Download Grade Six - Session 3 Applying Basic Properties of Lines, Angles

Mathematics – Grade Level Assessments and Content Expectations
Grade Six - Session 3
Applying Basic Properties of Lines, Angles, and Triangles
Participant Packet
Modules Developed by Macomb County Teachers
under the leadership of Marie Copeland
Developers
Mathematics – Grade Level Assessments and Content Expectations
M-GLAnCE Project Directors
Debbie Ferry
Macomb ISD
Mathematics Consultant
Carol Nowakowski
Retired
Mathematics Consultant
K-4 Project Coordinator
Marie Copeland
Warren Consolidated
Macomb MSTC
5-8 Project Coordinator
2004 Project Contributors
David Andrews
Chippewa Valley Schools
William Ashton
Fraser Public Schools
Lynn Bieszki
Chippewa Valley Schools
Sharon Chriss
Romeo Schools
Kimberly DeShon
Anchor Bay School District
Barbara Diliegghio
Retired, Math Consultant
Kimberly Dolan
Anchor Bay School District
Jodi Giraud
L’Anse Creuse Schools
Julie Hessell
Romeo Schools
Amy Holloway
Clintondale Schools
Barbara Lipinski
Anchor Bay School District
Linda Mayle
Romeo Schools
Therese Miekstyn
Chippewa Valley Schools
James Navetta
Chippewa Valley Schools
Gene Ogden
Anchor Bay School District
Rebecca Phillion
Richmond Comm. Schools
Charlene Pitrucelle
Anchor Bay School District
Shirley Starman
Van Dyke Public Schools
Ronald Studley
Anchor Bay School District
2005 and 2006 Session/Module Developers
Carol Nowakowski
Retired, Math Consultant
Deb Barnett
Luann Murray
Lake Shore Public Schools Genesee ISD
Kathy Albrecht
Retired, Math Consultant
Jo-Anne Schimmelpfenneg
Terri Faitel
Trenton Public Schools
Debbie Ferry
Macomb ISD
Retired, Math Consultant
Marie Copeland
Warren Consolidated
Grade 6 – Session #3 Applying Basic Properties of Lines, Angles, and Triangles
G.GS.06.01
G.SR.06.05
Understand and apply basic properties of lines,
angles, and triangles, including:
Triangle inequality
Relationships of vertical angles,
complementary angles, supplementary angles
Congruence of corresponding and alternate
interior angles when parallel lines are cut by a
transversal, and that such congruencies imply
parallel lines
Locate interior and exterior angles of any
triangle, and use the property that an exterior
angle of a triangle is equal to the sum of the
remote (opposite) interior angles
Know that the sum of the exterior angles of a
convex polygon is 360 degrees
Use paper folding to perform basic geometric
constructions of perpendicular lines, midpoints
of line segments and angle bisectors; justify
informally.
Instructional Sequence:
Develop basic terminology
of geometry
Triangle Inequality
Paper folding
constructions
Develop intersecting,
perpendicular, and parallel
lines
Parallel lines conjecture
Develop angle relationships
(eg. linear pair, adjacent, etc.)
Relationship between interior
and exterior angles of a
Sum of the measures of the
angles of a triangle
Sum of the exterior angles of a
convex polygon is 360 degrees
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Important Tips
•
•
•
•
•
•
•
•
•
Have student’s measure angles with a protractor as often as possible.
To help students understand which numbers to read on a protractor, make the relationship to
right, acute, and obtuse angles.
When students are given examples of intersecting lines, make sure that they are exposed to
situations where the intersecting part is not visible. In other words, the lines have to be extended
to see the intersecting portion.
Have the students use some sort of graphic organizer to help them with all the terms they are
learning.
Vary the experiences students have by sometimes using rulers and protractors, patty paper, and
the Geometer’s Sketchpad. Keep in mind, even if the students are using the Geometer’s
Sketchpad, which can perform the measurements, they can still be required to print a copy of
their work and measure the figures with a ruler and protractor.
When students are exploring the sum of the angles of a triangle, ask them questions such as, if
you have an obtuse triangle, give me a description of possible angle measures of the three
angles.
When investigating the parallel line conjecture, do not always present the situation in the typical
fashion. Example:
Sometimes when students are using manipulatives with investigating the triangle inequality, they
try to bend the materials to force a triangle.
Corresponding angles cause difficulties for students. Relate corresponding angles to
translations.
Common Misconceptions:
•
•
•
•
•
•
When students extend the sides of polygons to investigate exterior angles, they make inaccurate
extensions.
