Download Geometry and Measurement of Plane Figures Activity

Geometry and Measurement
of Plane Figures
Activity Set 4
Trainer Guide
GEOMETRY AND MEASUREMENT OF PLANE FIGURES—Activity Set 4 Int_PGe_04_TG
Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
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Amazing Angles
In this activity, participants explore angle concepts
in polygon shapes.
Materials
• Transparency/Page: Angle Types
• Transparency/Page: Measuring Angles
• Transparency/Page: A Circle of Measure
• Transparency/Page: Polygon Angles Chart
• Transparency/Page: Polygon Angles Chart Answer Key
• plain 3  5 cards (4 per participant)
• ruler for each participant
• protractor for each participant
• scissors for each participant
• pens/pencils (multicolor pens)
• blank transparency
Vocabulary
• degree
• angle
• vertex
• right angle
• straight angle
Time:
30 minutes
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GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
Introduce
angle types
interior
exterior
adjacent
A
A
A
C
C
C
B
B
•Remind participants that one aspect of geometry is the
application of angles to various shapes and figures.
B
D
interior angle
an angle formed by two sides
of a polygon
adjacent angles angles that share a common
side and a common vertex
between them, but that do
not share any interior points
exterior angle
an angle adjacent to, but
outside of, a polygon–formed
by extending one side of the
polygon
GEOMETRY AND MEASUREMENT OF PLANE FIGURES/ 31
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Transparency: Angle Types
Measuring angles
1
2
angle 1
angle 3
Teaching Tip: It may help to clarify the definitions
if you explain the meaning of adjacent—having a
common side or border and, in mathematics, a
common endpoint.
•Display Transparency: Measuring Angles and have
participants take out their matching pages.
•Demonstrate on angle 1 how to measure an angle.
◆
4
angle 4
•Go over the angle descriptions and names.
•Take out a protractor.
3
angle 2
•Display Transparency: Angle Types.
5
angle 5
acute angle—an angle less than 90º
obtuse angle—an angle more than 90º
right angle—an angle equal to 90º
straight angle—an angle of 180º
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Transparency: Measuring Angles
◆
Align the 0˚ mark and line with the right-hand side
of the angle, making sure that the vertex of the
angle is aligned with the center mark of the 0˚ line.
(There is usually a small hole at this location to
enable you to place the vertex appropriately.)
Locate the left-hand side of the angle and trace the
line to the degree mark on the protractor.
•Have participants measure the remaining angles and
write the degrees that they find in the appropriate
blanks on their pages.
•Go over the answers with the participants and
demonstrate the measurement process, if necessary, to
address any questions.
•Review the definitions at the bottom of the page.
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GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
Teaching Tip: If time permits, have participant
volunteers come to the front to measure the angles
and record the results on the transparency.
Teaching Tip: If the group is advanced, have them
also identify the angle type after they measure.
• 1—acute angle
• 2—acute angle and adjacent angle (adjacent to
angle 3)
• 3—obtuse angle and adjacent angle (adjacent to
angle 2)
• 4—straight angle (straight line)
• 5—right angle (formed by perpendicular lines)
Ask why none of the angles are interior or exterior.
(They are not part of, or adjacent to, polygons.)
a circle of Measure
•Ask participants how many degrees are around the
center of a circle.
•Display Transparency: A Circle of Measure.
•Point out to participants that the distance around
the center of the circle (360˚) is the basis for all angle
measure.
It is a mathematics convention that the unit of
1 of a complete
angle measure (degree) is 360
revolution around the center of a circle.
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Transparency: A Circle of Measure
•Point out on the transparency that the diameter of
the circle (a straight line) divides the revolution in
half, creating a straight angle, or 180˚.
•Explain to participants that they will use this
information to help them find the number of degrees
in the interior angles of a triangle.
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GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
Discuss and Do
•Distribute to each participant four 3  5 cards and a
pair of scissors.
•Have each participant draw on one card a large
triangle.
Teaching Tip: Have participants use a
straightedge, ruler, or card side to draw all figures.
Straightedges are required to achieve accuracy for
the activity. Also, no shape that they create can
have overlapping edges.
•Have participants cut out their triangles.
•Have participants use a pen or pencil to color in the
angles about 21 out from each vertex.
•Have them cut the triangle into 3 pieces, with each
piece containing 1 angle.
•Tell them to lay the 3 angles together with the
vertices joining and their sides touching.
•Point out that they now have a straight line or a
straight angle, which is defined as 180˚.
•Point out that they all made different kinds of
triangles.
•Explain that the angles of all triangles sum to 180˚.
•Have participants use their rulers to draw another
triangle on one of their 3  5 cards.
•Have them make the triangles as large as possible for
ease of measurement.
•Tell them to measure each angle in their triangles,
using their protractors, and add the three angles
together.
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GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
•Display a blank transparency.
•Have various volunteer participants share the angle
measures within their triangles.
