.
A
List the coordinates of the vertices of !:lABe.
)
10 ~~-:+,~
9 +--;".'1:'"
8 +--I---¥lIr,
7 +-"-~c'\"'f-"''"''
8
Trace the triangle on paper and cut it out.
C Place the cutout triangle on the original triangle.
Translate, or slide, the cutout triangle 4 units to the
right. List the coordinates of the vertices of the trans­
lated triangle.
>­
c
'"
Co
E
0
u
C>
6 -r-i~;"";H\.L-r5 +--'~'!,-"""!-,,I-:;;;
4
3
2 +---1--+­
2 3 4 5 6 7 8 9 10
Compare the coordinates of the translated
triangle with the coordinates of the original triangle. How do the
x-coordinates compare? How do the y-coordinates compare?
c
~
jj
"
0"
~
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't
.!:
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c
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.s:
0 Once again, place the cutout triangle on the original triangle. This time, translate the cutout
triangle 5 units down. List the coordinates of the vertices of the translated triangle.
0
)
C>
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J:
@
Compare the coordinates of the translated triangle with the coordinates of the original
triangle. How do the x-coordinates compare? How do the y-coordinates compare?
Chapter 8
331 Lesson 5
1a. Place the cutout triangle on the original triangle. Translate the cutout triangle 3 units to the
right and then 1 unit up. List the coordinates of the vertices of the translated triangle.
1b. What If...1 What if you translated t,.ABC 2 units to the left? How would the x-coordinates of
the vertices change? How would the y-coordinates of the vertices change?
When you translate t.ABC, is the translated triangle congruent to the original triangle? How
do you know?
A
is a transformation that flips a figure across
~~.itl~Ji is a transformation that turns a figure
about a point.
Reflecting and Rotating a Figure
Reflection
Rotation
t
I
F+'i
I
I
Y
Reflect and rotate t,.JKL on the coordinate plane as
described below.
. • ~... list the coordinates of the vertices of t,.JKL.
~l,
)
B Trace the triangle on paper and cut it out.
:C Place the cutout triangle on the original triangle. Reflect,
or flip, the cutout triangle across the y-axis. The arrow from
vertex K shows where vertex K should be in the reflected triangle.
List the coordinates of the vertices of the reflected triangle.
)
Compare the coordinates of the reflected triangle with the coordinates of the
original triangle.
Chapter 8
332 Lesson 5
~,; Once again, place the cutout triangle on the original triangle. This time, rotate, or turn,
the cutout triangle 900 clockwise about the origin. (A 900 tum is of a full tum.) The arrow
from vertex L shows where vertex L should be in the rotated triangle.
t
List the coordinates of the vertices of the rotated triangle.
1), L(
)
Compare the coordinates of the rotated triangle with the coordinates of the
original triangle.
!IIEFI..£Cr
'<";- "
;,,--,,,~;;;YY,.,;·,,",·
<e_._~,,,·
2. When you reflect or rotate t:.JKL, is the reflected or rotated triangle congruent to the original
triangle? How do you know?
Describing a Translation
A computer artist translates 6.PQR to produce 6.STU. Describe this
translation.
Determine how many units left or right 6.PQR was
translated.
t:.STUis _ _ _ units to the right I left of t:.PQR.
Determine how many units up or down t:.PQR was
translated.
t:.STU is _ _ _ units up I down from t:.PQR.
So, the artist translated t:.PQR
units to the
right I left and
units up I down.
s
u
3. Compare the coordinates of the vertices of t:.STU with the coordinates of the
vertices of t:.PQR.
Chapter 8
333 Lesson 5
PRACTICE
Trace t:,FGH on paper and cut it out. Use the cutout triangle
for each problem.
1. Translate t:,FGH 5 units to the left. What are the coordinates of
the translated triangle?
2. Translate t:,FGH 1 unit to the right and 4 units down. What are
the coordinates of the translated triangle?
3. Reflect t:,FGH across the x-axis. The arrow from vertexH shows where vertexH should be in
the reflected triangle. What are the coordinates of the reflected triangle?
~2,
), G(
4. Rotate t:,FGH 1800 (or ~ of a full tum) about the origin. The arrow from vertex F shows where
vertex F should be in the rotated triangle. What are the coordinates of the rotated triangle?
), G(
5. Compare the coordinates of the vertices of the rotated triangle with the coordinates of the
vertices of the original triangle.
