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Geometry and
Measurement
FLORIDA
CHAPTER
9
Name
Class
Date
Lesson
Student
Textbook
MA.8.G.2.2
9-1 Angle Relationships
345 –352
400 – 404
MA.8.G.2.2
9-2 Parallel and Perpendicular Lines
353 –360
405 – 408
MA.8.G.2.3
9-3 Triangles
361 –368
409 – 412
MA.8.G.2.3
9-4 Polygons
369 –376
413 – 417
Remember It?
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Worktext
379 –380
Rev. MA.7.G.2.1
9-5 Volume of Prisms and Cylinders
420 – 424
Rev. MA.7.G.2.1
9-6 Volume of Pyramids and Cones
425 – 429
Rev. MA.7.G.2.1
9-7 Surface Area of Prisms and
Cylinders
430 – 433
Rev. MA.7.G.2.1
9-8 Surface Area of Pyramids and
Cones
434 – 437
MA.8.G.5.1
9-9 Scaling Three Dimensional
Figures
381 –388
438 – 441
MA.8.G.5.1
9-10 Measurement in ThreeDimensional Figures
389 –396
442 – 445
Study It!
399 – 401
Write About It!
402
Chapter 9 Geometry and Measurement 343
CHAPTER
Benchmark
9
Chapter at a Glance
Vocabulary Connections
LA.8.1.6.5 The student will relate new vocabulary to familiar words.
Key Vocabulary
Vocabulario
Vokabilè
equilateral triangle
triángulo equilátero
triyang ekilateral
indirect measurement
medición indirecta
mezi endirèkt
lateral surface
superficie lateral
sipèfisi lateral
parallel lines
rectas paralelas
liy paralèl
perpendicular lines
rectas perpendiculares
liy pèpandikile
polygon
polígono
poligòn
surface area
área total
sòm total sipèfisi
transversal
transversal
transvèsal
volume
volumen
volim
CHAPTER
To become familiar with some of the vocabulary terms in the chapter, consider the
following. You may refer to the chapter, the glossary, or a dictionary if you like.
1. The word equilateral contains the prefix equi-, which means “equal,” and lateral,
which means “of the side.” What do you suppose an equilateral triangle is?
3. The word lateral is from the Latin laterals, meaning “of the side” A lateral in
football is a pass to the side. What do you think the lateral surface of a cone or
cylinder might be?
344 Chapter 9 Geometry and Measurement
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9
2. The Greek prefix poly- means “many,” and the suffix -gon means “angle.” What do
you suppose a polygon is?
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9-1
Date
MA.8.G.2.2 Classify and determine
the measure of angles,....
Angle Relationships
Investigate Angle Pairs
When two lines intersect, they form pairs of angles
that have special relationships with one another.
Activity
1 Use a protractor to measure the angles in the figures below to the nearest degree.
Record the measures in the table.
Figure 1
Figure 2
1
4
1
4
2
3
2
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3
Measure of Angle 1 Measure of Angle 2 Measure of Angle 3 Measure of Angle 4
(in degrees)
(in degrees)
(in degrees)
(in degrees)
Figure 1
Figure 2
2 Add the measures of the angles and record the sums in the table.
Sum of Angles 1
and 2 (in degrees)
Sum of Angles 3
and 4 (in degrees)
Sum of Angles 1
and 4 (in degrees)
Sum of Angles 2
and 3 (in degrees)
Figure 1
Figure 2
9-1 Angle Relationships 345
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Try This
1. Compare the angle measures you recorded for Figure 1. What do you notice?
2. Compare the angle measures you recorded for Figure 2. What do you notice?
Draw Conclusions
3. If two lines intersect, they form two pairs of vertical
angles. Vertical angles do not share any sides. Name the
two pairs of vertical angles in the figure at the right.
6
7
8
9
4. What can you say about the measures of a pair of vertical angles?
For 5 –7, use the figure at the right.
5. Without using a protractor, find the measure of ∠1.
Explain how you found your answer.
1
50°
7. Describe two ways you could find the measure of ∠3.
346 9-1 Angle Relationships
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6. Without using a protractor, find the measure of ∠2. Explain how you found
your answer.
2
3
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9-1
Date
MA.8.G.2.2 Classify and determine
the measure of angles,….
Angle Relationships (Student Textbook pp. 400–404)
Lesson Objective
Classify angles and find their measures
Vocabulary
angle
right angle
acute angle
obtuse angle
straight angle
complementary angles
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supplementary angles
adjacent angles
vertical angles
congruent angles
Example 1
Use th
U
the diagram
di
to name each figure.
A. two acute angles
B. two obtuse angles
m∠SQP =
S
T
°
, m∠RQT =
°
43° 90 47°
°
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P
Q
R
9-1 Angle Relationships 347
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C. a pair of complementary angles
Apply It!
S
T
m∠TQP + m∠RQS
°
=
°
+
°
43° 90 47°
P
= 90°
Q
R
D. two pairs of supplementary angles
°
m∠TQP + m∠TQR =
°
m∠SQP + m∠SQR =
Check It Out!
+
+
°
°
= 180°
= 180°
A
Use the diagram to name each figure.
B
1a. two acute angles
1b. two obtuse angles
58°
E
32°
C
D
1c. a pair of complementary angles
1d. two pairs of supplementary angles
1
Use th
the di
diagram to find each angle measure.
U
A. If m∠1 = 37°, find m∠2.
°
°
+ m∠2 =
-37°
−−−−−−−−−−
-37°
−−−−−
m∠2 =
348 9-1 Angle Relationships
4
3
B. Find m∠3.
°
m∠1 + m∠2 =
2
°
.
°
+ m∠3 =
-143°
−−−−−−−−−−
°
°
m∠2 + m∠3 =
m∠2 =
.
-143°
−−−−−
°
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Example 2
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U the diagram to find each angle measure.
Use
2a. If m∠3 = 142°, find m∠4.
2b. Find m∠1.
2
3
1
4
Example 3
A ttraffic
ffi engineer
i
designed a section of roadway where three streets intersect.
Based on the diagram, what is the measure of ∠DBE?
A
26°
C
B
F
D
E
Step 1: Find m∠CBD.
∠CBD ∠
angles are congruent.
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m∠CBD = m∠
°
m∠CBD =
Step 2: Find m∠
Congruent angles have the same measure.
°
Substitute
for m∠ABF.
.
m∠CBD + m∠
=
°
+ m∠DBE = 90°
m∠DBE =
°
Substitute
°
.
The angles are
Subtract
°
°
for m∠CBD.
from both sides.
Check It Out!
3. Based on the map, what is the measure of ∠BGC ?
1st St.
A
B
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F
Main St.
42°
G
C
E
D
9-1 Angle Relationships 349
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9-1
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Name
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Class
Date
LA.8.2.2.3 The student will organize
information to show understanding
or relationships among facts…
(e.g., representing key points…through…
summarizing…)
Summarize It!
Angle Relationships
Think and Discuss
1. Draw a pair of angles that are adjacent but not supplementary.
2. Explain why vertical angles must always be congruent.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing the
definition and making a sketch of each angle relationship.
Angle
Relationships
Supplementary
Angles
Adjacent
Angles
Vertical
Angles
Definition
Definition
Definition
Definition
Sketch
Sketch
Sketch
Sketch
350 9-1 Angle Relationships
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Complementary
Angles
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9-1
Date
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MA.8.G.2.2 Classify and determine
the measure of angles,….
Angle Relationships
Use the figure at the right for Exercises 1–7.
B
1. Name a right angle in the figure.
C
2. Name two acute angles in the figure.
A
D
F
G
3. Name two obtuse angles in the figure.
E
4. Name an angle adjacent to ∠CFD.
5. Name a pair of complementary angles in the figure.
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6. Name three pairs of supplementary angles in the figure.
7. Suppose that ∠CFD measures 49°. What is the measure of ∠GFE? Justify your
answer.
Use the figure at the right for Exercises 8–9.
8. If m∠1 = 24°, find m∠2.
1
2
4
3
9. Find m∠3.
9-1 Angle Relationships 351
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9-1
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Date
MA.8.G.2.2 Classify and determine
the measure of angles…
Angle Relationships
The figure shows a ray of light being reflected
off a mirror. The angle of incidence is
congruent to the angle of reflection. Use the
figure for 1−5.