When investigating triangles inequality situations, students get confused when it is an equality
situation which forms a straight line.
Students think that if their angles in a triangle add up to 177 degrees, it is close enough to 180
degrees. They do not realize that somewhere they made a mistake in measuring and they have
to remeasure the angles.
Students confuse the vocabulary.
Students think if the rays of an angle extend further than another angle, that is what makes the
angle measure larger
Students do not realize that an angle of a triangle extends beyond the triangle
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Grade 6: Participant – Applying Basic Properties of Lines, Angles, and Triangles
Name of Activity
Description of Activity
Materials/Transparencies/
Handouts
Vocabulary Building
• Students will use Twizzlers
• Twizzlers
and dot paper to begin the
• Colored Pencils
development of the
• Magnet Tape
vocabulary. For example,
• Dot Paper
they will use Twizzlers to
• Scissors
build a convex and non
• Rulers
convex polygon, etc. They
• Lab Packet pp 5 – 11
will also make graphic
organizers to benefit their
understanding.
Sum of the angles of a triangle • Students will make triangles
• 2 pieces of paper
and rip off the angles to
• Rulers
discover the sum of the
• Scissors
angles of a triangle.
• Colored Pencils
Connections to exterior
• Lab Packet pp 12 - 14
angles will also be made.
Patty paper investigations of
special angles
• Students will investigate
linear pairs, adjacent angles,
and the angles formed by 2
parallel lines cut by a
transversal
Euclidean Constructions
• Students will perform the 7
• Lab packet pp 21 – 24
basic Euclidean constructions • Rulers
• Lab packet pp 15– 20
• Patty paper
• Rulers
• Mechanical pencils
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
Key Tips for
The Teacher
• Allow some time for students
to work with a partner or
group.
• Have magnet tape available
so students can put their
finished products on the board
for group discussion.
• Have students make assorted
triangles that are large so they
will be easier to work with but
still allows for an extension of
the exterior angle. They will
also duplicate their triangle.
• Go around the room holding
up the various triangles to
bring out the terms such as
obtuse triangle, acute triangle
and right triangle.
• When students cut their
parallel lines by a transversal,
make sure to tell them not to
make the transversal
perpendicular.
• When tracing angles, no free
hand tracing. Use rulers.
• Keep reinforcing that even
though some of these
3
Grade 6: Participant – Applying Basic Properties of Lines, Angles, and Triangles
Name of Activity
Description of Activity
Materials/Transparencies/
Handouts
with paper folding.
• Patty paper
• Mechanical pencils
Triangle Inequality
• Students test different lengths • Commercial products for this
of line segments to determine
activity work best however,
which ones will form a
straws or line segments
triangle and why some work
made out of card stock are
and others do not work.
also suitable
• Fasteners
• Lab packet pp 25 – 33
Sum of exterior angles of a
polygon
• Students will use assorted
regular polygons and
measure interior, exterior,
and central angles to
investigate the relationships.
Also investigate symmetry in
the regular polygons.
• Lab packet pp 34 - 40
• Protractors
• Rulers
• Miras
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
Key Tips for
The Teacher
constructions seem too
simple, the reason is because
paper folding is being used to
introduce Euclidean
constructions that only use a
straightedge and compass.
• Make sure students build the
triangles but be careful so they
do not “force” a triangle,
especially when the
measurements are close.
• The majority of the work in the
investigation portion will be
spent with regular polygons,
however, also discuss nonregular polygons and nonconvex polygons.
4
Basic Ideas of Geometry
1. Points are labeled with capital letters
B
Point A, Point B, Point C, Point D
C
A
D
2. A line segment shows a part of a line with two distinct end points: written EF or FE
E
F
3. Since a line extends to infinity in each direction, a line has arrows going in opposite
directions to symbolize the extension. Lines are labeled in two different ways.
I. One method of labeling a line is to use a small letter. It is written as line j
j
II. Another way to name a line is to use any two points that fall on the line.
H
G
This line is written either GH or HG
4. A ray is a part of angle. A ray contains an endpoint at one end and an arrow at the
other end. The arrow symbolizes that the ray extends to infinity. It does not indicate the
direction the arrow is going. The ray contains not only an endpoint, but also a point
somewhere on the ray. The ray below is called ray QR and is written, QR . Notice that
the endpoint always comes first when naming a ray.
