Download Triangles for Sorting

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Triangles for Sorting
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A
E
D
C
I
G
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F
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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DATE
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Two Pentagons
A
B
a) How many right angles does each pentagon have?
b) Fold the pentagons so that you can see that the remaining angles are all equal.
c) Does each side on pentagon A have a side of the same length on pentagon B?
d) Is there the same number of sides of each length on both pentagons?
e) If you place one pentagon on top of the other, do they match?
f) Are they the same shape?
g) Can we say that these pentagons have the same sides and angles?
h) Are the pentagons congruent?
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i) What makes the answers to g) and h) different?
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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Congruence, Area, and Logic (1)
1. a) Find the area and/or the perimeter of the shapes.
A
B
h
h
a
3
5
4
4
5
5
3
a
9
Area =
Area =
C
3
3
5
Perimeter =
4
24
Area =
Perimeter =
b
D
b
h
h
a
Perimeter =
E
26
12
Perimeter =
3
10
Area =
Area =
Area =
F
Area =
Perimeter =
a
25
20
15
Perimeter =
b)The following statements are false. Match the counter-examples from a)
to the statements.
i) Two triangles with the same perimeter are always congruent.
ii) Two triangles with the same perimeter always have the same area.
Counter-example:
v) Two trapezoids with the same area always have the same perimeter.
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Counter-example:
iv) Two trapezoids with the same bases and height are always congruent.
Counter-example:
iii) Two triangles with the same base and height are always congruent.
Counter-example:
Counter-example:
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
COPYRIGHT © 2010 JUMP MATH: TO BE COPIED
4
Area =
5
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Congruence, Area, and Logic (2)
2. a) Find the area and perimeter of each shape.
A
6
Area =
Perimeter =
B
4
Area =
4
6
C
5
3
D
5
7
Perimeter =
Area =
3
8
Perimeter =
Area =
Perimeter =
b)Decide whether each statement is true or false. For each true statement, explain
your thinking. For each false statement, find at least one counter-example among
the shapes above.
i) Two rectangles with the same area are always congruent.
False. Counter-examples: A and D
ii) Two congruent parallelograms always have the same area.
iii) Two congruent triangles always have the same perimeter.
iv) Two parallelograms with the same base and height always have the
same area.
v)Two parallelograms with the same base and height always have the
same perimeter.
f)Two parallelograms with the same sum of base and height have the
same area.
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3. True or False?
For each true statement, explain your thinking.
For each false statement sketch a counter-example.
a) Two triangles with the same side lengths are always congruent.
b) Two triangles with the same side lengths always have the same area.
c) Two triangles with the same base and height always have the same area.
d) Two triangles with the same base and height always have the same angles.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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Congruence Rules on Geometer’s Sketchpad (1)
1.Using the polygon tool, construct a triangle ABC. Measure the sides and the angles
of your triangle.
2. a) Construct a point D. Using a command for constructing circles and the length
of AB as the radius, construct a circle with centre D. Construct line segment
DE = AB. Hide the circle.
b) Using a command for constructing circles and the length of BC as the radius,
construct a circle with centre E.
c) Using a command for constructing circles and the length of AC as the radius,
construct a circle with centre D.
d) Construct a point that is on both circles. Label it F. Use the polygon tool to
construct triangle DEF. Hide the circles.
3. a) Which sides are equal in ∆ABC and ∆DEF?
b) Measure the angles of ∆DEF. What can you say about ∆ABC and ∆DEF?
4. Try to move the vertices of ∆DEF around.
a) How does your triangle change? Which transformations can you make: rotations,
reflections, translations?
b) Can you move ∆DEF onto ∆ABC to check whether they are congruent?
Do you need to reflect the triangle to do that?
5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around.
a) Are the changes you can make in this triangle different from the changes you
could make to ∆DEF? Why?
b) What happens to ∆DEF when you modify ∆ABC? What can you say about the
triangles ∆ABC and ∆DEF?
6. Are three side lengths enough to determine a unique triangle?
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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DATE
NAME
Congruence Rules on Geometer’s Sketchpad (2)
1.Using the polygon tool, construct a triangle ABC. Measure the sides and the angles
of your triangle.
2. a) Construct a point D. Using a command for constructing circles and the length
of AB as the radius, construct a circle with centre D. Construct line segment
DE = AB. Hide the circle.
b) Using a command for constructing circles and the length of BC as the radius,
construct a circle with centre E.
c) Select E as a centre of rotation. Use a command in the Transformation menu
to mark ∠ABC as the angle of rotation. Rotate point D around E by the angle
chosen. Construct a ray from E through the image of D.
d) Construct a point that is on the ray and the circle. Label it F. Use the polygon tool
to construct triangle DEF. Hide the circle and the ray EF.
3. a) Which sides and angles are equal in ∆ABC and ∆DEF?
b) Measure the rest of the sides and angles of ∆DEF. What can you say about
∆ABC and ∆DEF?
4. Try to move the vertices of ∆DEF around.
a) How does your triangle change? Which transformations can you make: rotations,
reflections, translations?
b) Can you move ∆DEF onto ∆ABC to check whether they are congruent?
Do you need to reflect the triangle to do that?
5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around.
a) Are the changes you can make in this triangle different from the changes you
could make to ∆DEF? Why?
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b) What happens to ∆DEF when you modify ∆ABC? What can you say about the
triangles ∆ABC and ∆DEF?
6. Are two sides and the angle between them enough to determine a unique triangle?
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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Congruence Rules on Geometer’s Sketchpad (3)
1.Using the polygon tool, construct a triangle ABC. Measure the sides and the angles
of your triangle.
