Download ANGLE PAIRS

Geometry
Name
Angle Pairs
In “Shapes in the Sand” on page 4, you used the degree formula to find unknown angle measures in a given polygon. You
can also use the properties of different angle pairs to solve for missing angle measures.
Types of Angle Pairs
Complementary angles: Angles whose measures have a sum of 90° (a right angle)
Supplementary angles: Angles whose measures have a sum of 180° (a straight line)
Vertical angles: Angles opposite one another, formed when two lines intersect; these angles
are always congruent (they have the same degree measure)
Adjacent angles: Angles that have a common side and a common vertex (corner point) and
do not overlap
Use these definitions to classify angle pairs and find missing angle measures in the questions below.
a
1
66°
24°
A. What is the sum of the angle measures above?
B. Are the angles complementary, supplementary, vertical,
and/or adjacent?
b
c
3
A. Are angles a and c complementary, supplementary,
vertical, and/or adjacent?
B. Angle a is 116°. What is the measure of angle c?
C. Are angles a and b complementary, supplementary,
vertical, and/or adjacent?
D. What is the measure of angle b?
2
75°
105°
A. What is the sum of the angle measures above?
B. Are the angles complementary, supplementary, vertical,
and/or adjacent?
E. What is the measure of angle d?
F. What is the sum of the measures of
angles a, b, c, and d?
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d
Geometry
Name
INTERIOR ANGLES
In “Shapes in the Sand” on page 4, you used the formula for degrees in a polygon to find the sum of the interior angles of a
given shape. Use what you learned and the formula below to answer five more questions about interior-angle sums.
Formula for Degrees in a Polygon:
180° 5 (number of sides – 2)
A. How many sides does the polygon pictured above
have?
B. Based on the degree formula, what will be the sum of
its interior angles?
3
What is the sum of the interior angles of the polygon
above?
4
The sum of the interior angles of a particular polygon
is 900°. How many sides does the figure have?
5
The sum of the interior angles of a particular polygon
is 1,800°. How many sides does the figure have?
2
What is the sum of the interior angles of the polygon
above?
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1
Algebraic Expressions
Name
COMBINING LIKE TERMS
In “Battle of the Bots” on page 6, you practiced writing and evaluating algebraic expressions to represent word problems. To
simplify algebraic expressions, you can combine like terms: terms that contain the same power of the same variable.
EXAMPLE: Simplify this algebraic expression: 3xy + 4x 2 + 7y 3 – 8x – 5y 3.
The expression above has 5 terms. Each term is associated with the addition or subtraction sign that comes before it:
3xy, + 4x 2, + 7y 3, – 8x, – 5y 3
The terms “+ 7y 3” and “– 5y 3” have the same power of the same variable: y 3. That means these terms can be combined.
To combine them, add the coefficients (the integers in front of those terms). Keep the power of the variable the same:
+7y 3 – 5y 3 = +2y 3
Rewrite the simplified expression using the combined like terms and the remaining terms:
3xy + 4x 2 + 2y 3 – 8x
1
7g – h + 2f + 3f – g – f
2
4x3 + 5x2 – x2 + 7x –2x + 8
3
5
A. During the morning shift, a coffee shop sold 25
cups of coffee and 18 muffins. During the afternoon
shift, the shop sold 16 cups of coffee and 13 muffins.
Write an expression with four terms that describes the
shop’s total coffee and muffin sales that day, where c
represents the cost of a cup of coffee, and m represents the
cost of a muffin.
B. Simplify the expression by combining like terms.
3p2 + 4n – 2p2 + p3 – 2n
4
C. If a cup of coffee costs $1.75 and a muffin costs
$1.95, how much revenue did the coffee shop make?
5ab + 8a4 + b2 – 6a2 + 3b2 + a – 3ab
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Solve the following questions by combining like terms to simplify the given algebraic expressions.
Algebraic Expressions
Name
BUILDING BOTS
In “Battle of the Bots” on page 6, you wrote and evaluated algebraic expressions to find unknown quantities in mathematical
scenarios involving a robotics challenge. Use what you learned to write five more algebraic expressions and solve for their
variables.
A. To build a dog-shaped robot to enter into a robotics
competition, you will need four robot legs, two internal
computers, and one stabilizing electronic “tail.” Write an
expression for the total cost of your dog-bot where l
represents the cost of each leg, c represents the cost of an
internal computer, and t represents the cost of an electronic
tail.