Q
R
5. An angle contains two rays with a common endpoint called the vertex.
K
The angle is written ∠KLM or ∠MLK
Notice since L is the vertex, it is written in the middle
L
M
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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6. An angle that measures 90 degrees is a right angle
∠NLM is a right angle
N
N
90 °
L
L
M
M
In a drawing, a right angle is indicated with a square
7. Since ∠NLM is a right angle
N
90 °
L
M
the rays LN and LM are perpendicular
The symbol for perpendicular is ⊥
So we can say LN
⊥
LM (ray LN is perpendicular to ray LM)
8. An angle which measures between 0º and 90º is an acute angle.
9. An angle which measures between 90º and 180º is an obtuse angle.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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10. When you compare the sizes (lengths or areas) of geometric figures the measures
are equal or not equal.
A
B
C
G
The length of segment AB is equal to
the length of segment BC (AB = BC)
The measure of ∠ DEG is equal to
the measure of ∠ GEF
(m ∠ DEG = m ∠ GEF)
D
F
E
11. When you compare the shapes of geometric figures they are congruent or not
congruent. (congruent is to have the same measure) Symbol ( ≅ )
A
B
C
G
D
Line segment AB is congruent line
segment BC ( AB ≅ BC )
∠ DEG is congruent to ∠ GEF
( ∠ DEG ≅ ∠ GEF)
F
E
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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12. Bisect is to divide into to equal parts
D
A
B
C
BD bisects AC because AB = BC
G
D
F
EG bisects ∠DEF because ∠ DEG ≅ ∠ GEF
E
13. Parallel lines are lines on the same surface that never intersect.
The symbol for parallel is
OQ
PR (line OQ is parallel to line PR)
Q
O
P
R
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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14. Two angles who share a common vertex and a common ray but no common interior
points are called adjacent angles.
B
A
C
D
3
∠ADB and ∠BDC are adjacent angles
4
∠3 and ∠4 are adjacent angles
15. Two adjacent angles whose sum is 180 degrees form a linear pair.
B
A
D
C
∠ADB and ∠BDC form a linear pair
16. Two non adjacent angles formed by intersecting lines are vertical angles.
3
1
2
4
∠1 and ∠2 are vertical angles
∠3 and ∠4 are vertical angles
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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17. Two angles whose measures add up to 90 degrees are complementary angles.
1
30°
2
60°
A
∠1 and ∠2 are complementary angles
B
∠A and ∠B are complementary angles
because their sum is 90º
because their sum is 90º
18. Two angles whose measures add up to 180 degrees are supplementary angles
B
150 °
A
30 °
100 °
80°
C
D
R
∠ADB and ∠BDC are supplementary angles
because their sum is 180º
S
∠R and ∠S are supplementary angles
because their sum is 180º
19. Two polygons are congruent if they have all of their corresponding angles and sides
congruent. The order of the letters in the statement of congruence indicates which
segments and angles are corresponding and congruent.
C
D
F
A
E
B
If Δ ABC ≅ Δ FED
Then:
∠A ≅ ∠F
∠B ≅ ∠E
∠C ≅ ∠D
AB ≅ FE
BC ≅ ED
AC ≅ FD
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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20. An exterior angle of a triangle forms a linear pair with an interior angle of a triangle
A
A
F
C
D
C
D
B
When ∠CDB is an exterior angle of ΔACD
∠C and ∠A are the remote interior angles
When ∠ADF is an exterior angle of ΔACD
∠C and ∠A are the remote interior
angles
H
G
A
A
D
C
D
C
When ∠GAD is an exterior angle of ΔACD
∠C and ∠D are the remote interior angles
When ∠HAC is an exterior angle of ΔACD
∠C and ∠D are the remote interior
angles
A
A
C
D
D
K
C
M
When ∠KCA is an exterior angle of ΔACD
∠A and ∠D are the remote interior angles
When ∠MCD is an exterior angle of ΔACD
∠A and ∠D are the remote interior angles
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Lab – Sum of the Measures of the Angles of a Triangle
1 – Distribute paper, rulers and scissors.
2 – Have everyone make two large congruent triangles of all sizes and shapes on the paper using a
ruler.
3 – Have the students shade in the three interiors of the interior angles using the same three colors for
both triangles
4 – Have the students cut one of the triangles out
5 – Go around the room holding up the various triangles so everyone can see the variety of shapes and
sizes
6 – Have the students rip off the shaded regions
7. Have the students place the three regions as adjacent angles along a ruler
8. Ask students: What is the sum of the measures of the three angles?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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9. Take the second triangle and extend one side to form a linear pair. An exterior angle forms a linear
pair with an interior angle
10. How is the exterior angle related to the other two angles of the triangle?
11. Repeat steps 9 and 10 for the other two sides.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Exterior Angles in Triangles
In any triangle, each exterior angle, is equal to the sum of the two nonadjacent
interior angles.
