Download Geometry and Measurement of Plane Figures Activity Set 3 Trainer

Geometry and Measurement
of Plane Figures
Activity Set 3
Trainer Guide
geometry and measurement of Plane figures—Activity Set 3 Mid_PGe_03_TG
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GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
NGSSS 6.A.2.1
NGSSS 6.A.2.2
NGSSS 7.A.1.1
It’s Like This
In this activity, participants will distinguish between
similar and congruent plane figures using ratios.
Materials
• Transparency/Page: It’s Like This
• Transparency/Page: It’s Like This Answer Key
• Transparency/Page: Image Dilation
• Transparency/Page: Similar Sides
• Transparency/Page: Similar Sides Answer Key
• blank transparencies
• ruler with centimeters (1 for each participant)
• protractor (1 for each participant)
Vocabulary
• congruent figures
• similar figures
• dilation
• scale factor
Time:
20 minutes
Introduce
it’s like this
Measure the angles and sides of each figure and
label the diagrams.
B
E
H
Fig. 1
A
Fig. 2
C
D
F
Fig. 3
G
•Explain to participants that they will investigate
the relationships between similar figures and
congruent figures.
I
•Display Transparency: It’s Like This and have
participants take out their matching pages.
1. How are Figures 1 and 2 the same?
How are Figures 1 and 2 different?
2. How are Figures 1 and 3 the same?
How are Figures 1 and 3 different?
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Transparency: It’s Like This
•Instruct participants to use their protractors and rulers to
measure and compare the three triangles on the page.
•Direct participants to write the measurements on the
figures and then describe the relationships between
Figures 1 and 2 and Figures 1 and 3.
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1
GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
•Ask participants to work individually.
•Allow participants 5–7 minutes to complete
the activity.
Discuss and Do
•Call the group together.
•Ask volunteers to share their angle and length
measurements for each figure.
•Record the measurements on the transparency.
•Ask volunteers to share their findings to the
following questions:
How are Figures 1 and 2 the same? (They are the
same size and shape; all corresponding angles and
all corresponding sides are equal.)
◆
How are they different? (They are exactly the same.)
◆
•Explain to participants that figures that are exactly the
same size and same shape are congruent.
•Write, on the transparency, ABC  DEF, and point
out that this means that the two triangles are congruent.
•Ask volunteers to share their findings to the
following questions:
How are Figures 1 and 3 the same? (They are
the same shape; the corresponding angles are the
same size.)
◆
How are they different? (The corresponding sides
are of different lengths.)
◆
•Tell participants that figures that are the same shape
and have congruent angles, but that have
corresponding side lengths that are proportional
rather than equal, are similar.
•Write, on the transparency, ABC  GHI and point
out that this means the two triangles are similar.
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2
GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
•Draw, on a blank transparency, two similar triangles
as illustrated below.
•Tell participants that these triangles are similar.
m
2c
m
m
3c
2c
3c
m
•Ask participants what observations can be made
about these triangles. (Their corresponding angles
are congruent and their corresponding sides
are proportional.)
2cm
3cm
•Draw another pair of similar triangles as
illustrated below.
Note: The angles are labeled for these scalene
triangles. Side measures are rounded to the nearest
tenth and angle measures are rounded to the
nearest degree.
•Ask participants if these triangles are similar. (yes)
55°
96°
4.7
6m
6.3
96°
m
29°
4m
3m
•Ask participants how they can tell that these triangles
are similar. (Their corresponding angles are congruent
and their corresponding sides are proportional.)
55°
m
29°
8m
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3
GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
Teaching Tip: Ask participants if all rectangles are
similar, as they all have congruent corresponding
angles. (No, their sides may not be in proportion to
each other. Different rectangles may have different
shapes, even though all angles are congruent.
Squares, however, are always similar because their
sides are always proportional.)
image Dilation
•Display Transparency: Image Dilation
•Explain to participants that you can enlarge the
image by creating a larger space that is proportional
to this one.
•Explain to participants that this process is often used
by artists and craftspeople to enlarge a sketch to its
full size when producing a painting or a mural.
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Transparency: Image Dilation
•Demonstrate how to create a proportional grid by
drawing a new box below the existing image. Make it
with a scale factor of 2 : 1. The new box will be twice
as wide and twice as tall.