N
Angle of
incidence
A
1
C
6. The figure shows a drawbridge when
the moving section of the bridge is fully
elevated. This section of the bridge moves
through 6° in 5 seconds. How long does it
take this section of the bridge to swing from
the horizontal position to the fully-elevated
position?
Angle of
reflection
65°
2 3
M
B
4
Mirror
D
108°
1. Find m∠3.
2. Find m∠4.
The figure shows the braces in a bookshelf.
Use the figure for 7−8.
3. What is m∠AMD ? What type of angle is
∠AMD ?
4 3
2
4. Name two pairs of supplementary angles in
the figure.
5. Keri states that ∠1 and ∠4 are adjacent
angles. Do you agree or disagree? Why?
352 9-1 Angle Relationships
7. The ratio of m∠2 to m∠1 is 5:1. What are
the measures of these two angles?
8. Gridded
Response
Angles 3 and 4 are
supplementary, and
m∠4 is 30° greater
than m∠3. What is the
measure, in degrees,
of ∠4?
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1
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9-2
Date
MA.8.G.2.2 Classify and determine
the measure of angles, including
angles created when parallel lines are
cut by transversals.
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Parallel and Perpendicular Lines
Investigate Parallel and Perpendicular Lines
Parallel lines are lines in a plane that never intersect. A transversal is a line that
intersects two or more lines. Angles formed by parallel lines and a transversal
have some special properties.
Parallel lines
Transversal
Activity 1
1 Use a straightedge to draw a non-vertical
transversal through the given lines. Eight
angles will be formed. Number the angles
from ∠1 through ∠8 from left to right and
then down.
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2 Use a protractor to measure the eight
angles. Record the measures in
the table.
Angle
∠1
∠2
∠3
∠4
∠5
∠6
∠7
∠8
Measure
3 Describe any relationships you notice in the table.
m
Try This
3
Lines m and n are parallel. Write the measure of each angle.
n
1. ∠1
2. ∠2
3. ∠3
4. ∠4
5. ∠5
6. ∠6
2
1
4
6
5
135° 7
7. ∠7
9-2 Parallel and Perpendicular Lines 353
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Draw Conclusions
8. If one of the angles formed when a transversal intersected two
parallel lines measures 38˚, how many other angles would you
expect to measure 38˚?
9. If a transversal intersects two parallel lines and you know one angle
measures 72˚, explain how you know every other angle measure.
Perpendicular lines intersect at 90˚ angles. Angles formed
by perpendicular lines also have unique relationships.
Activity 2
1 Use a protractor to tell whether the lines at the right
are perpendicular. Explain your reasoning.
m
Try This
Lines m and n are perpendicular. Write the measure of the angle.
11. ∠2
n
40°
1
5
12. ∠3
13. ∠4
14. ∠5
Draw Conclusions
15. Describe a method you could use to draw a line perpendicular to a given line p.
16. Describe a method you could use to draw a line parallel to a given line p.
354 9-2 Parallel and Perpendicular Lines
2
4
3
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10. ∠1
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9-2
Date
MA.8.G.2.2 Classify and determine
the measure of angles, including
angles created when parallel lines are
cut by transversals.
Learn It!
Parallel and Perpendicular Lines (Student Textbook pp. 405–408)
Lesson Objective
Identify parallel and perpendicular lines and the angles formed by a transversal
Vocabulary
parallel lines
perpendicular lines
transversal
Example 1
M
Measure
the
th angles formed by the transversal and
the parallel lines. Which angles seem to be congruent?
∠1, ∠3, ∠5, and ∠7 all measure 150°.
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∠2, ∠4, ∠6, and ∠8 all measure
a
2
1
3
4
6
°.
5
7
b
8
c
Check It Out!
1. Measure the angles formed by the transversal and
the parallel lines. Which angles appear to be congruent?
1
2
3 4
5
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6
7 8
9-2 Parallel and Perpendicular Lines 355
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Example 2
In the figure, line l || line m. Find the measure of each angle. Justify your answer.
124° 1
2 3
4 5
6 7
l
m
A. ∠4
°
m∠4 =
The 124° angle and ∠4 are
angles, so they are
.
B. ∠2
°
m∠2 + 124° =
-
°
∠2 is
to the 124° angle.
°
°
m∠2 =
C. ∠6
-124°
m∠6 =
°
∠6 and ∠4 are supplementary angles.
-124°
°
356 9-2 Parallel and Perpendicular Lines
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°=
m∠6 +
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In the figure, line n || line m. Find the measure of each angle.
Justify your answer.
1
3
144°
4
5
6
7
m
8
n
2a. ∠5
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2b. ∠7
2c. ∠8
2d. ∠6
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9-2 Parallel and Perpendicular Lines 357
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Date
LA.8.2.2.3 The student will organize
information to show understanding
or relationships among facts…
(e.g., representing key points…through…
summarizing…)
Parallel and Perpendicular Lines
Think and Discuss
1. Tell how many different angles would be formed by a transversal intersecting
three parallel lines. How many different angle measures would there be?
2. Explain how a transversal could intersect two other lines so that
corresponding angles are not congruent.
3. Get Organized Complete the graphic organizer. Lines m and n are parallel.
To fill in each box, tell whether the angle is congruent to ∠1 or ∠2 and
explain why.
m
∠2
358 9-2 Parallel and Perpendicular Lines
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∠1
n
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9-2
Date
MA.8.G.2.2 Classify and determine
the measure of angles, including
angles created when parallel lines are
cut by transversals.
Practice It!
Parallel and Perpendicular Lines
In the diagram, lines w and z are parallel. Use the diagram to answer each question.
Explain your reasoning.
1. Measure the angles in the figure. Which angles appear to be
congruent?
1
5
2
w
6
7
3
8
4
z
2. If m∠1 = 45°, what is m∠3?
3. If m∠2 = 132°, what is m∠7?
4. If m∠4 = 118°, what is m∠5?
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5. If m∠6 = 53°, what is m∠7?
6. If m∠8 = 28°, complete two different ways of finding m∠2.
Method 1: ∠8 and ∠6
and m∠6 =
angles, so they are
.
∠6 and ∠2 are
+
.
= 180°, so m∠2 = 180° -
.
angles, so m∠8 + m∠4 =
Method 2: ∠8 and ∠4
and m∠4 = 180° -
=
= 152°.
and ∠2 are corresponding angles, so m∠2 =
=
.
9-2 Parallel and Perpendicular Lines 359
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9-2
Learn It!
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Class
Date
MA.8.G.2.2 Classify and determine
the measure of angles, including
angles created when parallel lines are
cut by transversals.
Apply It!
Parallel and Perpendicular Lines
The figure shows painted lines marking
parallel parking spaces in a parking lot. Use
the figure for 1−5.
The figure shows several city streets. Grove
Street is parallel to Hayes Street. Use the
figure for 6−8.
Laguna St.
1 2
3 4
12 11 10 9
5 6
8 7
123°
1. Name all the angles that are corresponding
angles to ∠4.
2
1
Octavia St.
Grove St.
3
Hayes St.
Belden Lane
6. Suppose m∠2 is twice m∠1. Find m∠1.
2. What type of angles are ∠5 and ∠9? What
can you conclude about their measures?
4. Name all angles congruent to ∠1.
5. m∠1 = 118˚. Explain how to find m∠10.
360 9-2 Parallel and Perpendicular Lines
8. Extended Response Daryl notices
that Belden Lane is a transversal that
intersects Laguna Street and Octavia Street.
He sees that ∠2 and ∠3 are alternate
interior angles and he concludes that these
angles must be congruent. Do you agree or
disagree with his conclusion. Why?
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3. What type of angles are ∠3 and ∠7? What
can you conclude about their measures?
7. Angles 1 and 3 are complementary. Find
m∠3.
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9-3
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Class
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is 180
degrees and apply this fact to find
unknown measure of angles…
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Triangles
Investigate the Angles of a Triangle
In the following activities, you’ll see three different ways
that the sum of the measures of the angles of a triangle is 180°.
