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Chapter 10
Geometry: Angles and Polygons
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10
Geometry: Angles and Polygons
Lesson 10-1
Measuring Angles
Lesson 10-2
Problem-Solving Strategy: Draw a
Diagram
Lesson 10-3
Estimating and Drawing Angles
Lesson 10-4
Parallel and Perpendicular Lines
Lesson 10-5
Problem-Solving Investigation:
Choose a Strategy
Lesson 10-6
Triangles
Lesson 10-7
Quadrilaterals
Lesson 10-8
Drawing Three-Dimensional Figures
10-1
Measuring Angles
Five-Minute Check (over Chapter 9)
Main Idea and Vocabulary
California Standards
Key Concept: Types of Angles
Example 1
Example 2
Example 3
Example 4
10-1
Measuring Angles
• I will measure and classify angles.
• angle
• degree
• obtuse angle
• side
• right angle
• straight angle
• vertex
• acute angle
10-1
Measuring Angles
Standard 5MG2.1 Measure, identify, and
draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate
tools (e.g., straightedge, ruler, compass,
protractor, drawing software.)
10-1
Measuring Angles
10-1
Measuring Angles
Find the measurement of the angle. Then classify
as acute, right, obtuse, or straight.
Use a protractor to find the measure of the angle.
10-1
Measuring Angles
10-1
Measuring Angles
Answer: Since the measurement is 60 degrees,
it is an acute angle.
10-1
Measuring Angles
Classify the angle below.
A. acute
B. obtuse
C. straight
D. right
10-1
Measuring Angles
Find the measurement of the angle. Then classify
as acute, right, obtuse, or straight.
Use a protractor to find the measure of the angle.
10-1
Measuring Angles
10-1
Measuring Angles
Answer: Since the measurement of the angle is 140
degrees, the angle is obtuse.
10-1
Measuring Angles
Classify the angle below.
A. acute
B. obtuse
C. straight
D. right
10-1
Measuring Angles
Find the measurement of the angle. Then classify
as acute, right, obtuse, or straight.
Use a protractor to find the measure of the angle.
10-1
Measuring Angles
10-1
Measuring Angles
Answer: Since the measurement of the angle is 90
degrees, the angle is right.
10-1
Measuring Angles
Classify the angle below.
A. acute
B. obtuse
C. straight
D. right
10-1
Measuring Angles
Find the measurement of the angle. Then classify
as acute, right, obtuse, or straight.
Use a protractor to find the measure of the angle.
10-1
Measuring Angles
10-1
Measuring Angles
Answer: Since the measurement is 105 degrees,
the angle is obtuse.
10-1
Measuring Angles
Classify the angle below.
A. acute
B. obtuse
C. straight
D. right
10-2
Problem-Solving Strategy: Draw a Diagram
Five-Minute Check (over Lesson 10-1)
Main Idea
California Standards
Example 1: Problem-Solving Strategy
10-2
Problem-Solving Strategy: Draw a Diagram
• I will solve problems by drawing a diagram.
10-2
Problem-Solving Strategy: Draw a Diagram
Standard 5MR2.3 Use a variety of methods,
such as words, numbers, symbols, charts,
graphs, tables, diagrams, and models, to explain
mathematical reasoning.
10-2
Problem-Solving Strategy: Draw a Diagram
Standard 5MG2.1 Measure, identify, and
draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate
tools.
10-2
Problem-Solving Strategy: Draw a Diagram
The science club is going to plant flowers in the
school courtyard, which is 46 feet by 60 feet,
and has walls on each side. The flower beds will
be 6 feet by 6 feet and will be 8 feet apart and 6
feet from the walls. How many flower beds can
the science club make to fit in the school
courtyard?
10-2
Problem-Solving Strategy: Draw a Diagram
Understand
What facts do you know?
• The courtyard measures 46 feet by 60 feet.
• Each flower bed will be 6 feet by 6 feet and will
be 8 feet apart and 6 feet from the walls.
What do you need to find?
• How many flower beds can fit in the school
courtyard?
10-2
Problem-Solving Strategy: Draw a Diagram
Plan
Draw a diagram.