Let a, b, and c represent the measures of
the three interior angles of any triangle. Let
d represent the measure of an exterior angle
that forms a linear pair with the interior angle
of measure a.
d
a
Prove: d = b + c
b
c
1) a + b + c = 180 degrees
1) The sum of the 3 interior angles of any triangle is
180 degrees
2)
2) Angles that form linear pairs are supplementary,
therefore their sum is 180 degrees
a + d = 180 degrees
3) a + b + c = a + d
3) Substitution of equal quantities (Steps 1 and 2)
4)
4) Reflexive
5)
-a = -a
b+c=d
5) Addition of equalities (Steps 3 and 4)
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Looking at Special Angles using Patty Paper
I. Vertical Angles
Definition: The pairs of opposite angles formed by two intersection lines are called vertical angles. For
example, in the diagram, ∠ 1 and ∠ 3 are a pair of vertical angles, and ∠ 2 and ∠ 4 are a pair of vertical
angles.
Investigation:
Step 1: Fold a line segment on a patty paper. Unfold and fold a second line segment intersecting
the first line segment
Step 2: Label the angles as in the diagram above
Step 3: Place a second patty paper over the first and copy one angle of a pair of vertical angles.
Rotate the copy to see how well it fits over the second angle of the vertical angle pair. Repeat
with the other vertical angle pair.
Write a conjecture about the relationship between a pair of vertical angles.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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II. Adjacent Angles and Linear Pairs
Definition: Two angles are called adjacent angles if they share a common vertex and a common
side but no common interior pints.
In the diagram ∠ ABD is adjacent to ∠ DBC
A
D
C
B
Definition: A pair of adjacent angles formed by two intersecting lines is called a linear pair. For
example ∠ 1 and ∠ 2 are a linear pair of angles and ∠ 3 and ∠ 4 are a linear pair of angles.
Investigation:
Step 1: Fold a line segment on a patty paper. Unfold and fold a second line segment intersecting
the first line segment
Step 2: Label the angles as in the diagram above
Step 3: Use your protractor to find the sum of the angles of each linear pair of angles. How
many pairs are there?
Write a conjecture about the relationship between a pair of vertical angles.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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III. Parallel Lines Investigation of Angles
Step 1: Construct a pair of parallel lines and use a ruler and straight edge to highlight them
Step 2: Draw a third line called a transversal that intersects the parallel lines and is not
perpendicular to the parallel lines and label the angles as shown.
Step 3: Use a second piece of patty paper and trace ∠ 1 with a ruler. What angles have the
same measure?
Step 4: What angle is congruent with ∠ 1 after a translation of ∠ 1 ?
Step 5: What angle is congruent with ∠ 1 after a translation of ∠ 1 and a 180 degree rotation of
∠1 ?
Step 6: Using the second piece of patty paper, trace ∠ 3 with a ruler. What angles have the
same measure?
Step 7: What angle is congruent with ∠ 3 after a translation of ∠ 3 ?
Step 8: What angle is congruent with ∠ 3 after a translation of ∠ 3 and a 180 degree rotation of
∠3 ?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Write a definition for the following:
A. alternate interior angles
B. alternate exterior angles
C. corresponding angles.
Identify as many as are pictured above
Name of Angles
Numbered Pairs of Angles
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Finding missing Angles
m ∠ 1 = _______
m ∠ 2 = _______
m ∠ 3 = _______
°
m ∠ 4 = _______
1.
m ∠ 1 = _______
m ∠ 2 = _______
m ∠ 3 = _______
m ∠ 4 = _______
°
°
m ∠ 5 = _______
2.
°
m ∠ 1 = _______
m ∠ 2 = _______
m ∠ 3 = _______
m ∠ 4 = _______
3.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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4.
4
8 3
135 °
1
9
12
11
10
2
13
14
6
5
7
m ∠ 1 = _______
m ∠ 8 = _______
m ∠ 2 = _______
m ∠ 9 = _______
m ∠ 3 = _______
m ∠ 10 = _______
m ∠ 4 = _______
m ∠ 11 = _______
m ∠ 5 = _______
m ∠ 12 = _______
m ∠ 6 = _______
m ∠ 13 = _______
m ∠ 7 = _______
m ∠ 14 = _______
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
20
Euclidean Constructions by Paper Folding
Below are the seven basic constructions used to make geometric drawings. Euclid was the first to
organize geometry into a logical structure in his book the Elements.