Activity
B
1 Use a protractor to measure the angles of ABC.
m∠A =
m∠B =
m∠C =
A
C
2 What is the sum of the measures of the angles of ABC?
3 Draw a triangle on a sheet of paper. Make each
side at least 3 inches long. Label the angles 1, 2,
and 3, as shown in the triangle at the right.
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4 Cut out your triangle. Then tear off corners
containing ∠1 and ∠3.
5 Arrange the corners as shown. Place ∠1 adjacent
to ∠2. Place ∠3 on the other side of ∠2 so that
it is also adjacent to ∠2.
2
1
3
1 2
3
6 What type of angle do the rearranged angles appear to form?
7 Use your results from Steps 2 and 6 to write a rule that describes
the sum of the angle measures in a triangle.
9-3 Triangles 361
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Try This
Tell whether the three angle measures could be the angle measures of a triangle.
1. 5˚, 80˚, 90˚
2. 30˚, 70˚, 80˚
3. 40˚, 54˚, 86˚
4. 37˚, 42˚, 102˚
Activity 2
1 Choose three angle measures that you think could form a triangle. Use a
protractor to draw the triangle on a separate sheet of paper. Again, make
each side at least 3 inches long. Write the angle measures that you chose:
m∠1 =
m∠2 =
m∠3 =
2 Place a point in the middle of two sides of the triangle. Do this
by measuring the lengths of the two sides with a ruler and
dividing by 2. Draw a line connecting the two points.
3 Fold the triangle’s top vertex (A) down across the line you
have drawn so that the vertex touches the bottom side of the
triangle. Then fold the other two vertices (B and C) inward to
meet A in its new position.
4 Explain how your results confirm the sum of the
measures of the angles of your triangle.
A
B
B
5. One angle in a right triangle measures 25°.
What are the measures of the other two angles?
Draw Conclusions
6. How can you find the measures of each angle of an equilateral triangle?
Explain.
7. If you know the measures of two angles of a triangle, how can you find the
measure of the third angle?
362 9-3 Triangles
A
C
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Try This
C
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9-3
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is
180 degrees and apply this fact to
find unknown measure of angles….
Learn It!
Triangles (Student Textbook pp. 409–412)
Lesson Objective
Find unknown angle measures in triangles
Vocabulary
Triangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
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isosceles triangle
scalene triangle
Example 1
A. Find c° in the right triangle.
B. Find m° in the obtuse triangle.
c
62°
42°
°+
m°
23°
° + c° =
°
°+
° + c° =
°
° + m° =
- 132°
- 132°
c° =
° + m° =
- 85°
°
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°
°
- 85°
m° =
°
9-3 Triangles 363
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1a. Find b° in the right triangle.
1b. Find x ° in the right triangle.
25°
36°
38°
x
b
Example 2
B. Find the angle measures
in the scalene triangle.
t°
2x°
t°
62°
3x°
2t° + 62° =
°
2t° = 118°
°
10x° = _____
180°
_____
°
The angle measures are
59°, 59°, and 62°
364 9-3 Triangles
3x° + 5x° + 2x° =
10x° = 180°
2t° = _____
118°
_____
t° =
5x°
x° =
°
The angle measures are 3(18)° = 54°,
5(18)° = 90°, and 2(18)° = 36°.
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A. Find
A
Fi d the
h angle measures
in the isosceles triangle.
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2a. Find the angle measures
in the isosceles triangle.
2b. Find the angle measures
in the scalene triangle.
p° + 8°
8p°
34p°
36°
m°
m°
Example 3
Th second
The
d angle in a triangle is six times as large as the first. The third
angle is half as large as the second. Find the angle measures and draw
a possible picture.
Let x ° = the first angle measure. Then 6x ° = second angle measure,
and __12 (6x °) = 3x ° = third angle measure.
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x ° + 6x ° + 3x ° =
°
Triangle
10x °
180°
_____
= _____
x° =
The angles measure
Theorem
Simplify, then divide both sides by
.
°
54°
°,
°, and
°.
18°
108°
Check It Out!
3. The second angle in a triangle is twice as large as the first. The third angle
measure is the average of the first two angle measures. Find the angle
measures and draw a possible figure.
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9-3 Triangles 365
Explore It!
9-3
Summarize It!
Learn It!
Name
Practice It!
Class
Summarize It!
Apply It!
Date
LA.8.2.2.3 The student will organize
information to show understanding
or relationships among facts…
(e.g., representing key points… through…
summarizing…)
Triangles
Think and Discuss
1. Explain whether a right triangle can be equilateral. Can it be isosceles? scalene?
2. Explain whether a triangle can have 2 right angles. Can it have 2 obtuse angles?
3. Get Organized Complete the graphic organizer with three kinds of triangles.
Triangle Sum
Theorem
Example
366 9-3 Triangles
Example
Example
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Definition
Explore It!
Learn It!
Practice It!
Summarize It!
Name
Class
Apply It!
9-3
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is 180degrees and apply this fact to find
unknown measure of angles…
Practice It!
Triangles
Find the missing angle in the triangle and tell whether the triangle is acute,
obtuse, or right.
1.
56°
2.
50°
3.
x°
27°
47°
x°
44°
46°
x°
4.
5.
6.
x°
55°
35°
15°
x°
137°
x°
x°
Copyright © by Holt McDougal. All rights reserved.
x°
7. The angle measures of a triangle are x°, 2x°, and 12x°. Find each angle
measure.
8. The angle measures of a triangle are 2x + 10°, 3x°, and 3x + 10°. Find each
angle measure.
9. The second angle in a triangle is two thirds as large as
the first. The third angle is one half as large as the second
angle. Find the angle measures. Draw a possible picture
of the triangle.
9-3 Triangles 367
Explore It!
9-3
Learn It!
Name
Summarize It!
Apply It!
Practice It!
Class
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is 180degrees and apply this fact to find
unknown measure of angles…
Apply It!
Triangles
The figure shows the
sails of a sailboat. Use
the figure for 1−2.
68°
1
6. The sum of two side lengths of a triangle
must be longer than the third side. Elliott
states that the distance from Jacksonville to
Tampa is 210 miles. Do you think Elliott’s
statement is true? Why or why not?
132°
Jacksonville
2
125 mi
1. Find m∠1.
2. Find m∠2.
Orlando
77 mi
Tampa
3. While making a quilt, Larissa cuts a triangle
out of a larger piece of material. Two of
the triangle’s angles measure 51° and 36°.
Explain how you can classify this triangle.
62°
m
80°
1
4. A pennant has the shape of an isosceles
triangle. The two congruent angles both
have measures that are twice as large as
the remaining angle. What are the angle
measures of the pennant?
5. Corey makes an earring by bending a piece
of wire into a triangle. The second angle is 3
times as large as the first. The third angle is
2 times as large as the second. What is the
measure of the largest angle?
368 9-3 Triangles
The beams formed
by lines m and n are
parallel. What is the
measure, in degrees,
of ∠1?
n
Copyright © by Holt McDougal. All rights reserved.
7. Gridded Response The figure shows
part of the support structure of a bridge.
Explore It!
Learn It!
Summarize It!
Name
Practice It!
Apply It!
Class
9-4
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is
180 degrees and apply this fact to
find unknown measure of angles, and the sum
of angles in polygons.
Explore It!
Polygons
Investigate Angles of Polygons
You found that the measures of the angles of a triangle
have a sum of 180°. In this activity, you’ll explore whether
the angles of a quadrilateral also have a constant sum.
Activity 1
Use a protractor.
B
1 Measure the angles of quadrilateral ABCD.
m∠A
m∠B
m∠C
m∠D
C
A
D
Find the sum of the measures of the angles of quadrilateral ABCD.
F
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2 Measure the angles of quadrilateral. EFGH.
m∠E
m∠F
m∠G
m∠H
G
E
H
Find the sum of the measures of the angles of quadrilateral EFGH.
J
3 Measure the angles of quadrilateral. IJKL.
m∠I
m∠J
m∠K
m∠L
K
L
I
Find the sum of the measures of the angles of quadrilateral IJKL.
Try This
The measures of three angles of a quadrilateral are given. Predict the measure
of the fourth angle. You may want to draw the quadrilateral to check your
prediction.
1. 50°, 120°, 85°
2. 75°, 75°, 135°
3. 88°, 131°, 107°
9-4 Polygons 369
Explore It!