•Record on the blank transparency the angle measures
as they are shared.
•Point out that the angles differed individually,
but that the sum of the angles for any triangle
was always 180˚.
•Have participants take out their third cards.
•Point out that the card is a rectangle.
•Ask them to color the 4 corners and cut the card into
4 pieces (1 corner to each piece).
•Ask them to arrange the corners together and tell
you how many degrees there are in the angles of
a rectangle. (360˚)
•Explain that any quadrilateral has angles that sum
to 360˚.
•Draw a rectangle on a blank transparency.
•Draw a diagonal from one corner of the rectangle to
the corner opposite.
•Point out that the angles of the 2 triangles thus
formed also sum to 360˚.
•Have participants draw a 5-sided figure on their
last cards.
•Have them color the corner angles, cut out the shape,
and then cut it into 5 pieces—1 angle per piece.
•Ask participants to lay the angles together in such a
way that they can tell you the total number of
degrees.
•Suggest that, when participants complain that they
cannot match all the angles, they create more than
one figure.
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GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
•Ask participants how many degrees there are in the
angles of a pentagon. (540˚)
•Have one participant come up with his or her shapes
and illustrate on the overhead projector his or her
solution.
•Draw a 5-sided polygon on a blank transparency.
•Draw, from 1 vertex, lines to all opposing vertices for
which you can make triangles.
polygon angles chart
Polygon
Name
Number of
Sides
Number of Sides
Minus 2
Number of Degrees:
triangle
quadrilateral
pentagon
hexagon
•Point out that the angles of the 3 triangles thus
formed also sum to 540˚.
heptagon
octagon
nonagon
decagon
GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4
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Conclude
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Transparency: Polygon Angles Chart
•Display Transparency: Polygon Angles Chart and have
participants take out their matching pages.
•Fill in, along with the participants, the first three
rows of the Polygon Angles Chart using information
that they have collected during this activity.
•Encourage participants to create triangles of each
shape to help them.
•Ask participants the number of degrees that they
think the angles of a hexagon would total. (720˚)
•Complete the hexagon row on the chart.
•Ask participants if they recognize a pattern.
(The rule is (n – 2) • 180˚.)
•Ask participants how this rule is derived.
(n, the number of sides, less 2 is the number
of non-overlapping triangles in each shape.)
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GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
•Write the rule on the transparency in the fourth
column heading.
polygon angles chart
Answer Key
Polygon
Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
Number of
Sides
n
Number of Sides
Minus 2
n–2
Number of Degrees:
(n – 2) • 180°
3
1
180°
4
2
360°
5
3
540°
6
4
720°
7
5
900°
8
6
1,080°
9
7
1,260°
10
8
1,440°
GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4
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Transparency: Polygon Angles Chart
Answer Key
•Go down to the last figures on the chart.
•Ask participants for the number of degrees at
each row.
•Fill in the transparency at each step.
•Refer to Transparency: Polygon Angles Chart Answer
Key, as necessary.
End of Amazing Angles
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GEOMETRY AND MEASUREMENT OF PLANE FIGURES
Activity Set #4
Race to Place
In this activity, participants 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
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 #4
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 the 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
• 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.
•Display Transparency: Race to Place Directions.
•Go over the steps of the game.
•Have participants move into 4 or 5 equal-sized groups.
•Distribute the shape cards—all of one colored shape
to each group, one card per person.
•Call, “Go.”
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•Have the first group to finish send one member to the
front of the room to ring the bell.
Transparency: Race to Place Directions
Teaching Tip: If a team member cannot place his
or her shape 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 #4
Conclude
•Congratulate the 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:
triangle Facts
Answer Key
◆
• A scalene triangle has
no congruent sides and
no congruent angles.
• An isosceles triangle has
2 congruent sides and
2 congruent angles.
◆
• An equilateral triangle has
3 congruent sides and
3 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.
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Transparencies: Triangle Facts Answer Key,
Angle Facts Answer Key, Angles in Shapes
Answer Key, Line Facts Answer Key, Circle
Facts Answer Key
◆
◆
Transparency: Triangle Facts Answer Key
–equilateral triangle
–right triangle (especially hypotenuse)
Transparency: Angle Facts Answer Key
–straight angle
–vertical angles
Transparency: Angles in Shapes Answer Key
–triangle
–equilateral triangle
Transparency: Line Facts Answer Key
–alternate interior angles
Transparency: Circle Facts Answer Key
–circumference
End of Race to Place
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10
Angle Types
interior
exterior
adjacent
A
A
A
C
C
C
B
B
B
D
interior angle
an angle formed by two sides
of a polygon
adjacent angles angles that share a common
side and a common vertex
between them, but that do
not share any interior points
exterior angle
an angle adjacent to, but
outside of, a polygon—
formed by extending one
side of the polygon
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Measuring Angles
1
2
angle 1
3
angle 2
angle 3
4
angle 4
5
angle 5
acute angle—an angle less than 90º
obtuse angle—an angle more than 90º
right angle—an angle equal to 90º
straight angle—an angle of 180º
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A Circle of Measure
It is a mathematics convention that the unit of
1 of a complete
angle measure (degree) is 360
revolution around the center of a circle.