•Point out a small section of the image and sketch the
part contained in the section into the corresponding
section in larger grid by using the 2 : 1 ratio for
each line.
•Have students complete the process on their pages.
•Allow 5–7 minutes for this activity.
•Get the group’s attention.
•Explain to participants that when proportions are
used to create images larger or smaller than their
originals, the proportions are referred to as scale
factors. When a copy is five times the size of the
original, it has been increased by a scale factor of 5.
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GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
•Ask participants to describe the relationship between
the height of the original and that of the copy.
(The copy is twice as long. In terms of a ratio, the
height of copy: height of original is 2 : 1.
•Ask participants what the scale factor is. (2)
•Ask volunteer participants how the area of the copy
compares to the area of the original. (The copy is
4 times the area of the original. In terms of a ratio,
area of copy : area of original is 4 : 1.
•Explain that this scale factor is often expressed as 22
to indicate that it refers to area or square units.
Conclude
Similar Sides
1.
ABC ~ DEF.
Find the length of DF.
•Display Transparency: Similar Sides and have
participants take out their matching pages.
B
150
150
A
120
•Have participants work in pairs.
E
50
50
C
D
x
F
2. Rectangle M is similar to
rectangle N. The ratio of
rectangle N’s width to
rectangle M’s width is 3:2.
Rectangle M has a length of
24 cm and a width of 16 cm.
What is the perimeter of
rectangle N?
M
N
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Transparency: Similar Sides
•Tell participants to use what they have learned about
ratios to find the missing dimensions of the figures on
their pages.
•Allow 5–7 minutes for this activity.
•Call the group together.
•Ask for volunteers to share their processes and their
findings. Write the steps for each solution on the
displayed transparency.
•Refer to Transparency: Similar Sides Answer Key to
resolve any questions.
End of It’s Like This
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5
GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
Race to Place
In this activity, participants will use geometric
knowledge that they remember to match pictures of
angles and shapes with their definitions.
Materials
• Transparency/Page: Race to Place Directions
• Transparency/Page: Triangle Facts Answer Key
• Transparency/Page: Angle Facts Answer Key
• Transparency/Page: Angles in Shapes Answer Key
• Transparency/Page: Line Facts Answer Key
• Transparency/Page: Circle Facts Answer Key
• Race to Place Cards
• 5 pocket charts
• bell
Time:
15 minutes
Teaching Tip: Post the pocket charts with their
titles and the definitions before the beginning of
the activity. Use the Facts transparencies as a guide
for the definitions that go with each title. Space
the charts around the room with a lot of room
between them.
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GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
Introduce
•Suggest to participants that over time they have
accumulated a lot of knowledge about the way lines,
shapes, and angles work.
•Point out the five charts and their definitions.
•Explain to participants that they will compete as
teams to match geometric definitions with pictures
that illustrate the concepts defined.
Teaching Tip: If you have a large group, assign
pairs instead of single people to each card.
DISCUSS AND DO
race to place
Directions
• Distribute your team cards evenly among the
members of your team.
• Have team members play their cards in relay fashion.
• Have a player:
• race to the chart that holds the definition of the
picture on his or her card
•Display Transparency: Race to Place Directions.
•Go over the steps of the game.
•Have participants move into 4 or 5 equal-sized groups.
• place the card next to the definition
• race back to the team and sit down
• Have the next person race to the chart and place his
or her card.
• Have one team member race to the front and ring the
bell when all the team’s cards are correctly placed.
•Distribute the cards—all of one colored shape to each
group, one card per person.
•Call, “Go.”
•Have the first group to finish send one member to the
front of the room to ring the bell.
geometry and measurement of Plane figures—activity set 8
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Transparency: Race to Place Directions
Teaching Tip: If a team member cannot place his
or her card, he or she should go to the end of the
line and wait to place the card after other team
members have placed their cards.
Teaching Tip: If the group is inexperienced,
permit them a few moments to look at the definition
sheets (Answer Keys) before the game.
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GEOMETRY AND MEASUREMENT OF Plane FIGURES
Activity Set 3
Conclude
•Congratulate participants for being able to remember
so many geometry concepts and definitions.
•Display the answer key transparencies in turn, quickly
reviewing the definitions.