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Draw Conclusions
4. What can you conclude about the sum of the measures of the angles of a
quadrilateral?.
Activity 2
You can use the fact that the sum of the measures of the angles of a triangle is 180°
to find the sums of the measures of the angles of other polygons. Draw diagonals
from any one vertex to the other vertices in each figure and fill in the table.
Number of Number of
180° sum
sides
triangles per triangle
4
2
· 180°
Sum of angle
measures
2 · 180° = 360°
· 180°
5
Find the missing angle from the polygon with the angles given.
5. 100°, 170°, 25°, 115°, x°
6. 145°, 55°, 117°, 159°, 90°, x°
Draw Conclusions
7. Describe how you could find the sum of the measures of the angles of a
heptagon.
370 9-4 Polygons
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Try This
Explore It!
Learn It!
Summarize It!
Name
Class
Practice It!
Apply It!
9-4
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is
180 degrees...apply this fact to find
unknown measure of angles, and the sum of
angles in polygons
Learn It!
Polygons (Student Textbook pp. 413–417)
Lesson Objective
Classify and find angles in polygons
Vocabulary
polygon
regular polygon
trapezoid
parallelogram
rectangle
rhombus
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square
Example 1
A. Find
A
Fi d the
h sum of the angle measures in a hexagon.
triangles.
Divide the figure into
· 180° =
°
B. Find the sum of the angle measures in an octagon.
triangles.
Divide the figure into
· 180° =
°
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9-4 Polygons 371
Learn It!
Explore It!
Summarize It!
Practice It!
Apply It!
Check It Out!
1. Find the sum of the angle measures in a heptagon.
Example 2
Find the angle measures in each regular polygon.
A.
congruent angles
8x° = 180°(8 - 2)
8x° = 180°(
8x° =
x° x°
x°
x°
x°
x°
x° x°
)
°
congruent angles
y°
y°
y°
y°
4y° = 180°(4 - 2)
4y° = 180°(
)
°
4y° =
4y°
360°
_____
= _____
8x° = _____
1080°
_____
°
°
y° =
Check It Out!
2. Find the angle measures in the regular polygon.
y°
y°
y°
y°
y°
y°
y°
y°
y°
372 9-4 Polygons
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x° =
B.
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Example 3
Give all
Gi
ll th
the names that apply to each figure.
A.
-sided polygon
6 cm
6 cm
2 pairs of
6 cm
4
6 cm
angles
4
sides
4
angles and
sides
4
B.
2 ft
J
sided polygon
K
2 ft
2 pairs of
2 ft
M
Check It Out!
Copyright © by Holt McDougal. All rights reserved.
3a.
4
L
2 ft
sides
sides
side
Give all the names that apply to each figure.
G
B
A
D
C
AB || DC
3b.
F
1 cm
1 cm
G
E
1 cm
1 cm
H
___
___ ___
____
FG || EH, EF || HG
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9-4 Polygons 373
Explore It!
9-4
Summarize It!
Learn It!
Name
Practice It!
Class
Apply It!
Date
LA.8.2.2.3 The student will organize
information to show understanding
or relationships among facts…(e.g.,
representing key points…through…
summarizing…)
Summarize It!
Polygons
Think and Discuss
1. Choose which is larger, an angle in a regular heptagon or an angle in a
regular octagon. Justify your answer.
2. Explain why all rectangles are parallelograms and why all squares are
rectangles.
3. Explain why it is not possible to draw a diagonal of a triangle.
Definition
Examples
374 9-4 Polygons
Property
Polygon
Nonexamples
Copyright © by Holt McDougal. All rights reserved.
4. Get Organized Complete the graphic organizer. To fill in the sections,
write the definition of a polygon and write a property of polygons. Then
sketch examples and nonexamples of polygons.
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Name
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Class
Apply It!
9-4
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is 180
degrees and apply this fact to find
unknown measure of angles, and the
sum of angles in polygons.
Practice It!
Polygons
Find the sum of the angle measures in each polygon.
1.
2.
3.
Find the angle measures in each regular polygon.
4. regular octagon
5. regular triangle
6. regular decagon
Copyright © by Holt McDougal. All rights reserved.
Honeybees store their honey in honeycombs. The honeycomb is made of
many small wax compartments that are regular polygons as shown.
7. What type of polygon is each compartment?
8. What is the sum of the angle measures in each compartment?
9. What is the measure of each angle in any compartment?
10. A campground site is in the shape of a quadrilateral. Three sides
of the campground form two right angles. The third interior angle
measures 10° less than the fourth angle. Find the measure of
each angle.
11. Three interior angles of a heptagon measure 125°, and two of the
angles measure 143°. The heptagon has one right angle. What is the
measure of the remaining angle?
12. Use the diagram to find the value of y.
x°
x°
x°
x°
x° y°
9-4 Polygons 375
Explore It!
9-4
Learn It!
Name
Summarize It!
Practice It!
Class
Apply It!
Date
MA.8.G.2.3 Demonstrate that the
sum of the angles in a triangle is 180
degrees and apply this fact to find
unknown measure of angles, and the
sum of angles in polygons.
Apply It!
Polygons
1. A stop sign is a regular octagon. What is the
sum of the angle measures in a stop sign?
What is the measure of each of the sign’s
angles?
2. One face of a gem is a quadrilateral with
a 38° angle, a 142° angle, and a 110° angle.
What is the measure of the remaining
angle?
5. The Pentagon building in Arlington,
Virginia, is a regular pentagon. Each side
of the building is 921 feet long. About how
long would it take to walk around the edge
of the building at 350 ft/min? Round to the
nearest tenth of a minute.
6. The figure shows the truss for a bridge.
Suppose m∠BCD = 156° and
∠ABC ∠CDE. Find m∠ABC.
C
3. In baseball, home plate is a pentagon with
three right angles. The other two angles are
congruent. What is the measure of each of
the two congruent angles?
115°
A
R
S
376 9-4 Polygons
115°
E
7. To make string art, Eric places nails at each
vertex of a regular hexagon. Then he strings
a piece of colored wire along each diagonal
of the hexagon. Along how many diagonals
does he string the wire?
Q
P
D
8. Gridded Response
Jim is making a frame
in the shape of a
rhombus. One side is
18 cm long. When he
buys the wood for the
frame, he allows 10%
extra for waste. How
many centimeters of
wood should he buy?
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4. Jasmine is making a kite as shown in the
figure. She wants to make the kite so that
m∠R is two times m∠P and m∠Q is three
times m∠P. She also wants ∠S and ∠Q to
be congruent. What measures must she use
for each angle?
B
9-1
Name
Class
Got It?
THROUGH
Date
9-4
Ready to Go On?
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Quiz for Lessons 9-1 through 9-4
9-1
Angle Relationships (Student Textbook pp. 400–404)
Use the diagram to name each figure.
E
1. two pairs of complementary angles
F
D
2. three pairs of supplementary angles
A
75° 55°
15°
35°
B
C
3. two right angles
9-2
Parallel and Perpendicular Lines (Student Textbook pp. 405–408)
In the figure, line m || line n. Find the measure of each angle.
Justify your answer.
125°
1 2
4. ∠1
3 4
m
n
5. ∠3
9-3
Triangles (Student Textbook pp. 409–412)
Copyright © by Holt McDougal. All rights reserved.
Find x° in each triangle.
6.
60°
9-4
7.
83°
74°
x°
x°
x°
Polygons (Student Textbook pp. 413–417)
8. Give all of the names that apply to the figure.
D
A
C
AB || CD
B
Find the angle measures and the sum of the angle measures in each regular polygon.
9. a regular nonagon
10. a regular 11-gon
Chapter 9 Geometry and Measurement 377
9-1
THROUGH
Name
9-4
Class
Date
Connect It!
MA.8.G.2.2; MA.8.G.2.3
Connect the concepts of Lessons 9-1 through 9-4
Angle Rummy
Find a partner and follow these steps to play angle rummy.
1. Prepare a deck of 30 cards. Write the following angle measures on
two index cards each: 10°, 20°, 30°, 40°, ... , 130°, 140°, 150°
2. The dealer shuffles the deck and deals 7 cards to each player. The
remaining cards are placed face down in a stack. The player who is not
the dealer goes first.
