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Mathematics: Applications and Concepts, Course 2 Interactive Chalkboard
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Lesson 10-1
Angles
Lesson 10-2
Making Circle Graphs
Lesson 10-3
Angle Relationships
Lesson 10-4
Triangles
Lesson 10-5
Quadrilaterals
Lesson 10-6
Similar Figures
Lesson 10-7
Polygons and Tessellations
Lesson 10-8
Translations
Lesson 10-9
Reflections
Example 1 Classify Angles
Example 2 Classify Angles
Example 3 Draw an Angle
Classify the angle as acute, obtuse, right, or straight.
Answer: The angle is exactly 180, so it is a straight angle.
Classify the angle as acute, obtuse, right, or straight.
Answer: right
Classify the angle as acute, obtuse, right, or straight.
Answer: The angle is less than 90, so it is an acute angle.
Classify the angle as acute, obtuse, right, or straight.
Answer: obtuse
Draw an angle with a measure of 140.
Classify the angle as acute, obtuse, right, or straight.
Draw a ray.
Place the center point of the protractor on the
endpoint of the ray. Align the ray with 0°.
Use the scale that begins with 0°. Locate the mark
labeled 140. Then draw the other side of the angle.
Answer: Since the angle is greater than 90°, it is obtuse.
Draw an angle with a measure of 60.
Classify the angle as acute, obtuse, right, or straight.
Answer: acute
Example 1 Construct a Circle Graph
Example 2 Construct a Circle Graph
Example 3 Interpret a Circle Graph
SPORTS In a survey, a group of middle school
students were asked to name their favorite sport. The
results are shown in the table. Make a circle graph of
the data.
Sport
football
basketball
baseball
tennis
other
Percent
30%
25%
22%
8%
15%
Find the degrees for each part. Round to the nearest
whole degree.
football:
basketball:
baseball:
tennis:
other:
Use a compass to draw a circle with a radius marked as
shown.
Then use a protractor to draw the first angle, in this
case 108.
Repeat this step for each section.
Check To draw an accurate circle graph, make sure the
sum of the angle measures equals 360°.
Label each section of the graph with the category
and percent. Give the graph a title.
Answer:
Students’ Favorite Sports
ICE CREAM In a survey, a group of students were
asked to name their favorite flavor of ice cream. The
results are shown in the table. Make a circle graph of
the data.
Flavor
chocolate
cookie dough
Percent
30%
25%
peanut butter
strawberry
15%
10%
other
20%
Answer:
MOVIES Gina has the following types of movies in her
DVD collection. Make a circle graph of the data.
Type of Movie
action
comedy
science fiction
Find the total number of DVDs:
Number
24
15
7
Find the ratio that compares each number with the total.
Write the ratio as a decimal number rounded to the nearest
hundredth.
Find the number of degrees for each section of the graph.
:
=
:
=
=
Draw the circle graph.
Check After drawing the first two sections, you can
measure the last section of the circle graph to
verify that the angles have the correct measures.
Answer:
MARBLES Michael has the following colors of
marbles in his marble collection. Make a circle
graph of the data.
Color
Black
Green
Red
Gold
Number
12
9
5
3
Answer:
VOTING The circle graph below shows the percent of
voters in a town who are registered with a political
party. Use the graph to describe the voters in this town.
Answer: The majority of
voters are Democrats.
Only a very small portion
are Independents.
SPORTS The circle graph below shows the responses
of middle school students to the question, “Should
teens be allowed to play professional sports?”. Use the
graph to describe the opinions of the middle school
students.
Answer: The majority of the middle
school students felt that teens should
be allowed to play professional
sports. A small portion of the middle
school students had no opinion on
this issue.
Example 1 Classify Angles
Example 2 Classify Angles
Example 3 Find a Missing Angle Measure
Example 4 Use Angles to Solve A Problem
Classify the pair of angles as complementary,
supplementary, or neither
Answer: The angles are supplementary.