10-2
Problem-Solving Strategy: Draw a Diagram
Solve
Answer: The diagram shows that 12 flower beds
will fit inside the courtyard.
10-2
Problem-Solving Strategy: Draw a Diagram
Check
Look back at the problem. Add the total distances along
the width to check that the sum is 46 feet. 6 + 6 + 8 + 6 +
8 + 6 + 6 = 46
Add the total distances across the length to check that
the sum is 60 feet. 6 + 6 + 8 + 6 + 8 + 6 + 8 + 6 + 6 = 60
Since the distances match the information in the problem,
the answer is correct.
10-3
Estimating and Drawing Angles
Five-Minute Check (over Lesson 10-2)
Main Idea
California Standards
Example 1
Example 2
Estimating Angles
10-3
Estimating and Drawing Angles
• I will estimate measures of angles and draw
angles.
10-3
Estimating and Drawing Angles
Standard 5MG2.1 Measure, identify, and
draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate
tools (e.g., straightedge, ruler, compass,
protractor, drawing software.)
10-3
Estimating and Drawing Angles
Estimate the
measure of the
following angle.
The angle is a
little less than
halfway between
180 degrees and
90 degrees.
Answer: 125 degrees is a reasonable estimate.
10-3
Estimating and Drawing Angles
Estimate the measure of the following angle.
A. 90 degrees
B. 45 degrees
C. 180 degrees
D. 170 degrees
10-3
Estimating and Drawing Angles
Draw a 39 degree angle.
Step 1 Draw one side of the angle. Then mark the
vertex and draw an arrow at the opposite end.
10-3
Estimating and Drawing Angles
Step 2 Place the center point of the protractor on
the vertex. Align the mark labeled 0 on the
protractor with the line. Count from 0° to
39° on the correct scale and make a dot.
10-3
Estimating and Drawing Angles
Step 3 Remove the protractor and use a straightedge
to draw the side that connects the vertex and
the dot.
10-3
Estimating and Drawing Angles
Choose the angle that shows 45 degrees.
A.
B.
C.
D.
10-4
Parallel and Perpendicular Lines
Five-Minute Check (over Lesson 10-3)
Main Idea and Vocabulary
California Standards
Example 1
Example 2
Example 3
Example 4
10-4
Parallel and Perpendicular Lines
• I will identify and measure parallel and
perpendicular lines.
• intersecting lines
• vertical angles
• parallel lines
• congruent angles
• perpendicular lines
10-4
Parallel and Perpendicular Lines
Standard 5MG2.1 Measure, identify, and
draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate
tools (e.g., straightedge, ruler, compass,
protractor, drawing software.)
10-4
Parallel and Perpendicular Lines
Use the figure below to determine if MN and MQ
are parallel, perpendicular, or neither.
The red square at point
M indicates that MN
and MQ intersect at
right angles.
Answer: Therefore, MN and MQ are perpendicular
lines.
10-4
Parallel and Perpendicular Lines
Use the figure below to determine if PR and RS
are parallel, perpendicular, or neither.
A. perpendicular
B. parallel
C. neither
10-4
Parallel and Perpendicular Lines
Use the figure below to determine if NP and MQ
are parallel, perpendicular, or neither.
If you extend the
lengths of the lines NP
and MQ, the lines will
never intersect.
Answer: Therefore, NP and MQ are parallel.
10-4
Parallel and Perpendicular Lines
Use the figure below to determine if PR and QS
are parallel, perpendicular, or neither.
A. perpendicular
B. parallel
C. neither
10-4
Parallel and Perpendicular Lines
Use the figure below to determine if NP and PQ
are parallel, perpendicular, or neither.
Since NP and PQ intersect,
they are not parallel lines.
And since NP and PQ do
not intersect at right angles,
they are not perpendicular
lines either.
10-4
Parallel and Perpendicular Lines
Answer: Therefore, NP and PQ are neither parallel nor
perpendicular.
10-4
Parallel and Perpendicular Lines
Use the figure below to determine if PQ and QS
are parallel, perpendicular, or neither.
A. perpendicular
B. parallel
C. neither
10-4
Parallel and Perpendicular Lines
Find the value of y in
the figure.