3. Play as follows: If you hold a set, place those cards face up on the table. If not, take a card
from the top of the stack. If you can make a set
after taking the card, place the set on the table.
One set per turn!
Sets
2 cards with complementary angles
2 cards with supplementary angles
3 cards with angles that form
a triangle
4 cards with angles that form
a quadrilateral
4. Alternate turns until a player holds no cards or
when no cards remain in the stack. The player
who has placed the most cards on the table is
the winner.
5. Play the game a few times. Describe a helpful
strategy.
1. Find the values of the variables d, g, s, and w. When a variable appears in
more than one figure, it has the same value.
140 °
g°
d°
g°
l || m
149 °
36°
l
m
w°
s°
32°
s°
d =
d°
s =
s°
Think About The Puzzler
2. Which variable’s value did you find first? Why?
3. Can a quadrilateral have angle measures d°, g°, s°, and w°? Explain
378 Chapter 9 Geometry and Measurement
g =
w=
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What’s Your Angle?
Name
Class
9-5
Date
THROUGH
9-8
Remember It?
Review skills and prepare for future lesson
lessons.
9-5
Lesson
Volume of Prisms and Cylinders (Student Textbook pp. 420–424)
Rev. MA.7.G.2.1,
Rev. MA.7.G.2.2
3 cm
Find
Fi
d the
h volume
l
to the nearest tenth.
V = Bh = (πr2)h
4 cm
= π(32)(4)
= (9π)(4) = 36π cm3 ≈ 113.0 cm3
For 1–3, find the volume to the nearest tenth. Use 3.14 for π.
4.2 mm
13 cm
1.
2.
7 cm
7.5 mm
3.
rectangular prism with
length 8 ft, width 2 ft,
and height 5 ft
8 cm
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4. A can has a diameter of 3 in. and a height of 5 in. Explain whether doubling
the height would have the same effect on the volume as doubling the diameter.
9-6
Lesson
Volume of Pyramids and Cones (Student Textbook pp. 425–429)
Rev. MA.7.G.2.1
Find
Fi
d the
h volume.
l
7m
V = __13Bh = __13(5)(6)(7)
= 70 m3
6m
5m
For 5–7, find the volume to the nearest tenth. Use 3.14 for π.
5.
6.
4.2 yd
7. 46 mm
8 in.
31 mm
31 mm
9 in.
5 in.
Lesson Tutorial Videos @ thinkcentral.com
8 yd
Chapter 9 Geometry and Measurement 379
9-7
Lesson
Surface Area of Prisms and Cylinders (Student Textbook pp. 430–433)
Rev. MA.7.G.2.1
Find
Fi
d the
h surface
f
area of the rectangular prism.
S = 2B + Ph
4 in.
= 2(6) + (10)(4)
= 52 in2
2 in.
3 in.
For 8–11, find the surface area of each figure to the nearest tenth. Use 3.14 for π.
8.
9.
14.9 mm
21 ft
8 mm
20 mm
20 ft
10 mm
40 ft
12 mm
10. a rectangular prism with length 6 m, width 3 m, and height 3 m
11. a cylinder with radius 10 cm and height 5 cm
12. A tissue box has two square bases 11 cm on a side and a height of
14 cm. A pattern covers all surfaces. What is the area of the pattern?
9-8
Lesson
Surface Area of Pyramids and Cones (Student Textbook pp. 434–437)
Find
Fi
d the
h surface
f
area of the pyramid.
4 in.
4 in.
= 56 in2
For 13–16, find the surface area of each figure to the nearest tenth. Use 3.14 for π.
13.
14.
8 cm
6 cm
10 in.
15. 1 m
1m
6 cm
12 in.
16. a square pyramid with an 8 in. by 8 in. base and a height of 3 in.
17. Explain whether doubling the height and diameter of the cone in Exercise 15
would double the surface area.
380 Chapter 9 Geometry and Measurement
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S = B + __12 Pl
= 16 + __12(16)(5)
Rev. MA.7.G.2.1
5 in.
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Class
9-9
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between
different measurement systems
and dimensions including…area, volume, and
derived units to solve problems.
Explore It!
Scaling Three-Dimensional Figures
Scale Dimensions
When you multiply each dimension of a two- or threedimensional figure by one number, you can predict
how the area or volume will change.
Activity 1
1 Complete the table. Find the area of each square.
Square
Side length (s)
A
B
C
D
E
F
G
1
2
3
4
6
8
9
Area of a square with
side length s in units2
s
s
2 What patterns do you notice in the area of the squares?
3 The dimensions of square D are 2 times those of square B. How do the areas compare?
Copyright © by Holt McDougal. All rights reserved.
4 The dimensions of square G are 3 times those of square C. How do the areas compare?
Try This
For 1–2, use the table in Activity 1.
1. Find another pair of squares where one’s dimensions are twice the other’s. How
do these areas compare?
2. Find another pair of squares where one’s dimensions are 3 times the other’s.
How do these areas compare?
Draw Conclusions
3. When the dimensions of a square are multiplied by 2, the area is multiplied by
.
4. When the dimensions are multiplied by 3, the area is multiplied by
.
5. When the dimensions are multiplied by n, the area is multiplied by
.
9-9 Scaling Three-Dimensional Figures 381
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Learn It!
Summarize It!
Practice It!
Apply It!
Activity 2
1 Complete the table. Find the volume of each cube.
Cube
Side length (s)
A
B
C
D
E
1
2
3
4
6
F
G
Volume of a cube with
side length s in units3
s
s
s
2 What patterns do you notice in the volumes of the cubes?
3 Notice that the dimensions of cube D are 2 times those of cube B. How do
the volumes compare?
4 Notice that the dimensions of cube E are 3 times those of cube B. How do the
volumes compare?
Try This
For 6–7, use the table in Activity 2.
6. Find another pair of cubes where one’s dimensions are 2 times the other’s.
How do these volumes compare?
Draw Conclusions
8. When the dimensions of a cube are multiplied by 2, the volume is
multiplied by
.
9. When the dimensions are multiplied by 3, the volume is multiplied by
.
10. When the dimensions are multiplied by n, the volume is multiplied by
.
382 9-9 Scaling Three-Dimensional Figures
Copyright © by Holt McDougal. All rights reserved.
7. Find another pair of cubes where one’s dimensions are 3 times the other’s.
How do these volumes compare?
Learn It!
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Summarize It!
Name
Practice It!
Class
Apply It!
9-9
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between
different measurement systems
and dimensions including…area, volume, and
derived units to solve problems.
Learn It!
Scaling Three-Dimensional Figures (Student Textbook pp. 438–441)
Lesson Objective
Make scale models of solid figures
Vocabulary
capacity
Example 1
A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the
following values.
A. the edge lengths of the two cubes
3-cm cube
________
1-cm cube
cm
________ =
Ratio of
corresponding
cm
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The edges of the large cube are
times as long as the edges of the small cube.
B. the surface areas of the two cubes
3-cm cube
________
1-cm cube
(
6(
2
)
cm )
cm
6
cm2
___________= _________ =
Ratio of
corresponding
2
cm2
2
=
The surface area of the large cube is
times that of the small cube.
C. the volumes of the two cubes
3-cm cube
________
1-cm cube
(
(
3
)
cm )
cm
cm3
__________ = _________ =
3
cm3
Ratio of
corresponding
3
The volume of the large cube is
=
Lesson Tutorial Videos @ thinkcentral.com
times that of the small cube.
9-9 Scaling Three-Dimensional Figures 383
Learn It!
Explore It!
Summarize It!
Practice It!
Apply It!
Check It Out!
An 8 cm cube is built from small cubes, each 2 cm on an edge.
Compare the following values.
1a. the edge lengths of the two cubes
1b. the surface areas of the two cubes
1c. the volumes of the two cubes
Example 2
A. What is the scale of the model?
6 in.
____
4 ft
6 in. = _____ Convert and simplify.
_______
in.
The scale of the model is
.
B. What are the length and width of the model?
length: __18 · 3 ft = __18 ·
in. =
in.
width: __18 · 2 ft = __18 ·
in. =
in.
384 9-9 Scaling Three-Dimensional Figures
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A box is in the shape of a rectangular prism. The box is 4 ft tall, and its base
has a length of 3 ft and a width of 2 ft. For a 6 in. tall model of the box, find
the following.