•Emphasize the following definitions for each key
(these will be of use in subsequent activities):
triangle Facts
Answer Key
• A scalene triangle has
no congruent sides and
no congruent angles.
◆
• An isosceles triangle has
two congruent sides and
two congruent angles.
• An equilateral triangle has
three congruent sides and
three congruent angles.
◆
• The angles of an acute triangle
are all less than 90˚.
• One angle in an obtuse triangle
is greater than 90˚.
◆
• A right triangle has one angle equal
to 90˚. The side opposite the 90˚
angle is called the hypotenuse.
Transparency: Triangle Facts Answer Key
–equilateral triangle
–right triangle (esp. hypotenuse)
Transparency: Angle Facts Answer Key
–straight angle
–vertical angles
Transparency: Angles in Shapes Answer Key
–triangle
–equilateral triangle
geometry and measurement of Plane figures—activity set 3
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Transparencies: Triangle Facts, Angle Facts,
Angles in Shapes, Line Facts, and Circle
Facts Answer Keys
◆
◆
Transparency: Line Facts Answer Key
–alternate interior angles
Transparency: Circle Facts Answer Key
–circumference
End of Race to Place
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8
It’s Like This
Measure the angles and sides of each figure and
label the diagrams.
B
E
H
Fig. 1
A
Fig. 2
C
D
F
Fig. 3
G
I
1.How are Figures 1 and 2 the same?
How are Figures 1 and 2 different?
2.How are Figures 1 and 3 the same?
How are Figures 1 and 3 different?
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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It’s Like This
Answer Key
Measure the angles and sides of each figure and
label the diagrams.
B
E
H
90º
A
53º
37º
4 cm
3 cm
Fig. 1
53º
C
Fig. 2
90º
D
6 cm
3 cm
53º
37º
4 cm
F
Fig. 3
90º
G
37º
8 cm
I
1.How are Figures 1 and 2 the same?
The angle and side measurements are exactly the same.
How are Figures 1 and 2 different?
They are not different. They are exactly the same.
2.How are Figures 1 and 3 the same?
The angle measurements are exactly the same.
How are Figures 1 and 3 different?
The side measurements are different, but they are in
proportion to each other.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Image Dilation
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Similar Sides
1. ABC ~ DEF.
Find the length of DF.
B
150
150
E
50
50
A
120
C
D
x
F
2.Rectangle M is similar to
rectangle N. The ratio of
rectangle N’s width to
rectangle M’s width is 3:2.
Rectangle M has a length of
24 cm and a width of 16 cm.
What is the perimeter of
rectangle N?
M
N
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Similar Sides
Answer Key
1. ABC ~ DEF.
Find the length of DF.
AC
=
DF
AB
DE
120 = 150
x
50
B
120 • 50 = 150x
150
6,000 = 150x
150
E
50
50
A
120
C
D
x
F
6,000 = x
150
40 = x
2.Rectangle M is similar to
length of N = l
length
of M = 24
rectangle N. The ratio of
2l
rectangle N’s width to
rectangle M’s width is 3:2.
2l
Rectangle M has a length of
l
24 cm and a width of 16 cm.
width of N = w
What is the perimeter of
width of M = 16
rectangle N?
2w
= 32
= 3 • 24
= 72
= 36
= 32
= 3 • 16
2w = 48
w = 24
M
N
P = 2l + 2w
P = 2(36) + 2(24)
P = 72 + 48
P = 120 cm
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Race to Place
Directions
•Distribute your team cards evenly among the
members of your team.
•Have team members play their cards in relay fashion.
•Have a player:
• race to the chart that holds the definition of the picture on his or her card
• place the card next to the definition
• race back to the team and sit down
•Have the next person race to the chart and place his
or her card.
•Have one team member race to the front and ring the
bell when all the team’s cards are correctly placed.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Triangle Facts
Answer Key
• A scalene triangle has
no congruent sides and
no congruent angles.
• An isosceles triangle has
two congruent sides and
two congruent angles.
• An equilateral triangle has
three congruent sides and
three congruent angles.
• The angles of an acute triangle
are all less than 90˚.
• One angle in an obtuse triangle
is greater than 90˚.
• A right triangle has one angle equal
to 90˚. The side opposite the 90˚
angle is called the hypotenuse.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Angle Facts
Answer Key
• The measure of an acute angle is
less than 90˚.