Classify the pair of angles as complementary,
supplementary, or neither
Answer: complementary
Classify the pair of angles as complementary,
supplementary, or neither
x and y form a right angle.
Answer: So, the angles are complementary.
Classify the pair of angles as complementary,
supplementary, or neither
Answer: supplementary
Angles PQS and RQS are supplementary.
If mPQS 56, find mRQS.
Since PQS and RQS are supplementary,
Write the equation.
Replace mPQS with 56.
Subtract 56 from each side.
Answer: The measure of RQS is 124.
Angles MNP and KNP are complementary.
If mMNP 23, find mKNP.
Answer: 67
GEOMETRY The rectangle shown is divided by a
diagonal. Find the value of x.
The sum of the two angles created by the diagonal is 90
because the corner of a rectangle is a right angle.
Write the equation.
Subtract 70 from each side.
Answer: The value of x is 20.
GRAPHING In the circle graph shown below, find the
value of x.
Answer: 62
Example 1 Find Angle Measures of Triangles
Example 2 Find a Missing Measure
Example 3 Classify Triangles
Example 4 Classify Triangles
PLANES An airplane has wings that are shaped like
triangles. Find the missing measure.
The sum of the measures is 180.
Simplify.
Subtract 159 from each side.
Answer: The missing measure is 21.
SEWING A piece of fabric is shaped like a triangle.
Find the missing measure.
Answer: 49
ALGEBRA Find mA in ABC if mA
mC 80.
mB and
Draw the triangle. Then write and solve an equation to
find mA.
The sum of the measures
is 180.
Replace B with A because
the angles are equal.
Simplify and replace
mC with 80.
Subtract 80 from each side.
Simplify.
Divide each side by 2.
Answer: The measure of A is 50.
ALGEBRA Find mJ in JKL if mJ
mL 100.
Answer: 40
mK and
Classify the triangle by its angles and its sides.
Answer: The triangle has one obtuse angle and
two congruent sides. So, it is an obtuse,
isosceles triangle.
Classify the triangle by its angles and its sides.
Answer: right, scalene
Classify the triangle by its angles and its sides.
Answer: The triangle has all acute angles and no
congruent sides. So, it is an acute,
scalene triangle.
Classify the triangle by its angles and its sides.
Answer: acute, equilateral
Example 1 Classify Quadrilaterals
Example 2 Classify Quadrilaterals
Example 3 Find a Missing Measure
Classify the quadrilateral using the name that
best describes it.
Answer: The quadrilateral has 4 right angles and
opposite sides are congruent. It is a rectangle.
Classify the quadrilateral using the name that
best describes it.
Answer: parallelogram
Classify the quadrilateral using the name that
best describes it.
Answer: The quadrilateral has one pair of parallel sides.
It is a trapezoid.
Classify the quadrilateral using the name that
best describes it.
Answer: square
Find the missing angle measure in the quadrilateral.
The sum of the
measures is 360.
Simplify.
Subtract 240 from both sides.
Answer: The missing angle measure is 120.
Find the missing angle measure in the quadrilateral.
Answer: 134
Example 1 Find Side Measures of Similar Triangles
Example 2 Use Indirect Measurement
If ABC  DEF, find the length of
Since the two triangles are similar, the ratios of their
corresponding sides are equal. So, you can write and
solve a proportion to find
Write a proportion.
Let n represent the length of
Then substitute.
Find the cross products.
Simplify.
Divide each side by 3.
Answer: The length of
is 16.5 centimeters.
.
If JKL  MNO, find the length of
Answer: 13.5 in.
GRID-IN TEST ITEM A rectangular picture window
12-feet long and 6-feet wide needs to be shortened to
9 feet in length to fit a redesigned wall. If the
architect wants the new window to be similar to the
old window, how wide will the new window be?
Read the Test Item
To find the width of the new window, draw a picture and
write a proportion.
Solve the Test Item
Write a proportion.
Find the cross products.
Divide each side by 12.
So, the width of the new window will be 4.5 feet.