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Summarize It!
Practice It!
Apply It!
Check It Out!
A box is in the shape of a rectangular prism. The box is 5 ft tall,
and its base has a length of 6 ft and a width of 4 ft. For a 6 in.
tall model of the box, find the following.
2a. the scale of the model
2b. the length and width of the model
Example 3
It takes 30 s for a pump to fill a cubic container whose edge measures
1 ft. How long does it take to fill a cubic container whose edge measures 2 ft?
Vsmaller = 1 ft · 1 ft · 1 ft = 1 ft3
ft ·
Vlarger =
Find the volume of each container.
ft ·
ft =
Copyright © by Holt McDougal. All rights reserved.
xs
30 s = _______
____
1 ft3
ft3
Set up a proportion and solve.
ft3
·
=x
Cross multiply.
=x
It takes
s, or
min, to fill the larger container.
Check It Out!
3. It takes 8 s for a machine to fill a cubic box whose edge measures 4 cm.
How long would it take to fill a cubic box whose edge measures 10 cm?
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9-9 Scaling Three-Dimensional Figures 385
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9-9
Learn It!
Name
Summarize It!
Practice It!
Class
Summarize It!
Apply It!
Date
LA.8.2.2.3 The student will organize
information to show understanding
or relationships among facts…(e.g.,
representing key points…through…
summarizing…)
Scaling Three-Dimensional Figures
Think and Discuss
1. Describe how the volume of a model compares to the original object if the linear
scale factor of the model is 1:2.
2. Explain one possible way to double the surface area of a rectangular prism.
3. Get Organized Complete the graphic organizer. To fill in the table, tell how the
surface area and volume of a three-dimensional figure change when all of the
figure’s dimensions are multiplied by each value.
Scaling Three-Dimensional Figures
The surface area is
multiplied by...
2
3
4
5
1
2
1
3
2
5
n
386 9-9 Scaling Three-Dimensional Figures
Then its volume
is multiplied by...
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All dimensions of a
three-dimensional figure
are multiplied by...
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Practice It!
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9-9
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between
different measurement systems and
dimensions including...area, volume, and derived
units to solve problems.
Scaling Three-Dimensional Figures
A 16-cm. cube is built from small cubes, each 4 cm on a side. Find and compare
the following values.
1. the edge lengths of the two cubes
2. the surface areas of the two cubes
Copyright © by Holt McDougal. All rights reserved.
3. the volumes of the two cubes
An oil drum is in the shape of a cylinder. The drum is 97 cm tall with a
diameter of 58 cm. A model of the drum in a model railroad setup has a
diameter of 9.0625 mm. Find the following to the nearest tenth.
4. the scale of the model
5. the height of the drum in the model
6. Find the volume of the drum in the model, to the nearest tenth.
Use 3.14 for π.
7. A machine fills a 1 foot by 1 foot by 2 foot box with packing
material in 12 seconds. How long would it take to fill a box
with dimensions 3 times that of the smaller box?
8. A water pump can fill a cylindrical pool with a height of 4 feet
and radius of 8 feet in 18 hours. How long would it take to fill a
children’s pool with a height of 2 feet and a radius of 4 feet?
9-9 Scaling Three-Dimensional Figures 387
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9-9
Learn It!
Name
Summarize It!
Practice It!
Class
Apply It!
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between
different measurement systems
and dimensions including…area, volume, and
derived units to solve problems.
Apply It!
Scaling Three-Dimensional Figures
1. John is designing a shipping container to
hold boxes. The container he designs holds
24 boxes of the same size. Suppose he
doubles the side lengths of the container.
How many boxes will the larger container
hold?
2. Maria uses two boxes of sugar cubes to
create a solid building for a class project.
She decides that the building is too small
and that she will rebuild it so that each
dimension is tripled. How many more
boxes of sugar cubes will she need?
4. A building has a volume that is 125,000
times the volume of a model of the
building. What is the ratio of the building’s
dimensions to the model’s dimensions?
388 9-9 Scaling Three-Dimensional Figures
Tank
Height (ft)
Radius (ft)
A
12
6
B
10
12
5. It takes 6 cans of paint to paint the lateral
surface of Tank A. Patrick is going to paint
the lateral surface of Tank B. How many
cans of paint should he buy?
6. Both water tanks are filled at the same rate.
Tank A can be filled in 4 hours. How much
longer does it take to fill Tank B?
7. Short Response
A cereal box is 6 in. by 3 in. by 8 in. The
price of the cereal is $1.85. The cereal
is available in a larger box that is 9 in.
by 4.5 in. by 12 in. What price would be
proportional for the large box? Explain.
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3. It takes 8 minutes to drain Alicia’s
aquarium. Carmen has an aquarium with
each dimension 3 times that of Alicia’s
aquarium. How long does it take to drain
Carmen’s aquarium, assuming both
aquariums drain at the same rate? Justify
your answer. Express your answer in hours
and minutes.
The table shows the dimensions of two
cylindrical water tanks. Use the table for 5−6.
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Name
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Class
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9-10
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between
different measurement systems
and dimensions including…area, volume, and
derived units to solve problems.
Explore It!
Measurement in Three-Dimensional Figures
Relate Measurement Stystems
The basic units of length in the customary system are different from those in the
metric system. You can find the relationship between them, however, and then
extend the relationship to two and three dimensions.
Activity 1
1 Use a metric ruler and an inch ruler. Measure the length
and width of the rectangle in centimeters and in inches.
Round to the nearest tenth.
Length
Width
Customary
Metric
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2 To find the number of centimeters in 1 inch, calculate the ratios
metric length
metric width
_____________
and ____________
. Average the two results
customary length
customary width
and round to hundredths.
customary length
customary width
3 Repeat Step 2 using the ratios _____________ and ____________.
metric length
metric width
1 in. ≈
cm
1 cm ≈
in.
4 Find the area of the rectangle in square inches and
in square centimeters.
customary area:
in2
cm2
metric area:
5 Divide the metric area by the customary area to find the number of
square centimeters in 1 square inch. Round to the nearest tenth.
1 in2 ≈
cm2
6 What relationship can you find between your results in Step 2 and Step 5?
Try This
Use the values you found above to convert between customary and metric units.
Round to the nearest tenth.
1. 18 in.
2. 44 in.
3. 24 in2
4. 160 in2
5. 10 cm
6. 32.5 cm
7. 44 cm2
8. 115 cm2
9-10 Measurement in Three-Dimensional Figures 389
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
Activity 2
1 Look for a pattern in the conversions shown
in the box at the right. Then use the pattern
to make a conjecture about the relationship
between the customary and metric systems
with respect to volume.
length: 1 in. = 2.54 cm
area:
1 in2 = (2.54)2 cm2 = 6.45 cm2
volume: 1 in3 = (
)3
1 in3 =
2 To test your conjecture, find the volume of the
cube in the customary and metric systems.
9 in. =
customary volume:
cm3
cm
metric volume:
9 in.
9 in.
metric volume
3 Use the volumes in Step 2 find the ratio _____________
.
customary volume
9 in.
4 Compare your results in Step 3 with your
conjecture in Step 1. What do you find?
Try This
12 in.
Find the area of the triangle in square centimeters to the nearest tenth two ways.
10. Find the area in square inches.
9. Convert each dimension to cm first.
base =
cm height =
cm
A = _12bh =
in2
20 in.
A = _12 bh =
cm2
A=
cm2
Find the volume of the prism in cubic inches to the nearest tenth two ways.
11. Convert each dimension to inches first. Then find volume.
length ≈
V = lwh ≈
in. width ≈
in
in. height ≈
4 cm
in.
3
12. Find the volume in cm3. Then convert to in3 using 1 in3 ≈ 16.39 cm3
V = lwh =
cm3 V ≈
in3
Draw Conclusions
13. To convert surface area of a three dimensional figure between measurement
systems, would you use the methods in Activity 1, Activity 2, or neither? Explain.
390 9-10 Measurement in Three-Dimensional Figures
3 cm
6 cm
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Convert to cm2 using 1 in2 ≈ 6.45 cm2
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Name
Practice It!
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Class
9-10
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between
different measurement systems and
dimensions including...area, volume, and derived
units to solve problems.
Learn It!
