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U N I T Geometry
72
Learning Goals
• estimate, measure, and drawangles using a protractor
• make and apply generalizationsabout the sum of angles intriangles and quadrilaterals
• make and apply generalizationsabout diagonal properties ofquadrilaterals
• sort quadrilaterals according to properties
• make and interpret orthographicdrawings
• create cross-sections from solids• make generalizations about the
planes of symmetry of solids
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protractor
rhombi
Venn diagram
chevron
orthographic drawing
mat plan
cross-section
plane symmetry
Key Words
73
• What figures do you see on the faces of the scraps?
• Which scraps go with which pieces?How do you know?
Georgette and Graham found scraps ofwood cut from these pieces of lumber.
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Investigating Angles
L E S S O N
74 LESSON FOCUS Estimate, measure, and draw angles.
You will need a protractor.Your teacher will give you a large copy of these angles.
➤ Estimate the measure of each angle.Record your estimates.Then, find each angle measure.Record the angle measures.
➤ Find the sum of the angles in each pair.a and b c and d e and f
What do you notice about the angles in each pair?
Show and Share
Share your work with a classmate.How did you estimate the measure of each angle?Were your estimates reasonable? Explain.How did you use the measures of angles a and b to estimate the measures of angles c and d?
fig: G6_A_03-01-01
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Unit 3 Lesson 1 75
On a protractor, the measures go from 0° to 180° clockwise and counterclockwise.
Recall that you can use a protractor to draw or measure angles.
➤ To measure an angle using a protractor, follow these steps:
Step 1Estimate if the angle is less than or greater than 90°.
Step 2Place the protractor on top of the angle.The vertex of the angle is at the centre of the protractor.One arm of the angle lines up with the base line of the protractor.
Step 3Start at 0° on the arm along the base line.Find where the other arm of the angle meets the protractor.Recall the estimate, then read the appropriate measure.This angle measures 60°, not 120°.
I think this angleis less than 90°.
This angle is less than 90°.
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➤ Follow these steps to construct an angle that measures 145°.
Step 1With a ruler, draw one arm of the angle.
Step 2Place the protractor on top of the arm.One end of the arm is at the centre of the protractor.The arm lines up with the base line of the protractor.Start at 0° on the arm along the base line.Make a mark at 145° so the angle is greater than 90°.
Step 3Remove the protractor.Join the mark to the end of the arm placedat the centre of the protractor.Label the angle with its measure of 145°.
➤ The angle on a straight line measures 180°.
76 Unit 3 Lesson 1
1. What is the measure of each angle? Explain how you know.a) b) c)
2. For each angle:• Estimate the angle measure.• Use a protractor to find the angle measure.a) b) c)
d) e) f)
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Draw an angle.Explain how to use a protractor to measure the angle.
ASSESSMENT FOCUS Question 6
3. Use a ruler and a protractor.Construct an angle with each measure.a) 80° b) 30° c) 100°d) 150° e) 180° f) 10°
4. Name 4 objects in your classroom that have:a) an angle greater than 100°b) an angle less than 60°Use a protractor to check your answers.
5. Draw an angle you think measures 140°.Use a protractor to check your angle measure.How close was your angle to 140°?
6. Without using a protractor, draw an angle that is 90° greater than each of these angles.a) b)
Measure each angle with a protractor to check.Explain how you drew each angle.
7. A student measured this angle and said it measured 60°.Do you agree? Explain.
Number Strategies
Order the numbers in each set from least to greatest.
• 123 321, 121 232, 123 231, 113 321
• 4 432 344, 4 344 342, 4 242 444, 4 432 413
Numbers Every Day
Unit 3 Lesson 1 77
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L E S S O N
78 LESSON FOCUS Make and apply generalizations about the sums of angles in figures.
Sum of the Angles in Trianglesand Quadrilaterals
You will need a ruler, scissors, and a protractor.Your teacher will give you 2 copies with three different triangles.
➤ Cut out the acute triangle from one copy.Tear off the 3 angles.Place the vertices of the 3 angles together,so adjacent sides touch.What do you notice?
Repeat the activity with the right triangle and the obtuse triangle.What do you notice?What can you say about the sum of the angles in each triangle?