1. To duplicate a given segment
2. To duplicate a given angle
→
→
3. To construct the bisector or an angle
4. To construct a perpendicular
from a point to a line
5. To construct a perpendicular through
a point on a line
6. To construct a perpendicular
bisector of a given segment
7. To construct a line through a given point parallel to a given line.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
21
Each of the seven constructions using patty paper.
1. To duplicate a given segment
Trace the segment onto a new piece of patty paper.
2. To duplicate a given angle
Trace the angle onto a new piece of patty paper
3. To construct the bisector or an angle
I. Use your straightedge to draw an angle on a piece of patty paper
II. Fold one of the rays of the angle onto the other ray starting at the vertex
III. Open up the paper and make the ray from the vertex along the crease darker by tracing it with a ruler
and pencil.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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4. To construct a perpendicular from a point to a line
I. Fold a line segment on a patty paper from one side of the paper to the other. Use a ruler and pencil
to draw in the line segment. Put a point on the patty paper that is not on your line segment.
II. Fold the line segment on top of itself so that the fold contains the given point. Unfold the paper and
draw a line through the crease with a ruler and pencil.
5. To construct a perpendicular through a point on a line
I. Fold a line segment on a patty paper from one side of the paper to the other. Use a ruler and pencil
to draw in the line segment. Put a point on the patty paper that is on your line segment.
II. Fold the line segment on top of itself so that the fold contains the given point. Unfold the paper and
draw a line through the crease with a ruler and pencil.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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6. To construct a perpendicular bisector of a given segment
I. Use a straightedge to draw a line segment on a piece of patty paper and put the endpoints on the
ends as shown below.
II. Fold the paper so that one endpoint lies on top of the other end point. Unfold the paper and draw in
the line segment
7. To construct a line through a given point parallel to a given line.
I. Fold a line segment on a patty paper from one side of the paper to the other. Use a ruler and pencil to
draw in the line segment. Put a point on the patty paper that is not on your line segment.
II. Fold the line segment on top of itself so that the fold contains the given point. Unfold the paper and
draw a line through the crease with a ruler and pencil.
III. Fold the new line onto itself through the given point. Unfold the paper and draw a line through the
crease with a ruler and pencil.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
24
Are These Triangles?
Group #1
Given three possible measurements for the side of a triangle, use the bars and connectors to find out if
they do make a triangle.
Case
Length(cm)
Length(cm)
Length(cm)
1
31
22
11
2
15.5
7.75
20
3
31
11
13.5
4
20
11
27
5
7.75
17
27
6
17
13.5
27
7
31
15.5
17
8
31
7.75
20
9
15.5
11
13.5
10
7.75
29
13.5
11
22
20
17
12
31
13.5
27
Is it a Triangle?
What conjecture can you make about the relationship between the sides of a triangle that would
guarantee you that your three lengths always made a triangle?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
25
Are These Triangles?
Group #2
Given three possible measurements for the side of a triangle, use the bars and connectors to find out if
they do make a triangle.
Case
Length(cm)
Length(cm)
Length(cm)
1
22
20
11
2
31
17
13.5
3
15.5
22
7.75
4
31
20
11
5
20
13.5
27
6
7.75
17
13.5
7
15.5
20
27
8
31
15.5
22
9
7.75
13.5
27
10
31
20
13.5
11
22
17
27
12
15.5
22
13.5
Is it a Triangle?
What conjecture can you make about the relationship between the sides of a triangle that would
guarantee you that your three lengths always made a triangle?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
26
Are These Triangles?
Group #3
Given three possible measurements for the side of a triangle, use the bars and connectors to find out if
they do make a triangle.
Case
Length(cm)
Length(cm)
Length(cm)
1
15.5
20
11
2
22
17
11
3
31
22
7.75
4
31
22
27
5
20
17
11
6
31
7.75
13.5
7
15.5
22
17
8
7.75
11
13.5
9
22
7.75
11
10
31
17
27
11
17
11
27
12
22
7.75
20
Is it a Triangle?
What conjecture can you make about the relationship between the sides of a triangle that would
guarantee you that your three lengths always made a triangle?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
27
Are These Triangles?
Group #4
Given three possible measurements for the side of a triangle, use the bars and connectors to find out if
they do make a triangle.
Case
Length(cm)
Length(cm)
Length(cm)
1
31
15.5
27
2
22
11
13.5
3
7.75
20
17
4
11
13.5
27
5
31
11
27
6
15.5
7.75
13.5
7
22
7.75
13.5
8
20
11
13.5
9
15.5
20
13.5
10
31
17
11
11
31
15.5
7.75
12
22
11
27
Is it a Triangle?