Answer:
GRID-IN TEST ITEM Tom has a rectangular garden
which has a length of 12 feet and a width of 8 feet. He
wishes to start a second garden which is similar to the
first and will have a width of 6 feet. Find the length of
the new garden.
Answer:
Example 1 Classify Polygons
Example 2 Classify Polygons
Example 3 Angle Measures of a Polygon
Example 4 Tessellations
Determine whether the figure is a polygon. If it is,
classify the polygon and state whether it is regular.
If it is not a polygon, explain why.
Answer: The figure is not a polygon since it has
a curved side.
Determine whether the figure is a polygon. If it is,
classify the polygon and state whether it is regular.
If it is not a polygon, explain why.
Answer: pentagon, not regular
Determine whether the figure is a polygon. If it is,
classify the polygon and state whether it is regular.
If it is not a polygon, explain why.
Answer: This figure has 6 sides which are not all of equal
length. It is a hexagon that is not regular.
Determine whether the figure is a polygon. If it is,
classify the polygon and state whether it is regular.
If it is not a polygon, explain why.
Answer: not a polygon, sides overlap
ALGEBRA Find the measure of each angle of a regular
heptagon. Round to the nearest tenth of a degree.
Draw all of the diagonals from one vertex and count the
number of triangles formed.
Find the sum of the measures of the polygon.
number of triangles
formed 180°
sum of angle measures
in polygon
Find the measure of each angle of the polygon.
Let n represent the measure of one angle in the heptagon.
There are seven congruent angles.
Divide each side by 7.
Answer: The measure of each angle in a regular
heptagon is about 128.6.
Find the measure of each angle in a regular hexagon.
Answer: 120
PATTERNS Ms. Pena is creating a pattern on her wall.
She wants to use triangles with angles 120, 30, and
30. Can Ms. Pena tessellate with these triangles?
The sum of the measures of the angles where the vertices
meet must be 360
Both 30 and 120 divide evenly into 360. Therefore,
Ms. Pena can arrange the triangles in a way that the
angles where the vertices meet make 360. She can
tessellate with these triangles.
Check
You can check if your answer is correct by drawing
a tessellation of triangles with angles measuring 120°, 30°,
and 30°.
Answer: Yes, they can be arranged in a way that the
angles where the vertices meet make 360.
QUILTING Emily is making a quilt using fabric pieces
shaped as equilateral triangles. Can Emily tessellate
the quilt with these fabric pieces?
Answer: Yes, they can be arranged in a way that the
angles where the vertices meet make 360.
Example 1 Graph a Translation
Example 2 Find Coordinates of a Translation
Example 3 Naming Translations with Ordered Pairs
Translate ABC 5 units left and 1 unit up.
Move each vertex of the figure
5 units left and 1 unit up.
Label the new vertices A, B,
and C.
Connect the vertices to draw
the triangle.
Answer:
The coordinates of the vertices
of the new figure are A(–4, –1),
B(–1, 2), and C(0, –1).
Translate DEF 3 units left and 2 units down.
Answer:
Trapezoid GHIJ has vertices G(–4, 1), H(–4, 3), I(–2, 3),
and J(–1, 1). Find the vertices of trapezoid GHIJ
after a translation of 5 units right and 3 units down.
Then graph the figure and its translated image.
(x + 5, y – 3)
Vertices of
trapezoid GHIJ
G(–4, 1)
H(–4, 3)
(–4 + 5, 1 – 3)
(–4 + 5, 3 – 3)
I(–2, 3)
J(–1, 1)
(–2 + 5, 3 – 3)
(–1 + 5, 1 – 3)
Vertices of trapezoid
GHIJ
G(1, –2)
H(1, 0)
I(3, 0)
J(4, –2)
Answer: The coordinates of trapezoid GHIJ are
G(1, –2), H(1, 0), I(3, 0), and J(4, –2).