Since the two given
angles are vertical
angles, they are
congruent.
Answer: So, the value of y is 65 degrees.
10-4
Parallel and Perpendicular Lines
Find the value of y in the figure.
A. 25 degrees
B. 105 degrees
C. 15 degrees
D. 75 degrees
10-5
Problem-Solving Investigation: Choose a Strategy
Five-Minute Check (over Lesson 10-4)
Main Idea
California Standards
Example 1: Problem-Solving Investigation
10-5
Problem-Solving Investigation: Choose a Strategy
• I will choose the best strategy to solve a problem.
10-5
Problem-Solving Investigation: Choose a Strategy
Standard 5MR1.1 Analyze problems by
identifying relationships, distinguishing relevant
from irrelevant information, sequencing and
prioritizing information, and observing patterns.
Standard 5MG2.1 Measure, identify, and draw
angles, perpendicular and parallel lines, rectangles,
and triangles by using appropriate tools.
10-5
Problem-Solving Investigation: Choose a Strategy
EMELIA: I recently made my own quilt
pattern. I pieced together triangles
to make squares of different sizes.
The first square is made from 2
triangles, the second square is made
from 8 triangles, and the third
square is made from 18 triangles. The
quilt will have squares of 5
different sizes.
10-5
Problem-Solving Investigation: Choose a Strategy
YOUR MISSION: Find how many
triangles are in the fifth square.
10-5
Problem-Solving Investigation: Choose a Strategy
Understand
What facts do you know?
• You know how many triangles are in the first,
second, and third squares.
What do you need to find?
• You need to find how many triangles are in the
fifth square.
10-5
Problem-Solving Investigation: Choose a Strategy
Plan
Look for a pattern to find the number of triangles.
10-5
Problem-Solving Investigation: Choose a Strategy
Solve
Each square has twice as many triangles as small
squares.
First square 2 × 1 or 2 triangles
Second square 2 × 4 or 8 triangles
Third square 2 × 9 or 18 triangles
Continuing the pattern, the fourth square has 2 × 16
or 36 triangles and the fifth square has 2 × 25 or 50
triangles.
10-5
Problem-Solving Investigation: Choose a Strategy
Check
Draw the fifth square and count the number of
triangles. Since there are 50 triangles in the fifth
square, the answer is correct.
10-6
Triangles
Five-Minute Check (over Lesson 10-5)
Main Idea and Vocabulary
California Standards
Key Concept: Classify Triangles Using Angles
Key Concept: Classify Triangles Using Sides
Key Concept: Sum of Angle Measures in a Triangle
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the Lesson Menu
Angles in Triangles
10-6
Triangles
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Angles in Triangles
10-6
Triangles
• I will classify triangles and find missing angle
measures in triangles.
10-6
Triangles
• acute triangle
• congruent segments
• right triangle
• scalene triangle
• obtuse triangle
• isosceles triangle
• line segment
• equilateral triangle
10-6
Triangles
Standard 5MG2.1 Measure, identify, and
draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate
tools.
10-6
Triangles
Standard 5MG2.2 Know that the sum of
the angles of any triangle is 180° and the sum of
the angles of any quadrilateral is 360° and use
this information to solve problems.
10-6
Triangles
10-6
Triangles
10-6
Triangles
10-6
Triangles
Classify the triangle as acute, right, or obtuse.
Answer: There is a right angle, so this triangle is a
right triangle.
10-6
Triangles
Classify the triangle.
A. obtuse
B. acute
C. right
D. isosceles
10-6
Triangles
Classify the triangle as
acute, right, or obtuse.
All the angles are acute.
Answer: So, the triangle is an acute triangle.
10-6
Triangles
Classify the triangle as acute, right, or obtuse.
A. acute
B. right
C. obtuse
D. scalene
10-6
Triangles
ALGEBRA Find the value of x in the triangle.
10-6
Triangles
Since the sum of the angle measures in a triangle is
180 degrees, x + 85 + 24 = 180.
x + 85 + 24 =
180
Write the equation.
x + 109 =
180
Add 85 and 24.
– 109 = – 109
x
=
71
Subtract 109 from each side.