• The measure of an obtuse angle is
greater than 90˚ and less than 180˚.
• The measure of a straight angle is
equal to 180˚.
• The measure of a right angle is
equal to 90˚.
• Angles that share a common side
between them are adjacent.
• Two angles with measures that sum
to 180˚ are called supplementary.
• Nonadjacent angles formed by
two intersecting lines are called
vertical angles. They have the
same measure.
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Angles in Shapes
Answer Key
• A triangle has angles that sum
to 180˚.
• A rectangle has angles that sum
to 360˚.
• Angles inside a shape are
interior angles.
• Exterior angles are angles outside a
shape that are formed by extending
a side of the shape.
• The base angles and opposite
sides of an isosceles triangle are
congruent.
• The sides and angles of an
equilateral triangle are congruent.
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Line Facts
Answer Key
• A set of points, a straight path,
that extends indefinitely in
two opposite directions is a line.
• A line segment is two
endpoints and the straight
path between them.
• Perpendicular lines form
right angles.
• If a line intersects two parallel
lines, the alternate interior
angles are equal.
• Parallel lines are equidistant
from each other.
6 cm
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6 cm
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Circle Facts
Answer Key
• A complete revolution around the center
of a circle has 360º.
• A chord is a line segment that connects
two points on the circumference of a circle.
• The line segment joining the center of the
circle and a point on its circumference is
called a radius.
• A diameter is a chord that passes through
the center of a circle. Its length is twice that
of the radius of the circle.
• A circle is the set of all points in a plane
that are equidistant from a specified point.
• The distance around a circle is called
its circumference.
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Glossary
Geometry and Measurement of Plane Figures
acute angle An angle with a measure less than 90 degrees (°).
angle A geometric figure composed of two raysor line segments that share
the same endpoint,called a vertex.
area The numberof square units in a region.
circle The set of all points in a plane that are the same distance from a
fixedpoint (the center of the circle).
circumference The perimeter of (distance around) a circle. The
circumference can befound using the formula C = 2πr,where C is
the circumference of the circle and r is the radius of the circle.
congruent figures Two figures that haveidentical size and shape so that
when one is placed overthe other,theycoincide exactly.
coordinate pair An ordered pair of numbersthat indicates the position of
a point on a plane. The first numberof a coordinate pair givesthe
point’s location in relation to the x-axis.The second numberin a
coordinate pair givesthe point’s location in relation to the y-axis.
coordinate plane A plane containing an x-axisand a y-axis.Everypoint
on the plane can bedescribedusing a coordinate pair.
degree (°) A unit of measure for angles. 1° is
around a point.
1
360
of a complete revolution
equilateral The propertyof havingequal,or congruent,sides.
equilateral triangle A three-sided polygonwith all sides and with all
angles congruent.
hexagon A six-sidedpolygon.
irregular polygon A polygonin which not all the sides are congruent
and not all the angles havethe same measure.
GEOMETRY AND
AND MEASUREMENT
MEASUREMENT OF
OF PlANE
Plane FIGURES—AcTIvITY
FIGURES—ActivitySET
Set13
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Glossary
(continued)
isosceles triangle A triangle that has two congruent sides and two
congruent angles.
line The set of all contiguous (touching) points that form a straight path
extendingindefinitelyin two directions opposite each other.
line segment A part of a straight line that has two end points and a
fixedlength; a straight line segment marksthe shortest distance
betweentwo points.
linear unit A unit of measure for elements of a single dimension—length.
obtuse angle An angle with a measure greater than 90° and less than 180°.
parallel lines Lines that do not intersect and that are everywhere
equidistant from each other.
parallelogram A quadrilateral in which bothpairs of opposite sides
are parallel.
pentagon A five-sidedpolygon.
perimeter The distance around the outside of a plane shape or figure.
perpendicular At right angles to. Two lines are perpendicular if their
intersection creates right angles.
pi (π) The ratio of the circumference of anycircle to its diameter
(3.141592653 . . .). Pi is usuallyrepresented bythe Greekletter,π.
plane A flat surface that extendsforeverin all directions.
plane figure A figure that lies entirelyin one plane.
point A location in space.
polygon A simple,closed plane shape composed of a minimum of three
straight-line segments.