Triangle MNO has vertices M(–5, –3), N(–7, 0), and
O(–2, 3). Find the vertices of triangle MNO after
a translation of 6 units right and 3 units up.
Then graph the figure and its translated image.
Answer: M(1, 0)
N(–1, 3)
O(4, 6)
CLASSROOMS The squares below represent desks
in a classroom. Ana is seated at the square marked X.
She is moved to the seat marked Y. Describe this
translation as an ordered pair.
Answer: Ana moved 2 places right and 2 places up.
So, the translation can be written (2, 2).
CONVENTION The squares below represent booths
at a convention center. The Sail Maker Company was
located at the booth marked X last year. This year they
have been moved to the booth marked Y. Describe this
translation as an ordered pair.
Answer: (–3, –3)
Example 1 Identify Lines of Symmetry
Example 2 Identify Lines of Symmetry
Example 3 Identify Lines of Symmetry
Example 4 Reflect a Figure Over the x-axis
Example 5 Reflect a Figure Over the y-axis
Determine whether the figure has line symmetry.
If so, copy the figure and draw all lines of symmetry.
Answer: This figure has line symmetry.
There are two lines of symmetry.
Determine whether the figure has line symmetry.
If so, copy the figure and draw all lines of symmetry.
Answer: no symmetry
Determine whether the figure has line symmetry.
If so, copy the figure and draw all lines of symmetry.
Answer: This figure has line symmetry.
There is one line of symmetry.
Determine whether the figure has line symmetry.
If so, copy the figure and draw all lines of symmetry.
Answer: There are two lines of symmetry.
Determine whether the figure has line symmetry.
If so, copy the figure and draw all lines of symmetry.
Answer: This figure does not have line symmetry.
Determine whether the figure has line symmetry.
If so, copy the figure and draw all lines of symmetry.
Answer: There is one line of symmetry.
Quadrilateral QRST has vertices Q(–1, 1), R(0, 3),
S(3, 2), and T(4, 0). Find the coordinates of QRST after
a reflection over the x-axis. Then graph the figure and
its reflected image.
Vertices of
Quadrilateral QRST
Distance
from x-axis
Q(–1, 1)
R(0, 3)
1
S(3, 2)
T(4, 0)
3
2
0
Vertices of
Quadrilateral
QRST
Q(–1, –1)
R(0, –3)
S(3, –2)
T(4, 0)
Plot the vertices and connect to form the quadrilateral
QRST.
The x-axis is the line of symmetry. So, the distance from
each point on quadrilateral QRST to the line of symmetry
is the same as the distance from the line of symmetry to
quadrilateral QRST.
Answer: Q(–1, –1)
R(0, –3)
S(3, –2)
T(4, 0)
Quadrilateral ABCD has vertices A(–3, 2), B(–1, 5),
C(3, 3), and D(2, 1). Find the coordinates of ABCD
after a reflection over the x-axis. Then graph the figure
and its reflected image.
Answer: A(–3, –2)
B(–1, –5)
C(3, –3)
D(2, –1)
Triangle XYZ has vertices X(1, 2), Y(2, 1), and Z(1, –2).
Find the coordinates of XYZ after a reflection over the
y-axis. Then graph the figure and its reflected image.
Vertices of
XYZ
X(1, 2)
Y(2, 1)
Z(1, –2)
Distance
from y-axis
1
2
1
Vertices of
XYZ
X(–1, 2)
Y(–2, 1)
Z(–1, –2)
Plot the vertices and connect to form the triangle XYZ.
The y-axis is the line of symmetry. So, the distance from
each point on triangle XYZ to the line of symmetry is the
same as the distance from the line of symmetry
to triangle XYZ.
Answer: X(–1, 2),
Y(–2, 1), Z(–1, –2)
Triangle QRS has vertices Q(3, 4), R(1, 0), and S(6, 2).
Find the coordinates of QRS after a reflection over the
y-axis. Then graph the figure and its reflected image.
Answer: Q(–3, 4)
R(–1, 0)
S(–6, 2)
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