Simplify.
Answer: So, the value of x is 71 degrees.
10-6
Triangles
Find the value of x in the triangle.
A. 46 degrees
B. 45 degrees
C. 50 degrees
D. 40 degrees
10-6
Triangles
ALGEBRA A city park is in the shape of the triangle
shown. Find the value of x in the triangle.
10-6
Triangles
Since the sum of the angle measures in a triangle is
180 degrees, x + 36 + 36 = 180.
x + 36 + 36 =
180
Write the equation.
x + 72 =
180
Add 36 and 36.
– 72 =
=
– 72
108
Subtract 72 from each side.
x
Simplify.
Answer: So, the value of x is 108 degrees.
10-6
Triangles
A corner building
downtown is in the
shape of the triangle
shown. Find the value
of x in the triangle.
A. 50 degrees
B. 45 degrees
C. 43 degrees
D. 52 degrees
10-6
Triangles
Classify the triangle
shown as scalene,
isosceles, or
equilateral.
None of the sides are
congruent.
Answer: So, the triangle is a scalene triangle.
10-6
Triangles
Classify the triangle below as scalene, isosceles,
or equilateral.
A. scalene
B. isosceles
C. equilateral
D. obtuse
10-6
Triangles
Classify the triangle
shown as scalene,
isosceles, or
equilateral.
Only two of the sides are
congruent.
Answer: So, the triangle is an isosceles triangle.
10-6
Triangles
Classify the triangle below as scalene, isosceles,
or equilateral.
A. scalene
B. isosceles
C. equilateral
D. right
10-7
Quadrilaterals
Five-Minute Check (over Lesson 10-6)
Main Idea and Vocabulary
California Standards
Key Concept: Angles of a Quadrilateral
Key Concept: Classifying Quadrilaterals
Example 1
Example 2
Example 3
10-7
Quadrilaterals
• I will classify quadrilaterals and find missing angle
measures in quadrilaterals.
• quadrilateral
• parallelogram
• rectangle
• rhombus
• square
• trapezoid
10-7
Quadrilaterals
Standard 5MG2.1 Measure, identify, and
draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate
tools.
10-7
Quadrilaterals
Standard 5MG2.2 Know that the sum of
the angles of any triangle is 180° and the sum of
the angles of any quadrilateral is 360° and use
this information to solve problems.
10-7
Quadrilaterals
10-7
Quadrilaterals
10-7
Quadrilaterals
10-7
Quadrilaterals
Find the value of x in the quadrilateral shown.
You know that in
a parallelogram,
opposite angles
are congruent.
Answer: Since the angle opposite the missing angle
has a measure of 130°, x = 130°.
Check 50° + 130° + 50° + 130° = 360°
10-7
Quadrilaterals
Find the value of x in the quadrilateral shown.
A. 50 degrees
B. 60 degrees
C. 130 degrees
D. 120 degrees
10-7
Quadrilaterals
Classify the quadrilateral of each rug below.
Answer: This is a
parallelogram.
Answer: This is a
square.
10-7
Quadrilaterals
Classify the quadrilateral below.
A. rectangle
B. rhombus
C. trapezoid
D. square
10-7
Quadrilaterals
ALGEBRA What is the value of x in the
quadrilateral below?
10-7
Quadrilaterals
You know that in a quadrilateral, all angles add up to
360 degrees. We know that there are two right angles
that are each 90 degrees.
x + 90 + 90 + 73 =
360
Write the equation.
x + 253 =
360
Add 90, 90 and 73 together.
– 253 = – 253
x
=
107
Subtract 253 from each side.
Simplify.
Answer: So, the value of x is 107 degrees.
10-7
Quadrilaterals
ALGEBRA What is the value of x in the
quadrilateral below?
A. 108 degrees
B. 100 degrees
C. 90 degrees
D. 118 degrees
10-8
Drawing Three-Dimensional Figures
Five-Minute Check (over Lesson 10-7)
Main Idea and Vocabulary
California Standards
Key Concept: Prisms
Example 1
Example 2
Example 3
10-8
Drawing Three-Dimensional Figures
• I will draw two-dimensional views of threedimensional figures.