➤ Use the triangles from the other copy.Measure each angle with a protractor and label it with its measure.Find the sum of the angles in each triangle.Does this confirm your results from tearing off the angles? Explain.
Show and Share
Compare your results with those of another pair of classmates.What can you say about the sum of angles in a triangle? Explain.
➤ We can show that the sum of the angles in a triangle is the same for all triangles.
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Unit 3 Lesson 2 79
We label the angles in 3 congruent triangles.We arrange the triangles to make a tessellation.
The angles in each triangle are a, b, and c.The tessellation shows that these three angles make a straight angle.So, a + b + c = 180°The sum of the angles in a triangle is 180°.
➤ We can divide any quadrilateral into 2 triangles.
Since the sum of the angles in a triangle is 180°,the sum of the angles in a quadrilateral is 2 � 180° � 360°.
1. Draw a large obtuse triangle on dot paper. Measure each angle.Find the sum of the measures of the angles.
2. Determine the missing angle measure without measuring.a) b) c)
d) e) f)
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Explain how you know the sum of the angles inany triangle or quadrilateral.
Mental Math
Estimate each quotient.2343 � 995456 � 503620 � 608405 � 12
Numbers Every Day
80 ASSESSMENT FOCUS Question 3 Unit 3 Lesson 2
3. Find the measure of the third angle in each triangle.Then, draw the triangle.Explain how you found each measure.a) A triangle with two angles measuring 65° and 55°b) A triangle with two equal angles; each measures 45°c) A right triangle with a 70° angled) An isosceles triangle with one angle measuring 120°
4. a) What do you know about each angle in a square? A rectangle?b) How can you use your answer in part a to find the sum of
the angles in a square or a rectangle?
5. Jacques and Alicia hiked from their cottage to the river.They turned and hiked to their Grandma’s cottage.Through which angle should they turn to get back to their cottage? Explain.
6. Draw a pentagon.Draw diagonals from one vertex to divide the pentagon into triangles.Find the sum of the angles in a pentagon.Check your answer by measuring the angles.
7. How can you use the first diagram in Connectto verify that the sum of the angles in a quadrilateral is 360°?
50°
110°
Jacquesand Alicia’s
cottage
Grandma’s cottage
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L E S S O N
81LESSON FOCUS Make and apply generalizations about diagonals of quadrilaterals.
Properties of Diagonals of Quadrilaterals
You will need scissors, a ruler, and a protractor.Your teacher will give you copies of rhombi.
➤ Cut out the rhombi.Each partner works with 1 rhombus.
➤ Use a ruler to draw 2 diagonals.Measure the diagonals.What do you notice?
➤ Use your protractor to measure:• the angles where the
diagonals meet• the angles formed where each
diagonal meets a vertexWhat do you notice?
Show and Share
Share your results with another pair of students.How are your results the same? How are they different?Do you think these results will be true for all rhombi? Explain.
A diagonal of a quadrilateral joins 2 opposite vertices.
The plural of rhombusis rhombi.
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82 Unit 3 Lesson 3
➤ The diagonals of a rhombus have these properties:• They are perpendicular.• They bisect each other.• They form 4 congruent right triangles.• They lie on the 2 lines of symmetry of the rhombus.• They bisect the angles of the rhombus.
➤ We can use the properties of a diagonal of a rhombus to find the measures of all angles when we measure one angle.
A diagonal lies on a line of symmetry,so b = 66°.
A diagonal divides the rhombus into 2 congruent isosceles triangles.In each triangle, the two equal angles add up to 180° � 66° � 114°.So, each equal angle is 114° � 2 � 57°.So, a � 57° � 57° � 114°‚ and c � 114°
1. Here is a rhombus.
Find the measure of each unknown angle without using a protractor.Explain your thinking.
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How can you use the properties of thediagonals of a parallelogram to list theproperties of the diagonals of a squareand a rectangle? Include diagrams.
Number Strategies
Find each quotient.