Use the triangles on the coordinate plane for each problem. 6
6. How is t:,ABC translated to produce t:,JKL?
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c:
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s:
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7. How is t:,JKL translated to produce t:,XYZ?
x
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8
c:
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c:
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8. Error Analysis Blake says that t:,XYZ could be translated 4 units
to the right and 6 units up to produce t:,ABC. What error did he
make? §:
5°
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0
3
-0
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9. Reasoning t:,DEFhas sides that measure 3 inches, 4 inches, and 5 inches. t:,GHJhas sides that
measure 6 inches, 8 inches, and 10 inches. Could t::.GHlbe a translation of t::.DEF? Explain.
Chapter 8
334 Lesson 5
Name _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ _ _;
Describe each translation.
1.
2.
Graph each transformation.
3. Translate MBC 7 units to the right and 1 unit down.
4. Reflect MBC 7 across the x-axis.
Chapter 8
335
Practice and Problem Solving
1. Triangle OEF has vertices at 0(0, 0),
E(-2, -3), and F(-5, -3). Rotate
t,.OEF 90° clockwise about the vertex
O. Graph both triangles and write the
coordinates of the vertices of the
image.
Choose the letter for the best answer.
2. What transformation of triangle 1 created triangle 2? A translation 3 units right and 1 unit down B translation 8 units right and 1 unit down C rotation of 180° about the origin
D reflection across the y-axis
4. If you reflect triangle 1 across the
x-axis, what will be the coordinates
of the new triangles?
3. If you rotate triangle 2 90° clockwise
about vertex 0, what will be the
coordinates of the new triangle?
F O'{3, 1), E'(7, 1), F(3, -3)
A A'(5, 2), 8'(5, 6), C'(1, 2) G 0'(3, 1), E'(3, -3), F(7, 1)
B A'(-5, 0), 8'(-5, -4), C'{-1, 0) H 0'(3, 1), E'{-4, 1), F(-3, 3)
C A'(5, -2), 8'{5, -6), C'(1, -2) J 0'(3, 1), E'(-3,3), F(-7, 1)
D A'(-5, -2), 8'(-5, -6), C'(-1, -2) Chapter 8 336
Practice and Problem Solving
Name _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ _.,:;,
Problem Solving Connections What's the Pattern? Archeologists are exploring the ruins
of an ancient palace. One of the walls contains a tile pattern
that has been badly damaged. The archeologists find a scrap
of paper with information about one of the tiles. They want to
use the information to reconstruct the pattern. What will the
complete pattern look like?
D Find Angle Measures
The figure shows the scrap of paper that the archeologists found. The triangle at the center of the paper, .6.ABC, represents one of the tiles used to make the pattern. A· Name
all of the lines that are shown in the figure. B
Name three different rays that are shown in the figure.
,C
What type of angle pair are LBAC and LDAC? Why?
D
Let x represent the measure of LBAC. Show how to write and solve an
equation to find mLBAC.
,<"
E
What type of angle is LACG ? How do you know?
337 Let y represent the measure of LACB . Show how to write and solve an
equation to find mLACB.
What is mLABC? How do you know?
D
Draw the Triangle
The archeologists use what they have discovered to help them draw a full-size
version of one of the tiles.
lit, The archeologists find that the length of side AC in the tiles was 2 inches.
Use a ruler and protractor to draw a triangle that has a 39 angle, a 53 angle,
and an included side, AC, that is 2 inches long. Label the vertices of the
triangle A, B, and C.
0
0
....
~
..'Q.
Is there a different triangle that can be drawn with the given conditions?
Make another drawing of the triangle using a ruler and
protractor. Make the drawing on a piece of cardboard
or heavy paper. Label the vertices inside the triangle, as
shown at right. Then cut out the triangle. You will use
this triangle as a template to reproduce the tile pattern.
338 A
Problem· Solving· COll l1 ections
D; Begin the Pattern
v
n;u,'
The archeologists reproduce the tile pattern on a piece of t-inch graph paper. Use
the coordinate plane on the next page to help you recreate the pattern
A .. Place your template triangle
on the coordinate plane so that pOint A is at the
origin and side AC lies on the positive x-axis. Plot points on the coordinate
plane at A, B, and C. What are the coordinates of these points?
...;
B· Use the points you plotted to draw .6.ABC on the coordinate plane.