• three-dimensional figure
• vertex
• face
• prism
• edge
• base
10-8
Drawing Three-Dimensional Figures
Standard 5MG2.3 Visualize and draw twodimensional views of three-dimensional
objects made from rectangular solids.
10-8
Drawing Three-Dimensional Figures
10-8
Drawing Three-Dimensional Figures
Draw a top, a side, and a front view of the
prism below.
10-8
Drawing Three-Dimensional Figures
The front and side views of the prism are rectangles.
The top is also a rectangle.
10-8
Drawing Three-Dimensional Figures
Select the correct
top, side, and front
view descriptions for
the prism below.
A. top and front are rectangles, side is a square
B. all are rectangles
C. all are squares
D. top and side are squares, front is a rectangle
10-8
Drawing Three-Dimensional Figures
Draw a top, a side, and a front view of this plant
stand.
10-8
Drawing Three-Dimensional Figures
The top view is a rectangle. The side is a square.
The front is two rectangles.
10-8
Drawing Three-Dimensional Figures
Select the correct top, side, and front view
descriptions for the prism below.
10-8
Drawing Three-Dimensional Figures
Select the correct top, side, and front view
descriptions for the prism below.
A. top – rectangle, side – square, front – two
rectangles
B. top – two rectangles, side – square, front –
rectangle
C. top – two rectangles, side – square, front –
two rectangles
D. top – square, side – rectangle, front – two
rectangles
10-8
Drawing Three-Dimensional Figures
Draw a three-dimensional figure whose top, side,
and front views are shown. Use isometric dot paper.
10-8
Drawing Three-Dimensional Figures
Step 1 Use the top view to draw the base of the
figure, a rectangle that is 3 units long.
Step 2 Use side and
front views to
complete the
figure.
10-8
Drawing Three-Dimensional Figures
Draw a three-dimensional figure whose top, side,
and front views are shown. Use isometric dot paper.
10-8
Drawing Three-Dimensional Figures
Draw a three-dimensional figure whose top, side,
and front views are shown. Use isometric dot paper.
A.
B.
10-8
Drawing Three-Dimensional Figures
Draw a three-dimensional figure whose top, side,
and front views are shown. Use isometric dot paper.
C.
D.
10-8
Drawing Three-Dimensional Figures
Draw a three-dimensional figure whose top, side,
and front views are shown. Use isometric dot paper.
C.
10
Geometry: Angles and Polygons
Five-Minute Checks
Estimating Angles
Angles in Triangles
10
Geometry: Angles and Polygons
Lesson 10-1 (over Chapter 9)
Lesson 10-2 (over Lesson 10-1)
Lesson 10-3 (over Lesson 10-2)
Lesson 10-4 (over Lesson 10-3)
Lesson 10-5 (over Lesson 10-4)
Lesson 10-6 (over Lesson 10-5)
Lesson 10-7 (over Lesson 10-6)
Lesson 10-8 (over Lesson 10-7)
10
Geometry: Angles and Polygons
(over Chapter 9)
Estimate 23% of 120.
A. 1 × 100 = 20
5
B. 1 × 120 = 50
2
C. 1 × 120 = 30
4
10
Geometry: Angles and Polygons
(over Chapter 9)
Estimate 67% of 589.
A. 1 × 600 = 400
3
B. 2 × 500 = 300
3
C. 2 × 600 = 400
3
10
Geometry: Angles and Polygons
(over Chapter 9)
Estimate 78% of 243.
A. 3 × 200 = 150
5
B. 4 × 250 = 200
5
C. 3 × 250 = 225
5
10
Geometry: Angles and Polygons
(over Chapter 9)
The Lorenzo family wants to leave a 20% tip on a
restaurant bill of $48.64. Estimate how much they
should leave.
A. 1 × $50 = $10
5
B. 2 × $40 = $5
5
C. 1 × $50 = $10
6
10
Geometry: Angles and Polygons
(over Lesson 10-1)
Classify the angle.
159°
A. obtuse
B. right
C. acute
10
Geometry: Angles and Polygons
(over Lesson 10-1)
Classify the angle.