1530 � 98575 � 256344 � 522136 � 12
Numbers Every Day
ASSESSMENT FOCUS Question 2 Unit 3 Lesson 3 83
2. Use square dot paper.Draw a large parallelogram.a) Measure each diagonal.
Record your findings.b) What are the measures of the angles
where the diagonals meet?c) What are the measures of the angles
formed by a diagonal at each vertex?d) What do you notice about the triangles
that are formed?e) How many diagonals lie on lines of symmetry?
3. Use square dot paper.Draw a kite.Draw the diagonals.What are the properties of the diagonals of a kite?
4. List the properties of the diagonals of a rhombus that are the same as those of a kite.List the properties that are different.
5. Use square dot paper.Draw a trapezoid.Draw the diagonals.What are the properties of the diagonals of a trapezoid?
6. Draw a line segment that is 8 cm long.Use a Mira.Draw a rhombus that has this segment as one of its diagonals.Explain how you drew the rhombus.
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L E S S O N
Sorting Quadrilaterals
Show and Share
Discuss the strategies you used to choose figures with another pair of students.How were your strategies similar? How were they different?
Number Strategies
Round each number to thenearest thousand and to thenearest hundred thousand.
• 4 682 364• 803 091 531• 9 989 899
Numbers Every Day
84 LESSON FOCUS Sort quadrilaterals according to their properties.
Your teacher will give you a copy of a set of figure cards and a set of property cards.
➤ Player A chooses a property card and places it face up.Player B looks through the figure cards.She places each figure card with the property shown around the property card.The players discuss whether each figure chosen is appropriate and whether there are any figures missing.Players record the properties and figures.
➤ Player B chooses another property card, placing it over the previous property card.Player A determines which figure cards should be removed, and whether any figure cards should be added.Players discuss the appropriateness of the chosen figure cards.Players record the properties and figures.
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Unit 3 Lesson 4 85
➤ This table shows the properties of trapezoids, parallelograms, rhombi, and kites.
Name Properties Example
Trapezoid • 1 pair of parallel sides
Parallelogram • 2 pairs of parallel sides• opposite sides equal• opposite angles equal• diagonals that bisect each other• diagonals that form 2 pairs
of congruent triangles
Rhombus • all sides equal• opposite angles equal • 2 pairs of parallel sides• diagonals that are perpendicular
bisectors• diagonals that form 4 congruent
right triangles• diagonals that lie on 2 lines
of symmetry• diagonals that bisect the
angles of the rhombus
Kite • 2 pairs of equal adjacent sides• 1 pair of equal angles• diagonals that are perpendicular• diagonals that form 2 pairs of
congruent right triangles• one diagonal that is bisected• the other diagonal that lies on a
line of symmetry and bisects two opposite angles of the kite
Math LinkYour World
Kites have been used for thousands of years. The earliest writtenaccount of kite flying occurred about 200 B.C.E., when General HanHsin of the Han Dynasty flew a kite over the city walls. Han Hsinused the kite to measure the length of a tunnel needed to reach theenemy’s palace. Benjamin Franklin experimented with kites toinvestigate atmospheric electricity. Guglielmo Marconi launchedtransatlantic wireless communication with the help of a kite.
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86 Unit 3 Lesson 4
➤ We can use a Venn diagram to sort these quadrilaterals:
The properties are:• Diagonals are perpendicular. • Diagonals bisect each other.
Diagonals are perpendicular
A trapezoidhas neitherproperty.
Diagonals bisect each other
Parallelogram Rhombus Trapezoid Kite
A kite hasdiagonals that are
perpendicular.
A rhombus has diagonalsthat are perpendicularand bisect each other.
A parallelogramhas diagonals thatbisect each other.
Has 4 right angles
Has at least 1 pair of parallel sides
Has 4 congruent sides1. Copy this Venn diagram.
a) Sort these quadrilaterals: trapezoid,parallelogram, rectangle, square,rhombus, and kite.
b) Which quadrilateral has all 3 properties?Where is this quadrilateral on the Venn diagram?
2. Draw a Venn Diagram, similar to that in question 1.Choose 3 different properties.a) Sort these quadrilaterals: trapezoid, parallelogram,
rectangle, square, rhombus, and kite.b) Does any quadrilateral have all 3 properties? How do you know?