0
C
Now rotate .6.ABC 180 counterclockwise about the origin. Draw the rotated
triangle on the coordinate plane. What are the coordinates of the vertices of
the rotated triangle?
D·
How do the coordinates of the vertices of the rotated triangle compare with
the coordinates of the vertices of the original triangle?
Are the two triangles congruent? Why or why not?
One of the archeologists suggests that rotating .6.ABC 180 clockwise about
the origin might produce a different design. Do you agree or disagree? Why?
0
339 D Answer the Question
The archaeologists use additional transformations to complete the pattern.
Continue your work on the coordinate plane below.
A
Translate .6ABC 8 units to the left. Draw the translated triangle on the
coordinate plane. What are the coordinates of the vertices of the translated
triangle?
B
Now translate the rotated triangle 8 units to the right. Draw the translated
triangle on the coordinate plane.
C Now translate D.ABC 5 units left and 4 units down. Draw the translated
triangle on the coordinate plane.
·D
Continue in this way, filling in and extending the pattern by rotating
and translating D.ABC. Draw as much of the pattern as possible on the
coordinate plane.
340 Problem SolVIng Connections
Name _ _ _ _ _ _ _ _ _ _ Class _______ Date ___,,:i!f
Performance Task
.,1.
Philippe drew this figure in his notebook.
a. Name all pairs of vertical angles in the figure.
b. Find the value of x. Show your work.
. . 2. The diagram shows the intersections of
3 roads. Main Street and Newton Avenue are
perpendicular. If a car is traveling north on Newton
Avenue, it must turn 140 to take a right onto Fay
Drive. At what angle must a car turn to take a right
onto Main Street from Fay Drive? Does it matter
whether the car is coming from north or south of
Main Street? Explain.
0
341
Main 5t
Fay Dr
Newton Ave
'*
.,3.
Terry chalks the design at right onto
the asphalt for a game he's making up.
a. The measure of each interior angle of the pentagon is 108°.
What are the measures of the angles of the triangle that are
adjacent to the pentagon? Explain.
h. Find the measure of the third angle of the triangle. Show your work.
Cassandra is making a design for a logo. One part of the design is
a triangle with two congruent sides. She must draw the triangle with
at least one side with length 6 centimeters, and at least one side with
length 4 centimeters.
a. Sketch two possible figures that Cassandra could use. Label
the side lengths in both figures.
h. Suppose the side length 4 centimeters is changed to 2 centimeters.
How many triangles are possible? Explain using sketches.
.'.~
,..•....
342 Perforroance Task
CHAPTER 8 (~~C~~~N
.. .....
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ASSESSMENT READINESS Name _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ __ \
SELECTED RESPONSE 5. The sum of which two angle measures
equals the measure of LWLK?
Use the figure for problems 1-5.
A. LSLYand LnR
B. LSLWand LYLR
C. LRLK and LSLY
D. LRLKand LnR
6. If two angles are supplementary, what is
the sum of their measures?
1. Which pair of angles are adjacent angles?
A. LSLWand LRLK
B. LSLWand LWLK
7. Angle D is a vertical angle to LF. The
measure of LD is 53°. What is the
measure of LF?
C. LSLYand LWLK
D. LnRandLnK
2. Which pair of adjacent angles are
supplementary angles?
F. LRLK and LYLR
8. Which describes the transformation from
the original to the image?
G. LSLYand LnR
H. LRLKand LWLK
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J. LSLWand LWLR
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3. Which pair of angles are complementary
angles?
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A. LnS and LRLK
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B. LYLR and LYLS
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C. LSLWand LRLK
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D. LWLK and LRLK
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F. reflection across the x-axis
4. The measure of LRLK is 38°. What is the
measure of LSLY?
G. translation
H. reflection across the y-axis
J. rotation
Chapter 8 343
Assessment Readiness
11. Name two ways to describe angles TSU
and TSR. Explain.
9. Which describes the transformation from
the original to the image?
A. reflection across the x-axis
B. translation
12. Draw a triangle with angle measures of
C. rotation
32°, and 45°, and an included side with a
length of 2 inches.
D. reflection across the y-axis
CONSTRUCTED RESPONSE Use the figure for problems 10 and 11. 10. Write and solve an equation to find the
measure of LTSU.
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Chapter 8 344
Assessment Readiness