3
A. A Rubik’s Cube-solving robot can unscramble a
Rubik’s Cube in 5 seconds flat. Write an expression for
how long it would take the robot to solve a series of cubes,
where n represents the total number of cubes solved.
B. How many cubes could the Rubik’s Cube-solving robot
solve in 2 minutes?
B. How much will it cost to build your dog-bot if each leg
costs $250, an internal computer costs $200, and a tail
costs $50?
2
A. Your finished dog-bot can trot at a pace of 10
miles per hour. Write an expression for how far the
dog-bot can travel in x hours.
4
An updated model of the Rubik’s Cube robot can
unscramble a cube in 3 seconds. Write an expression
for how much longer it would take the older robot to solve
n cubes than it would take the newer robot.
5
How many Rubik’s Cubes can the newer model solve
in the time it takes the older model to solve 12 cubes?
B. How far can your dog-bot travel in 15 minutes?
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Scatter Plots
Name
CLUSTERS & OUTLIERS
In “Manatees on the Move” on page 8, you interpreted scatter plot data by drawing and analyzing trend lines. When
interpreting the meaning of data shown on a scatter plot, it is also helpful to identify clusters and outliers in the data.
Clusters are distinct groups of points that are close together within a scatter plot. These points may overlap or merely cluster
near one another.
Outliers are points that do not seem to follow the overall pattern of the data. These isolated points will be located far apart
from other points on the plot.
Analyze the scatter plot below to identify outliers and clusters.
Dog Weights by Height for Select Dogs
140
130
120
100
90
80
70
60
50
40
30
20
10
30
25
20
15
10
0
Dog height (inches)
1
What two variables are being compared in this scatter
plot?
4
Based on the clusters and outliers in the scatter plot,
how would you characterize the relationship between
dog height and weight? Use the data to support your
answer.
2
Circle any clusters you see in the data. What do these
clusters indicate?
3
Circle any outliers you see in the data. What do the
outliers indicate?
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Dog weight (pounds)
110
Scatter Plots
Name
MORE MANATEES!
In “Manatees on the Move” on page 8, you analyzed scatter plots by identifying and drawing trend lines through data points.
Use what you learned to answer five more questions based on the scatter plot below, which shows the combined totals of
east and west Florida manatee counts.
Manatee Counts in East and West Florida*
7,000
MANATEE COUNT
6,000
5,000
4,000
3,000
2,000
YEAR
15
20
10
20
20
05
00
20
95
19
19
90
0
SOURCE: Florida Fish and Wildlife CONSERVATION Commission
*Gaps in data indicate years in which manatee counts were not conducted.
1
Draw a trend line through the data in the scatter plot
above.
2
Does the trend line indicate a positive or a negative
relationship between the two variables?
3
4
In which year did the number of manatees sighted
decrease the most compared with the prior year?
5
Look back at the scatter plots on page 11 of your May
issue. In what ways does the scatter plot of combined
totals differ from the individual east and west scatter plots?
In what ways are the plots similar?
In which year did the number of manatees sighted
increase the most compared with the prior year?
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1,000
Mixed Skills
Name
MULTIPLYING & DIVIDING
DECIMALS
In “Super Skills” on page 14, you practiced solving word problems involving rational numbers. A rational number is a number
that can be expressed as the quotient of two integers, where the divisor is not zero. Rational numbers can be expressed as
integers, fractions, or decimals. When multiplying or dividing decimals in a problem, you can use an algorithm.
EXAMPLE: 3.25 5 4.8
EXAMPLE: 30.155 ÷ 3.7
STEP 1: Rewrite the expression vertically, leaving out the
STEP 1: Rewrite the expression using a division bracket.
decimal points.
The number outside the bracket is the divisor; the number
inside the bracket is the dividend.
325
48
5
)
3.7 30.155
STEP 2: Multiply as you would with whole numbers.
STEP 2: Move the decimal point in the divisor to the right
until the divisor is a whole number. Move the decimal
point in the dividend to the right the same number of
times.
325
5
48
2600
+ 13000
15600
)
37. 301.55
factors. This will be the total number of decimal places in
the product.