What conjecture can you make about the relationship between the sides of a triangle that would
guarantee you that your three lengths always made a triangle?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
28
Are These Triangles?
Group #5
Given three possible measurements for the side of a triangle, use the bars and connectors to find out if
they do make a triangle.
Case
Length(cm)
Length(cm)
Length(cm)
1
22
20
13.5
2
31
22
20
3
20
17
27
4
31
20
17
5
22
13.5
27
6
15.5
7.75
17
7
31
15.5
11
8
15.5
7.75
17
9
15.5
13.5
27
10
7.75
20
27
11
20
17
13.5
12
22
7.75
13.5
Is it a Triangle?
What conjecture can you make about the relationship between the sides of a triangle that would
guarantee you that your three lengths always made a triangle?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Are These Triangles?
Group #6
Given three possible measurements for the side of a triangle, use the bars and connectors to find out if
they do make a triangle.
Case
Length(cm)
Length(cm)
Length(cm)
1
7.75
11
27
2
15.5
20
17
3
31
15.5
20
4
15.5
22
11
5
31
7.75
11
6
15.5
22
20
7
22
20
27
8
17
11
13.5
9
7.75
17
11
10
22
17
13.5
11
15.5
17
27
12
15.5
11
27
Is it a Triangle?
What conjecture can you make about the relationship between the sides of a triangle that would
guarantee you that your three lengths always made a triangle?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Are These Triangles?
Group #7
Given three possible measurements for the side of a triangle, use the bars and connectors to find out if
they do make a triangle.
Case
Length(cm)
Length(cm)
Length(cm)
1
31
7.75
27
2
31
15.5
13.5
3
15.5
7.75
13.5
4
15.5
17
11
5
31
22
17
6
31
7.75
17
7
15.5
17
13.5
8
7.75
20
11
9
31
20
27
10
22
7.75
17
11
15.5
22
27
12
31
22
13.5
Is it a Triangle?
What conjecture can you make about the relationship between the sides of a triangle that would
guarantee you that your three lengths always made a triangle?
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Use the triangle conjecture to mathematically verify which lengths of sides, from the combinations in
your group, form triangles and which ones do not form triangles for the first five cases.
Case #1
Case #2
Case #3
Case #4
Case #5
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Number of
Sides and
Name of
Regular
Polygon
Measure of
one Central
Angle
Measure of
one Interior
Angle
Sum of all of Measure of Sum of all of
the Interior One Exterior the Exterior
Angles
Angle
Angles –
One at each
Vertex
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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Measure and complete the chart
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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PARALLEL LINE CONJECTURE
1.)
Go to the Edit Menu and select Preferences. Make sure the units for angle is set to degrees
and the precision for angles is set to units.
2.)
Use the line tool and make a horizontal line towards the bottom of the screen.
3.)
Use the selection arrow tool and click anywhere on the screen to deselect.
4.)
Use the point tool and make a point above the horizontal line.
5.)
Use the selection arrow tool and highlight the line. The point above the line should already be
highlighted. If the point is not highlighted, then click on it.
6.)
Go to the Construct Menu and select Parallel Line. Use the selection arrow tool and click to
deselect.
7.)
Use the line tool and make a transversal through the parallel lines.
8.)
Use the selection arrow tool and click to deselect.
9.)
Use the selection arrow tool and highlight the point on the parallel line. Go to the Display
Menu and select Hide Point.
10.)
Use the selection arrow tool and highlight one of the parallel lines and the transversal. Go to
the Construct Menu and select Intersection. Repeat with the other parallel line and the
transversal.
11,)
Use the selection arrow tool and measure all eight angles. Line up together the angles that
are congruent to each other.
12.)
Use the selection arrow tool and highlight all eight angle measurements. Go to the Graph
Menu and select Tabulate. A small table appears with columns for all eight angle
measurements.
13.)
Double click on the table and another row of the table will appear.
14.)
Use the selection arrow tool and move the transversal around to create different angle
measurements.
15.)
Repeat steps 13 and 14 until you have 10 different angle measurements.
16.)
Go to the File Menu and select Print Preview. Make sure your parallel lines cut by the
transversal and table will print on one page. If not, highlight the entire figure and move it
underneath the table.
17.)
Print two copies of your figure and table measurements.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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NAME:_________________________________________
1.)
Look at the angles formed by the parallel lines cut by the transversal you made on the
Geometer’s Sketchpad. Cut out and tape the picture of the parallel lines cut by the transversal
that is on your Geometer’s Sketchpad screen.