2. a) Construct a point D. Using a command for constructing circles and the length
of AB as the radius, construct a circle with centre D. Construct line segment
DE = AB. Hide the circle.
b) Select E as a centre of rotation. Use a command in the Transformation menu
to mark ∠ABC as the angle of rotation. Rotate point D around E by the angle
chosen. Construct a ray from E through the image of D.
c) Select D as a centre of rotation. Mark ∠BAC as the angle of rotation. Rotate point
E around D by the angle chosen. Construct a ray from D through the image of E.
d) Construct a point F that is the intersection point of both rays. Use the polygon
tool to construct triangle DEF. Hide the rays.
3. a) Which sides and angles are equal in ∆ABC and ∆DEF?
b) Measure the rest of the sides and angles of ∆DEF. What can you say about
∆ABC and ∆DEF?
4. Try to move the vertices of ∆DEF around.
a) How does your triangle change? Which transformations can you make: rotations,
reflections, translations?
b) Can you move ∆DEF onto ∆ABC to check whether they are congruent?
Do you need to reflect the triangle to do that?
5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around.
a) Are the changes you can make in this triangle different from the changes you
could make to ∆DEF? Why?
b) What happens to ∆DEF when you modify ∆ABC? What can you say about the
triangles ∆ABC and ∆DEF?
6. Are two angles and a side between them enough to determine a unique triangle?
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
COPYRIGHT © 2010 JUMP MATH: TO BE COPIED
NAME
DATE
Constucting Regular Polygons with a Compass and
a Straightedge (1)
A polygon is called regular if all its sides and all its angles are equal.
1. a) Draw a quadrilateral with all sides
b) Draw a quadrilateral with all angles
equal that is not regular. equal that is not regular.
2. Can you draw a triangle with all sides equal that is not regular? Explain.
3. What triangle is a regular triangle?
What is the size of its angles?
4. What special quadrilateral is a regular quadrilateral?
What is the size of its angles?
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5. Construct a regular quadrilateral using a compass and a straightedge.
A
6.Jason constructed a regular triangle using a compass and a straightedge.
What is the size of ∠ABD in his drawing?
D
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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Constucting Regular Polygons with a Compass and
a Straightedge (2)
A line of symmetry breaks a picture into two parts of the same shape and size.
The parts are mirror images of each other. In polygons, lines of symmetry are perpendicular bisectors
of sides or angle bisectors.
REMINDER
The sum of the angles around a point is 360°.
The sum of the angles in a triangle is 180°.
How could you construct a regular hexagon using a compass
and a straightedge? Start by determining the angles between the sides.
INVESTIGATION
A. Sketch a regular hexagon.
B.A regular hexagon has 6 lines of symmetry. They all
pass through the same point in the centre of the hexagon.
Sketch the lines of symmetry of your hexagon.
Label the intersection point of the lines of symmetry O.
C. How many non-overlapping angles are around the point O?
What is the total degree measure of all the angles around O?
What is the measure of the angles between the lines of symmetry?
How do you know?
A
D. The hexagon at right is a regular hexagon. The lines AO, BO, CO, and GO
are some of the lines of symmetry of the hexagon.
Use the information on your sketch to mark the equal line segments and
the equal angles on the diagram at right. Include any right angles.
B
Use your sketch to determine the size of these angles:
O
∠BOG =
∠AOB =
∠BGO =
Write this information on the diagram at right, too.
E. What is the size of ∠OGB?
How do you know?
What is the size of ∠OBC?
What is the size of ∠ABC?
What is the size of the angles between the sides of a regular hexagon?
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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DATE
NAME
Constucting Regular Polygons with a Compass and
a Straightedge (3)
7.Follow the steps below to construct a regular hexagon using a compass
and a straightedge.
a) Sketch a hexagon. Label it ABCDEF.
b)Start constructing ABCDEF by drawing a line segment AB. Then
construct a line that intersects AB at 120° and passes through B.
HINT: How could constructing an equilateral triangle help you?
See Question 3.
c) On the line you drew, mark a point C so that BC = AB, and ∠ABC = 120°.
d)The angle between AC and CD is 120° again. Which of the diagrams
at right is better for drawing a hexagon? Use your sketch to decide.
e)Repeat steps a) and b) with the new line segment. Repeat until you
close the polygon. Use your sketch to keep track of what you are doing.
120°
B
A
120°
120°
120°
120°
8.How could you use symmetry to construct a regular hexagon using a compass
and a straightedge?
a) Look at the picture of a regular hexagon at right.
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Mark 12 equal line segments and 18 equal angles in the picture.
b)Draw a circle with the centre at O and the radius equal to a side of the
hexagon. Do all the vertices of the hexagon lie on this circle?
Now you will construct your own hexagon. Use the picture above as a sketch.
c)The hexagon in the sketch consists of 6 equilateral triangles. Construct an
equilateral triangle using a compass and a straightedge.
d)Choose the side of the triangle that will be a side of your hexagon. Mark that side
on your sketch.
e)Where on your triangle is the point O? Label it. Label the other two vertices A
and B. Which sides of the triangle are the future lines of symmetry (diagonals)
of the hexagon?
f) Extend OA and OB beyond O. Using a compass, construct a circle as in b).
g)How can you use symmetry and the circle to construct two more vertices of
your hexagon?
h)How many vertices does a hexagon have? How many more vertices do you have
to construct?
i)Construct the last two vertices of the hexagon. How did you find the vertices?
Did you use parallel lines, construction of lines at 120°, construction of circles
with centres A and B, or another method?
j) Using a protractor and a ruler, check that your hexagon is regular.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.2
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