Measurement in Three-Dimensional Figures (Student Textbook pp. 442–445)
Lesson Objective
Convert units of measure in two and three dimensions within measurement
systems and between different measurement systems.
Example 1
A shoebox has a length of 13 inches, a width of 9 inches, and a height of
4__12 inches. Find the surface area of the box in square centimeters to the
nearest tenth.
Step 1: Find the surface area in square inches
S=2·
+ lateral area
S = 2 · (length)(width) + perimeter · (
+ 9)(4__12 )
= 2 · (13)(9) + 2(
)
Substitute
= 234 + 198 = 432 in2
Copyright © by Holt McDougal. All rights reserved.
Step 2: Find a conversion factor for inches to centimeters.
1 inch =
cm, so the conversion factor is
.
Step 3: Convert the surface area.
54m
432 in
( ____
1in.
2
2
) = 2787.0912 cm2
The surface area of the shoebox is about
.
Check It Out!
1. A cone has
h a radius
d of 3 centimeters, a height of 4 centimeters, and a slant
height of 5 centimeters. What is the surface area of the cone in square inches to
the nearest tenth? Use 3.14 for π.
Lesson Tutorial Videos
9-10 Measurement in Three-Dimensional Figures 391
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Example 2
A standard beverage can is a cylinder with a radius of 3.25 cm and a height of
10.7 cm. What is the volume of the can in cubic inches to the nearest tenth?
Step 1: Find the volume in
centimeters.
V = π r2h
2
) ·
V = 3.14(
Use 3.14 for π.
V = 354.88 cm
Step 2: Find a conversion factor for
to
1 inch = 2.54 cm, so the conversion factor is
Step 3: Convert the
to
.
.
.
1 in. 3= 21.6761063
354.88 cm3 ______
2.54 cm
(
)
The volume of the can is about
.
Check It Out!
2. Find
2
d the
h approximate volume of the cone in cubic feet.
1m
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2m
392 9-10 Measurement in Three-Dimensional Figures
Lesson Tutorial Videos
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Example 3
An archaeologist wants to apply a liquid solution to the lateral area of a square
pyramid as a protectant. Each side of the square base measures 12 meters and
the slant height is 10 meters. One gallon of solution covers 200 ft2.
About how many gallons of solution does the architect need to cover the
lateral area of the pyramid?
Step 1: Find the lateral area in m2. Do not include the area of the base.
L = __12 pℓ = __12 (4 ·
)(10) = 240 m2
Step 2: Find a conversion factor for meters to feet.
m, so the conversion factor is
1 foot =
.
feet.
Step 3: Convert to
240 m2
= 2583.3385 ft2, or about
ft2
Step 4: Find the number of gallons needed.
ft2 = 12.915.
ft2
About
gallons are needed.
3 ft
Copyright © by Holt McDougal. All rights reserved.
Check It Out!
3. The concrete tile shown is a hexagonal prism. A cubic yard of concrete 1 ft
weighs about 3600 pounds. What is the weight of 40 tiles in tons.
Lesson Tutorial Videos
2.6 ft
9-10 Measurement in Three-Dimensional Figures 393
Explore It!
9-10
Learn It!
Summarize It!
Name
Practice It!
Class
Summarize It!
Apply It!
Date
LA.8.2.2.3 The student will organize
information to show understanding
or relationships among facts…
(e.g., representing key points…through…
summarizing…)
Measurement in Three-Dimensional Figures
Think and Discuss
1. Explain how to convert an area in square feet to square centimeters.
2. Draw a diagram to show the number of cubic feet in a cubic yard.
3. Get Organized Complete the graphic organizer. To fill in the boxes, describe the
steps for finding the volume of a prism in cubic meters when the dimensions of the
prism are given in yards. You may use any number of steps.
1.
2.
3.
Volume of the
prism given in
cubic meters
394 9-10 Measurement in Three-Dimensional Figures
Copyright © by Holt McDougal. All rights reserved.
Dimensions of a
prism given in
cubic yards
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Practice It!
Summarize It!
Name
Apply It!
Class
9-10
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between
different measurement systems (US
customary or metric (SI)) and dimensions including...
area, volume, and derived units to solve problems.
Practice It!
Measurement in Three-Dimensional Figures
1. Find the approximate
area in in2.
2. Find the approximate
area in cm2.
3. Find the approximate
area in m2.
6 in.
160 ft
80 ft
20 cm
4 in.
20 cm
86 ft
Use the cylinder at the right for 4–5.
15 mm
4. Find the approximate surface area in in2.
10 mm
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5. Find the approximate volume in in3.
6. Find the approximate volume of the triangular prism
in cubic centimeters.
2 in.
1 in.
6 in.
7. A store sells tea leaves in cylindrical containers. A
certain tea costs $0.25 per teaspoon. What is the cost of
filling the container with tea? 1 in3 ≈ 3.3 tsp.
3.1 cm
12 cm
8. A cereal box is a rectangular prism that is 40 cm tall, 30 cm wide,
and 5 cm thick. A new box will have the same volume, a width of 12
inches and a thickness of 2 inches.
a. How tall is the new box in inches?
b. Is this taller or shorter than the original box? Justify your answer.
9-10 Measurement in Three-Dimensional Figures 395
Explore It!
9-10
Learn It!
Summarize It!
Name
Apply It!
Practice It!
Class
Date
MA.8.G.5.1 Compare, contrast, and
convert units of measure between different
measurement systems (US customary or
metric (SI)) and dimensions including…area, volume,
and derived units to solve problems.
Apply It!
Measurement in Three-Dimensional Figures
1. A crate in the shape of a rectangular prism
with linear dimensions 1 yd by _12 yd by
_3 yd is being filled with topsoil. How many
4
gallons of topsoil will fit? 1 in3 ≈ 0.0043 gal
5. Construction paper was used to make a
model of a pyramid for a class project. The
dimensions are shown.
4m
2. The diagram shows a stage floor.
2m
3m
28 ft
Find the volume in cubic feet.
26 ft
40 ft
Estimate the cost to varnish the stage floor if
the cost is $1.99 per square meter.
4. The glass cone below is to be filled with
colored sand at $0.02 per tablespoon. If
1 cm3 is about 0.0676 tbsp, how much will
the sand cost? (Hint: Convert cubic inches
to cubic centimeters to tablespoons.)
3 in.
7. Gridded Response
Nora is measuring fabric
in cubic yards. The price
is given per cubic meter.
She knows that 1 yard
is 0.9144 meter. What
number completes the
conversion factor she
will use? Round to three
decimal places.
m3
1 yd3 = _________
3
1 yd
6 in.
396 9-10 Measurement in Three-Dimensional Figures
Copyright © by Holt McDougal. All rights reserved.
3. Water covers 5983 square miles of Florida
and land covers 53,927 square miles. Find
the difference between the land and water
areas, in square kilometers.
6. Air passes through an air purifier at a rate
of 30 m3/h. How long would it take for the
purifier to process the air in a room with
dimensions 12 ft by 20 ft by 8 ft?
9-9
Name
Class
Got It?
THROUGH
Date
9-10
Ready to Go On?
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Quiz for Lessons 9-9 through 9-10
9-9
Scaling Three-Dimensional Figures (Student Textbook pp. 438–441)
A cube 12 inches on a side is made up of small cubes 4 inches on a side.
Find and compare the following values.
1. the side lengths of the two cubes
2. the surface areas of the two cubes
4. A skating arena is 400 ft long, 280 feet wide, and 100 feet tall. The
scale model used to build the arena is 20 inches long. Find the
width and height of the model.
5. A cube 6.5 inches on a side has a cube 5 inches on a side inside
it. Which volume is greater, the remaining space inside the cube
or the volume of the small cube? By what percent is the volume
greater?
9-10
Measurement in Three Dimensional Figures (Student Textbook pp. 442–445)
Find the area of a square with side length 130 centimeters to the nearest tenth.
6. in square meters
7. in square feet
Find the volume of the cone to the nearest tenth. Use 3.14 for π.
5 ft
Copyright © by Holt McDougal. All rights reserved.
3. the volumes of the two cubes
8. in cubic inches
9 ft
9. in cubic meters
10. An industrial popcorn making machine makes 175
1-ounce servings per hour. One ounce is about 2000 cm3.