NAMES OF ANGLES
2.)
PAIRS OF ANGLES
In each of the nine additional situations of parallel lines cut by the transversal, what do you
notice about the angle measurements?
________________________________________________________________________
3.)
State the Parallel Line Conjecture.
________________________________________________________________________
________________________________________________________________________
4.)
State the Converse of the Parallel Line Conjecture.
________________________________________________________________________
________________________________________________________________________
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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THE SUM OF THE ANGLES OF A TRIANGLE
1.)
Go to the Edit Menu and select Preferences. Make sure the units for angle is set to degrees
and the precision for angles is set to units.
2.)
Use the line segment tool and make a triangle.
3.)
Use the selection arrow tool and click anywhere on the screen to deselect.
4.)
Use the text tool and label the triangle A, B, and C by clicking on the points of the triangle.
5.)
Use the selection arrow tool and select points BAC in order to measure angle A. Next, go to
the Measure Menu and select Angle.
6.)
Click anywhere on the screen to deselect.
7.)
Use the selection arrow tool and select points ABC in order to measure angle B. Next, go to
the Measure Menu and select Angle.
8.)
Click anywhere on the screen to deselect.
9.)
Use the selection arrow tool and select points BCA in order to measure angle C. Next, go to
the Measure Menu and select Angle.
10.)
Click anywhere on the screen to deselect.
11.)
Go to the Measure Menu and select Calculate.
12.)
Click on the measurement for angle A, click on the addition sign, click on the measurement for
angle B, click on the addition sign, click on the measurement for angle C, and click OK.
13.)
Use the selection arrow tool and select all three angle measurements and the sum of all three
angle measurements. (Four items should be highlighted.)
14.)
Go to the Graph Menu and select Tabulate. A small table will appear on your screen with
columns for the three angle measurements and the sum of the angles of the triangle.
15.)
Double click on the table and another row of the table will appear.
16.)
Use the selection arrow tool and move the vertices around to create a different triangle.
17.)
Repeat steps 15 and 16 until you have 10 different triangle measurements.
18.)
Go to the File Menu and select Print Preview. Make sure your triangle and table will print on
one page. If not, highlight the entire triangle and move it underneath the table.
19.)
Print a copy of your triangle and table measurements.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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NAME: ___________________________________
1.)
In your original triangle you made on the Geometer’s Sketchpad, what is the sum of the
measurements of the three angles?
________________________________________________________________________
2.)
In each of the nine triangles you created, what is the sum of the measurements of the three
angles?
________________________________________________________________________
3.)
State a generalization of the sum of the angles of all triangles.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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ADJACENT ANGLES AND LINEAR PAIRS
1.)
Go to the Edit Menu and select Preferences. Make sure the unit for angles is set to degrees
and the precision for angles is set to units.
2.)
Use the line segment tool and make a line segment.
3.)
Use the text tool and label the endpoints, Point C and Point D. If necessary, double click on
the label the computer gave the points and rename them.
4.)
Use the line segment tool and highlight the line segment CD. Go to the Construct Menu and
select Point On Segment.
5.)
Use the test tool and label the point B. If necessary, double click on the label the computer
gave the point and rename it.
6.)
Use the line segment tool and make a line segment from point B.
7.)
Use the text tool and label the endpoint, Point A. If necessary, double click on the label the
computer gave the point and rename it.
8.)
Use the selection arrow tool and click on the points, A, B, and C. Go to the Measure Menu
and select Angle.
9.)
Use the selection arrow tool and click on the points, A, B, and D. Go to the Measure Menu
and select Angle.
10.)
Use the selection arrow tool and go to the Measure Menu and select Calculate. Click on one
of the angle measurements, click on the addition sign, click on the other angle measurement,
and click OK.
11.)
Use the selection arrow tool and highlight the two angle measurements and the sum of the
angle measurements. Go to the Graph Menu and select Tabulate. A small table will appear
on your screen with columns for the two angle measurements and the sum of the two angles.
12.)
Double click on the table and another row of the table will appear.
13.)
Use the selection arrow tool and move the endpoints around to create different angle
measurements.
14.)
Repeat steps 12 and 13 until you have 10 different angle measurements.
15.)
Go to the File Menu and select Print Preview. Make sure your linear pair and table will print on
one page. If not, highlight the entire figure and move it underneath the table.
16.)
Print a copy of your figure and table measurements.
17.)
Go to the File Menu and select New Sketch. Use the line segment tool and make two line
segments that intersect.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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18.)
Use the selection arrow tool and click anywhere on the screen to deselect.