71°
A. right
B. obtuse
C. acute
10
Geometry: Angles and Polygons
(over Lesson 10-1)
Classify the angle.
180°
A. acute
B. straight
C. perpendicular
10
Geometry: Angles and Polygons
(over Lesson 10-1)
Classify the angle.
90°
A. straight
B. acute
C. right
10
Geometry: Angles and Polygons
(over Lesson 10-1)
Find the measure of the angle and then classify
the angle.
A. 90°; right
B. 165°; obtuse
C. 180°; straight
10
Geometry: Angles and Polygons
(over Lesson 10-2)
Jay walks north 3 blocks; turns right and walks 2
blocks; turns right and walks 4 blocks; then turns
right and walks 2 blocks. Where is Jay in
relationship to where he started?
A. 3 blocks north of his starting point
B. 1 block south of his starting point
C. 1 block north of his starting point
10
Geometry: Angles and Polygons
(over Lesson 10-2)
What is the perimeter of a square pool with an area
of 400 ft2?
A. 80 ft
B. 40 ft
C. 20 ft
10
Geometry: Angles and Polygons
(over Lesson 10-3)
Estimate the measure of the angle.
A. 120°
B. 45°
C. 90°
10
Geometry: Angles and Polygons
(over Lesson 10-3)
Estimate the measure of the angle.
A. 95°
B. 180°
C. 45°
10
Geometry: Angles and Polygons
(over Lesson 10-4)
Describe the lines as parallel, perpendicular, or
intersecting.
A. parallel
B. interesecting
C. perpendicular
10
Geometry: Angles and Polygons
(over Lesson 10-4)
Describe the lines as parallel, perpendicular, or
intersecting.
A. intersecting
B. parallel
C. perpendicular
10
Geometry: Angles and Polygons
(over Lesson 10-4)
Describe the lines as parallel, perpendicular, or
intersecting.
A. intersecting and
perpendicular
B. perpendicular
C. parallel
10
Geometry: Angles and Polygons
(over Lesson 10-4)
When are pairs of vertical angles formed?
A. when two lines are parallel
B. when two lines intersect
C. when two lines are obtuse
10
Geometry: Angles and Polygons
(over Lesson 10-5)
Solve. The sum of two numbers is 12. When you
subtract the smaller number from the larger number,
the difference is 10. What are the two numbers?
What strategy did you use?
A. 1 and 10
B. 5 and 7
C. 1 and 11
10
Geometry: Angles and Polygons
(over Lesson 10-6)
Classify each triangle with the given angle
measures or side lengths.
65°, 75°, 55°
A. acute
B. obtuse
C. scalene
10
Geometry: Angles and Polygons
(over Lesson 10-6)
Classify each triangle with the given angle
measures or side lengths.
35°, 35°, 110°
A. acute
B. obtuse
C. scalene
10
Geometry: Angles and Polygons
(over Lesson 10-6)
Classify each triangle with the given angle
measures or side lengths.
6 cm, 8 cm, 10 cm
A. scalene
B. isosceles
C. right
10
Geometry: Angles and Polygons
(over Lesson 10-6)
Classify each triangle with the given angle
measures or side lengths.
5 in, 8 in, 5 in
A. obtuse
B. scalene
C. isosceles
10
Geometry: Angles and Polygons
(over Lesson 10-7)
What type(s) of quadrilateral has (have) the following
properties:
two pairs of parallel sides
A. parallelogram, rectangle, rhombus, square
B. trapezoid, square, prism, parallelogram
C. cone, square, circle, triangle
10
Geometry: Angles and Polygons
(over Lesson 10-7)
What type(s) of quadrilateral has (have) the following
properties:
four right angles
A. cone, prism
B. triangle, prism
C. rectangle, square
10
Geometry: Angles and Polygons
(over Lesson 10-7)
What type(s) of quadrilateral has (have) the following
properties:
all congruent sides
A. triangle, prism
B. rhombus, square
C. cone, trapezoid
10
Geometry: Angles and Polygons
(over Lesson 10-7)
What type(s) of quadrilateral has (have) the following
properties:
exactly one pair of parallel sides
A. cone, prism
B. trapezoid
C. square, cone
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