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Polygon Angles Chart
Polygon
Name
Number of
Sides
Number of Sides
Minus 2
Number of Degrees:
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
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Polygon Angles Chart
Answer Key
Polygon
Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
Number of
Sides
n
Number of Sides
Minus 2
n–2
Number of Degrees:
(n – 2) • 180°
3
1
180°
4
2
360°
5
3
540°
6
4
720°
7
5
900°
8
6
1,080°
9
7
1,260°
10
8
1,440°
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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.
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Triangle Facts
Answer Key
• A scalene triangle has no congruent sides and no congruent angles.
• An isosceles triangle has 2 congruent sides and 2 congruent angles.
• An equilateral triangle has 3 congruent sides and 3 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.
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Angle Facts
Answer Key
• An acute angle is less than 90˚.
• An obtuse angle is greater than 90˚ and less than 180˚.
• A straight angle is equal to 180˚.
• A right angle is equal to 90˚.
• Angles that share a common side between them are adjacent.
• Two angles 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.
• Angles outside a shape are exterior angles.
• 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 2 opposite directions is a line.
• A line segment is 2 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
GEOMETRY AND MEASUREMENT OF PLANE FIGURES—Activity Set 4 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
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
angle
Geometric figure made of 2 rays or 2 line segments
that share the same endpoint, called a vertex.
area
The number of square units in a region.
congruent
Having the same shape, size, and/or measure.
degree
A unit for measuring angles.
irregular polygon
A polygon in which not all the sides are congruent
and/or not all the angles have the same measure.
line
A set of points forming a straight path in 2 directions that are opposite each other.
perimeter
The distance around the outside of a shape or
figure.
plane
A flat surface that extends forever in all directions.
point
A location in space.
polygon
A closed shape made up of a minimum of 3 line
segments.
quadrilateral
A polygon with 4 sides.
rectangle
A quadrilateral with 4 right angles.
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Glossary
(continued)
regular polygon
A polygon in which all the sides are congruent and
all the angles have the same measure.
triangle
A polygon with 3 sides.
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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)
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Race to Place Cards (6 of 20)
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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 4 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (11 of 20)
A complete revolution
around the center of a
circle has 360º.
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Parallel linesare
equidistantfrom
eachother.
Aright angle is
equalto90°.
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Race to Place Cards (12 of 20)
Ifalineintersectstwo
parallellines,the
alternate interior angles
areequal.
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Aline segment has
twoendpoints.
Perpendicular lines
formrightangles.
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Race to Place Cards (13 of 20)
Asetofpointsthatextend
indefinitelyin2opposite
directionsisaline.
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Thebaseanglesand
oppositesidesofan
isosceles triangleare
congruent.
Thesidesandanglesof
anequilateral triangle
arecongruent.
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Race to Place Cards (14 of 20)
Anglesoutsideashape
areexterior angles.
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Arectangle hasangles
thatsumto360˚.
Anglesinsideashapeare
interior angles.
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Race to Place Cards (15 of 20)
Atriangle hasangles
thatsumto180˚.
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Angles that share a
common side between
them are adjacent.
Two angles that sum to
180˚ are called
supplementary.
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Race to Place Cards (16 of 20)
Non adjacent angles
formed by two intersecting
lines are called vertical
angles. They have the
same measure.
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Anobtuse angle is
greaterthan90˚and
lessthan180˚.
Astraight angle is
equalto180˚.
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Race to Place Cards (17 of 20)
Anacute angle is
lessthan90˚.
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Oneangleinan
obtuse triangle is
greaterthan90˚.
Aright triangle hasone
angleequalto90˚.
GEOMETRY AND MEASUREMENT OF PLANE FIGURES—Activity Set 4 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (18 of 20)
Theanglesofan
acute triangle areall
lessthan90˚.
Int_PGe_04_PM
Acircle isthesetofall
pointsinaplanethatare
equidistantfroma
specifiedpoint.
Thedistancearounda
circleiscalledits
circumference.
GEOMETRY AND MEASUREMENT OF PLANE FIGURES—Activity Set 4 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (19 of 20)
Thediameterisachord
thatpassesthroughthe
centerofacircle.
Int_PGe_04_PM
Anisosceles triangle
has2congruentsides
and2congruentangles.
Anequilateral triangle
has3congruentsides
and3congruentangles.
GEOMETRY AND MEASUREMENT OF PLANE FIGURES—Activity Set 4 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development
Race to Place Cards (20 of 20)
Ascalene triangle has
nocongruentsidesand
nocongruentangles.
Int_PGe_04_PM