19.)
Use the selection arrow tool and highlight both line segments, go to the Construct Menu and
select Intersection.
20.)
Use the text tool and label the intersection point E. Label one line segment AB and the other
segment CD.
21.)
Use the selection arrow tool and select points BED. Next, go to the Measure Menu and select
Angle.
22.)
Click anywhere on the screen to deselect.
23.)
Use the selection arrow tool and select points AEC. Next, go to the Measure Menu and select
Angle.
24.)
Click anywhere on the screen to deselect.
25.)
Use the selection arrow tool and select points AED. Next, go to the Measure Menu and select
Angle.
26.)
Click anywhere on the screen to deselect.
27.)
Use the selection arrow tool and select points CEB. Next, go to the Measure Menu and select
Angle.
28.)
Click anywhere on the screen to deselect.
29.)
Identify each linear pair of angles.
30.)
Use the selection arrow tool and go to the Measure Menu and select Calculate. Focus on one
linear pair at a time. Click on one of the angles in the linear pair, click on the addition sign,
click on the other angle, click OK.
31.)
Repeat step 30 with the other linear pair.
32.)
What is the sum of each of the linear pairs?
33.)
Use the selection arrow tool and select both angles of one linear pair and their sum, then
select both angles of the other linear pair and their sum.
34.)
Go to the Graph Menu and select Tabulate. A small table will appear on your screen with
columns for the measurements of each angle of the linear pair and their sum.
35.)
Double click on the table and another row of the table will appear.
36.)
Use the selection arrow tool and move the endpoints around to create different angle
measurements.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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37.)
Repeat steps 35 and 36 until you have 10 different angle measurements.
38.)
Go to the File Menu and select Print Preview. Make sure your intersecting lines and table will
print on one page. If not, highlight the entire figure and move it underneath the table.
Print a copy of your figure and table measurements.
39.)
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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NAME: _________________________________________
1.)
Look at the angles formed by the intersecting lines you made on the Geometer’s Sketchpad.
Identify and name each of the linear pairs of angles.
________________________________________________________________________
2.)
What do you notice about the sum of each linear pair of angles?
________________________________________________________________________
3.)
In each of the nine additional situations of intersecting lines, what do you notice about the sum
of each linear pair of angles?
________________________________________________________________________
4.)
State the conjecture about the sum of the measures of a linear pair of angles.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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INTERSECTING LINES AND VERTICAL ANGLES
1.)
Go to the Edit Menu and select Preferences. Make sure the units for angle is set to degrees
and the precision for angles is set to units.
2.)
Use the line segment tool and make two line segments that intersect.
3.)
Use the selection arrow tool and click anywhere on the screen to deselect.
4.)
Use the selection arrow tool and highlight both line segments, go to the Construct Menu and
select Intersection.
5.)
Use the text tool and label the intersection point E. Label one line segment AB and the other
segment CD.
6.)
Use the selection arrow tool and select points BED. Next, go to the Measure Menu and select
Angle.
7.)
Click anywhere on the screen to deselect.
8.)
Use the selection arrow tool and select points AEC. Next, go to the Measure Menu and select
Angle.
9.)
Click anywhere on the screen to deselect.
10.)
Use the selection arrow tool and select points AED. Next, go to the Measure Menu and select
Angle.
11.)
Click anywhere on the screen to deselect.
12.)
Use the selection arrow tool and select points CEB. Next, go to the Measure Menu and select
Angle.
13.)
Click anywhere on the screen to deselect.
14.)
Use the selection arrow tool and select all four angle measurements.
15.)
Go to the Graph Menu and select Tabulate. A small table will appear on your screen with
columns for the four angle measurements.
16.)
Double click on the table and another row of the table will appear.
17.)
Use the selection arrow tool and move the endpoints around to create different angle
measurements.
18.)
Repeat steps 16 and 17 until you have 10 different angle measurements.
19.)
Go to the File Menu and select Print Preview. Make sure your intersecting lines and table will
print on one page. If not, highlight the entire figure and move it underneath the table.
20.)
Print a copy of your figure and table measurements.
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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NAME: _________________________________________
1.)
Look at the angles formed by the intersecting lines you made on the Geometer’s Sketchpad.
Identify and name the pairs of vertical angles.
________________________________________________________________________
2.)
What do you notice abut each pair of vertical angles?
________________________________________________________________________
3.)
In each of the nine additional situations of intersecting lines, what do you notice about each
pair of vertical angles?
________________________________________________________________________
4.)
State the Vertical Angle Conjecture.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet
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