GEOMETRY AND
AND MEASUREMENT
MEASUREMENT OF
OF PlANE
Plane FIGURES—AcTIvITY
FIGURES—ActivitySET
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Glossary
(continued)
quadrilateral A four-sided polygon.
radius A segment connecting the center of a circle to anypoint on the
circle; the length of the radius.
ray A subsetof a line that includes one endpoint and that extendsinfinitely
from that endpoint in one direction.
rectangle A quadrilateral that includes four interior right angles.
regular polygon A polygonin which all the sides are congruent and all the
angles havethe same measure.
rhombus A parallelogram in which all sides are congruent.
right angle An angle with a measure of 90°.
right triangle A triangle with one right angle.
scalene triangle A triangle in which no sides are congruent and no angles
havethe same measure.
similar figures Figures that havecongruent corresponding angles and in
which corresponding sides are proportional.
square A quadrilateral in which all sides and all angles are congruent.
square unit A unit of measure used to describethe surface (area) of figures
of two dimensions—length and width.
straight angle An angle with a measure of 180°.
trapezoid A quadrilateral in which onlyone pair of sides is parallel.
triangle A three-sided polygon.
vertex (pl. vertices) The intersection point shared bytwo sides of a
polygonor the two sides (rays)of an angle. Also the intersection point
shared bythree or more edges of a polyhedron.
GEOMETRY AND
AND MEASUREMENT
MEASUREMENT OF
OF PlANE
Plane FIGURES—AcTIvITY
FIGURES—ActivitySET
Set13
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Race to Place Cards (1 of 20)
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Race to Place Cards (2 of 20)
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6 cm
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Race to Place Cards (3 of 20)
6 cm
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Race to Place Cards (4 of 20)
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Race to Place Cards (5 of 20)
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Mid_PGe_03_PM
Race to Place Cards (6 of 20)
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Race to Place Cards (7 of 20)
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Race to Place Cards (8 of 20)
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Race to Place Cards (9 of 20)
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Race to Place Cards (10 of 20)
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A chord is a line segment
that connects two points
on the circumference
of a circle.
The line segment joining
the center of the circle and
a point on its circumference
is called a radius.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Race to Place Cards (11 of 20)
A complete revolution
around the center of a
circle has 360º.
Mid_PGe_03_PM
Parallel lines are
equidistant from
each other.
The measure of a
right angle is
equal to 90°.
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Race to Place Cards (12 of 20)
Ifa line intersects two
parallel lines,the
alternate interior angles
are equal.
Mid_PGe_03_PM
A line segment has
two endpoints.
Perpendicular lines
form right angles.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
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Race to Place Cards (13 of 20)
A set of points that extend
indefinitelyin two opposite
directions is a line.
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The base angles and
opposite sides of an
isosceles triangle are
congruent.
The sides and angles of
an equilateral triangle
are congruent.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (14 of 20)
Exterior angles are angles
outside a shape that are
formed by extending a
side of the shape.
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A rectangle has angles
that sum to 360˚.
Angles inside a shape are
interior angles.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (15 of 20)
A triangle has angles
that sum to 180˚.
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Angles that share a
common side between
them are adjacent.
Two angles that sum to
180˚ are called
supplementary.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (16 of 20)
Nonadjacent angles
formed by two intersecting
lines are called vertical
angles. They have the
same measure.
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The measure of an
obtuse angle is greater
than 90˚ and
less than 180˚.
The measure of a
straight angle is
equal to 180˚.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (17 of 20)
The measure of an
acute angle is
less than 90˚.
Mid_PGe_03_PM
Oneangle in an
obtuse triangle is
greater than 90˚.
A right triangle has one
angle equal to 90˚.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (18 of 20)
The angles of an
acute triangle are all
less than 90˚.
Mid_PGe_03_PM
A circle is the set of all
points in a plane that are
equidistant from a
specified point.
The distance around a
circle is called its
circumference.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (19 of 20)
The diameter is a chord
that passes through the
center of a circle.
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An isosceles triangle has
two congruent sides and
two congruent angles.
An equilateral triangle
has three congruent sides
and three congruent angles.
GEOMETRY AND MEASUREMENT OF Plane FIGURES—Activity Set 3
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (20 of 20)
A scalene triangle has
no congruent sides and
no congruent angles.
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