How long would it take to fill a 24 inch by 20 inch by
36 inch popcorn cart?
Chapter 9 Geomentry and Measurement 397
9-9
Name
THROUGH
9-10
Class
Date
Connect It!
MA.8.G.2.4; MA.8.G.5.1
Connect the concepts of Lessons 9-9 through 9-10
A Fishy Problem
Javier is looking for an aquarium that fits in the corner of a room.
He wants an aquarium that holds at least 20 gallons of water.
1. Javier sees the aquarium at right in a store. The base of the
aquarium is an isosceles right triangle. Write and solve an
equation to find the length x to the nearest centimeter.
180 cm
2. Find the volume of the aquarium.
x
3. Show how to convert the volume to cubic feet.
(Hint: 1 ft = 30.48 cm)
x
42 cm
4. One cubic foot is approximately 7.48 gallons. Should Javier buy the
aquarium? Why or why not?
In the Thick of It
1. In each box, circle the letter of the measurement that is closer to the
measurement shown in blue.
600 in2
2
120 m2
2
2
8 in3
2
50 ft
4 ft
1292 ft
394 ft
131 cm
A
R
T
N
O
3
1600 ft3
3
52 cm
149 m
I
S
3
2. Arrange the circled letters to find the animal
that has up to a million hairs per square inch.
Think About The Puzzler
3. Explain how you can use estimation to help solve the puzzle.
398 Chapter 9 Geometry and Measurement
45 m
T
390 cm3
3
24 in
E
3
60 in
H
3
Copyright © by Holt McDougal. All rights reserved.
Which animal has the thickest fur? Solve the puzzle to find out!
FLORIDA
Name
Class
Study It!
Vocabulary
CHAPTER
Date
9
Multi-Language
Glossary
Go to thinkcentral.com
(Student Textbook page references)
acute angle . . . . . . . . . . . . . (400)
obtuse angle . . . . . . . . . . . . (400)
scalene triangle . . . . . . . . . (410)
acute triangle . . . . . . . . . . . (409)
parallel lines. . . . . . . . . . . . (406)
slant height. . . . . . . . . . . . . (434)
adjacent angles . . . . . . . . . (401)
parallelogram . . . . . . . . . . (415)
square . . . . . . . . . . . . . . . . . (415)
angle. . . . . . . . . . . . . . . . . . . (400)
perpendicular lines . . . . . (406)
straight angle . . . . . . . . . . . (400)
capacity . . . . . . . . . . . . . . . . (438)
polygon . . . . . . . . . . . . . . . . (414)
surface area . . . . . . . . . . . . (430)
complementary angles . . (400)
rectangle . . . . . . . . . . . . . . . (415)
supplementary angles . . . (400)
congruent angles . . . . . . . (401)
regular polygon . . . . . . . . . (415)
transversal . . . . . . . . . . . . . (406)
equilateral triangle. . . . . . (410)
regular pyramid . . . . . . . . (434)
trapezoid . . . . . . . . . . . . . . . (415)
isosceles triangle. . . . . . . . (410)
rhombus . . . . . . . . . . . . . . . (415)
Triangle Sum Theorem . . (402)
lateral face . . . . . . . . . . . . . (430)
right angle. . . . . . . . . . . . . . (400)
vertical angles . . . . . . . . . . (401)
lateral surface . . . . . . . . . . (430)
right cone . . . . . . . . . . . . . . (434)
volume . . . . . . . . . . . . . . . . . (420)
obtuse triangle. . . . . . . . . . (409)
right triangle . . . . . . . . . . . (409)
Complete the sentences below with vocabulary words from the list above.
1. Lines in the same plane that never meet are called
.
Copyright © by Holt McDougal. All rights reserved.
Lines that intersect at 90° angles are called
.
2. A quadrilateral with 4 congruent angles is called a
.
A quadrilateral with 4 congruent sides is called a
.
3. The nonadjacent angles formed by two intersecting lines are called
Lesson 9-1
.
Angle Relationships (Student Textbook pp. 400–404)
A
Find the angle measure.
m∠1
m∠1 + 122° =
180°
122°
-122°
−−−−−−−−
−−−−
m∠1
=
58°
1
2
MA.8.G.2.2
122°
3
Find each angle measure.
4. m∠1
5. m∠2
3
2
68°
1
6. m∠3
Lesson Tutorial Videos @ thinkcentral.com
Chapter 9 Geometry and Measurement 399
Lesson 9-2
Parallel and Perpendicular Lines (Student Textbook pp. 405–408)
P
Line j line k. Find each angle measure.
m∠1
m∠1 = 143°
MA.8.G.2.2
Alternate interior angles are .
143° 2
m∠2
m∠2 + 143° =
180°
- 143°
- 143°
−−−−−−−−
−−−−−
m∠2
=
37°
1
k
j
Line p line q. Find each angle measure.
q
7. m∠1
8. m∠2
9. m∠3
10. m∠4
p
1
114°
3
2
4
5
11. m∠5
Lesson 9-3
Triangles (Student Textbook pp. 409–412)
T
MA.8.G.2.3
Find n°.
n° + 50° + 90°
n° + 140°
- 140°
−−−−−−−−−−
n°
=
=
180°
180°
- 140°
−−−−−
=
40°
n°
50°
13. Find p°.
m°
p°
128°
Lesson 9-4
m°
47°
Polygons (Student Textbook pp. 413–417)
P
78°
MA.8.G.2.3
Find the angle measures in a regular 12–gon.
12x° = 180°(12 - 2)
12x° = 180°(10)
12x° = 1800°
x° = 150°
Find the angle measures in each regular polygon.
14. a regular hexagon
400 Chapter 9 Geometry and Measurement
15. a regular 11–gon
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12. Find m°.
Lesson 9-9
Scaling Three-Dimensional Figures (Student Textbook pp. 438–441)
S
A 4-in. cube is built from small cubes, each 2 in. on a side. Compare the
volumes of the large cube and the small cube.
vol. of large cube
____________
=
vol. of small cube
MA.8.G.5.1
3
64 in
4 in 3
= _____
=8
( ___
2 in )
8 in
3
The volume of the large cube is 8 times that of the small cube.
A 9-ft cube is built from small cubes, each 3 ft on a side. Compare the indicated
measures of the large cube and the small cube.
16. side lengths
17. surface areas
18. volumes
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19. A rectangular prism that is 4 in. deep, 12 in. wide, and 16 in. long
is scaled down so that the new prism is 3 in. wide.
How do the surface area and volume compare to the original surface area
and volume?
Lesson 9-10
Measurement in Three-Dimensional Figures
M
MA.8.G.5.1
((Student Textbook pp. 442–445)
How many square meters of sheet metal was used to construct the water tank
to the nearest tenth?Use 3.14 for π.
S = 2π(4)2 + 2π(4)(30)
= 32π + 240 π
= 272π = 854.08 ft2
1m 2
≈ 79.4 m2
= 854.08 ft2 _____
3.28 ft
(
4 ft
)
2
About 79.4 m of sheet metal was used to construct the water tank.
30 ft
20. What is the volume of the water tank above in cubic feet? in cubic
meters?
21. If the tank fills at a rate of 3 gallons per second and 1 cubic meter is
about 264 gallons, how long does the tank take to fill in minutes?
Lesson Tutorial Videos @ thinkcentral.com
Chapter 9 Geometry and Measurement 401
Name
Class
Write About It!
Date
LA.8.3.1.2 The student will prewrite
by making a plan for writing that
addresses purpose, audience, main
idea, logical sequence, and time frame for
completion.
Think and Discuss
Answer these questions to summarize the important concepts from Chapter 9
in your own words.
1. If m∠1 is 50°, explain how to find m∠5.
2
3
7
1
4
5 6
8
2. If the measures of two angles in a triangle are 43° and 74°, explain how to find
the measure of the third angle.
4. If the side lengths of a rectangular prism are measured in centimeters, its
surface area is A cm2 and its volume is V cm3, what is its surface area in in2
and volume in in3?
Before The Test
I need answers to these questions:
402 Chapter 9 Geometry and Measurement
Copyright © by Holt McDougal. All rights reserved.
3. Explain how triangles help you find the angle measures in a regular polygon
with more than 3 sides.