3.25 5 4.8
➔
three decimal places
in the factors
15.600
three decimal places
in the product
So 3.25 5 4.8 = 15.6
the decimal point in the quotient (your answer) directly
above the decimal point in the dividend.
8.15
37. 301.55
– 296
55
– 37
185
– 185
0
)
So 30.155 ÷ 3.7 = 8.15
Use the algorithms above to solve the following questions. Use another piece of paper to show your work.
1
1.936
5
2.7
2
94.17 ÷ 7.3
3
4
72.15
5
0.03
5
261.6 ÷ 1.2
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369 ÷ 1.5
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STEP 3: Divide as you would with whole numbers. Place
STEP 3: Count the total number of decimal places in the
Mixed Skills
Name
MIGHTY MEASURES
1
Bruce Banner weighs 128 pounds. The Hulk weighs
1,400 pounds. By what factor does Banner’s weight
increase when he transforms into the Hulk? Round to the
nearest whole number.
2
The Hulk can lift 200,000 pounds. A common
American field ant can lift 5,000 times its weight.
Based on the Hulk’s weight from question 1, who can lift
more in proportion to its weight: the Hulk or the ant?
4
Thor’s bones are 4 times denser than human bone,
which means they can withstand more weight before
they break. If Thor’s bones can handle a load of 76,000
pounds, how many pounds can normal human bone
withstand?
5
Iron Man’s suit is not made of iron. This metal is very
heavy and rusts easily. Instead, Tony Stark uses a mix
of metals, also called an alloy. Iron’s density is 0.284
pounds per cubic inch. If the alloy’s density is 0.233
pounds per cubic inch, what percent less dense is the alloy
compared with iron? Round to the nearest percent.
3
Hawkeye’s vision is 8 times better than normal human
vision. If he can spot a target from a distance of 56
feet, how close to the target would a person with normal
vision have to be to see it?
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In “Super Skills” on page 14, you used mixed skills to uncover fun facts about the Avengers’ incredible powers and abilities. Use
what you learned to answer five more questions about these superheroes’ superstrengths!
Geometry
Name
PYTHAGOREAN TRIPLES
In “Triple Threat!”on page 16, you used the Pythagorean theorem, a2 + b2 = c2, to find the length of an unknown side in a
right triangle. When all three sides of a right triangle are integers, the set of these numbers is known as a Pythagorean triple.
Use what you learned and the right-triangle diagram below to find the missing side in each of the following sets of Pythagorean
triples.
c
a
1
a = 11, b = 60, c = _____
2
a = 12, b = 35, c = _____
4
a = 4, b = _____, c = 5
5
a = _____, b = 15, c = 17
3
a = 20, b = 21, c = _____
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b
Practice Test
Name
ISSUE SKILLS REVIEW
The superhero Quicksilver travels at breakneck speeds,
moving even faster than the speed of sound! If he travels
675,000 meters in 30 minutes, what is his speed in meters
per second?
2
Bruce Banner’s height is 5'11". As the Hulk, he shoots
up to 8 feet tall. What is Banner’s percent increase in
height when he transforms into the Hulk? Round to the
nearest whole percent.
3
Tickets to a robotics competition are sold at three
different price points. Floor seats cost $20, upper-level
seats cost $10, and standing room costs $5. Write an
expression for the total cost of tickets purchased in a batch,
where f represents the number of floor-seat tickets,
l represents the number of upper-level tickets, and
s represents the number of standing-room tickets.
4
Use the expression from question 3 to find the total
cost of tickets if you were to purchase 10 floor tickets,
7 upper-level tickets, and 8 standing-room tickets.
6
If you are unable to draw a trend line, what does this
tell you about the variables on the plot?
7
What is the sum of the interior
angles in the figure to the right?
Hint: Use the formula for degrees in
a polygon:
180° 5 (number of sides – 2).
8
What is the sum of the interior
angles in the figure to the right?
9
Use the Pythagorean theorem to find the length of
the hypotenuse c for a right triangle with legs a = 5
inches and b = 4 inches. Round to the nearest tenth.
10
Find the length of the hypotenuse c for a right
triangle with legs a = 7 inches and b = 9 inches.
Round to the nearest tenth.
5
If a trend line drawn on a scatter plot has a slope of 2,
what type of relationship must the plotted values have?
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1