3. Name the quadrilateral. It has:• opposite angles equal• diagonals that are perpendicular bisectors of each other• diagonals that form four congruent right triangles• 2 lines of symmetry
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ASSESSMENT FOCUS Question 6
4. Name the quadrilateral. It has:• diagonals that are perpendicular and form
two pairs of congruent right triangles• one diagonal that bisects the other• one line of symmetry
5. Copy this table. Fill in each blank with Yes or No.
6. Use a geoboard and geobands.a) Make quadrilaterals with each property:
• two obtuse angles • two acute angles• exactly one right angle • diagonals that are perpendicularDraw your quadrilaterals on square dot paper.
b) How many different quadrilaterals did you make for each property? Name each quadrilateral if you can.
7. Is it possible for a quadrilateral to have:• more than 2 obtuse angles?• opposite angles equal and no lines of symmetry?Use words and pictures to explain.
8. A chevron is a concave kite.a) Draw a chevron on dot paper.
Make sure adjacent sides are equal.b) What do you have to do to the diagonals so that the
properties of the diagonals of a kite apply to the chevron?
How can you use geometric properties to sort quadrilaterals?Use words and pictures to explain.
Unit 3 Lesson 4 87
Type of Do the diagonals Are the diagonals Do the diagonals Quadrilateral bisect each other? perpendicular? form two pairs of
congruent triangles?
Rectangle
Square
Parallelogram
Rhombus
Trapezoid
Kite
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88 LESSON FOCUS Check and reflect.
L E S S O N
Look at Paolo’s quiz answers.Are Paolo’s answers correct? Explain.
Create a list of five true or false statements based on the properties of quadrilaterals.Write your true or false answers on a separate page.
Show and Share
Trade statements with a classmate.Identify each of your classmate’s statements as true or false.Discuss your answers with your classmate.
Strategiesfor Success
• Get unstuck.
• Check and reflect.
• Focus on the problem.
• Represent your thinking.
• Select an answer.
• Do your best on a test.
• Explain your answer.
Homework
1. All squares are quadrilaterals.
True
2. All rectangles are parallelograms.
True
3. All parallelograms are trapezoids.
False
4. The diagonals of rhombi are of equal length.
False
5. If a figure is a square, it is a trapezoid.
True
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Unit 3 Lesson 5 89
1. Check each statement.Is it true or false? Explain.a) All rhombi are parallelograms.b) All parallelograms are rectangles.c) All squares are rhombi.d) All parallelograms have equal diagonals.
Marg considered this statement: “All squares are rectangles.”
To find out if the statement was true or false,Marg recorded these properties of a rectangle:
• exactly four right angles• two pairs of parallel sides• opposite sides equal• two lines of symmetry• diagonals equal • diagonals that form 2 pairs of congruent triangles
Marg looked at the list.She placed a checkmark beside each property that applied to a square.She concluded that all the properties applied to a square.So, the statement that all squares are rectangles is true.
Why is it important to always check your solution?
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L E S S O N
90 LESSON FOCUS Make and interpret orthographic drawings.
Orthographic drawings are 2-D views of a 3-D object.The views may be from the top, left, front, right, or back.
A mat plan is a top view that indicates the height of the cubes in the object.The mat plan, below, represents the object on the right.
Orthographic Drawings
You will need Snap Cubes,grid paper, and a ruler.
➤ Build an object using Snap Cubes.On grid paper, draw a mat plan of your object.Hide the object.
➤ Trade mat plans with a classmate.Build the object shown on your classmate’s mat plan.
➤ Compare your object with the object your classmate hid.Are the objects the same? Explain.
Show and Share
Use square dot paper.Work together to draw as many different views of your object as possible.
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Unit 3 Lesson 6 91
The orthographic views below represent this object.
To draw different views, it may be helpfulto use a building mat.Place the object on the building mat.Move the mat to draw each view.
You will need Snap Cubes and grid paper. Use a building mat when it helps.
1. Use 5 Snap Cubes to build an object.Draw the top, front, left, right, and back orthographic views.Label each view.
2. Use Snap Cubes to build each object.Draw a mat plan for each object.Draw 5 orthographic views of each object.a) b)
The left and right ofthe object are relative
to the front.
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92 ASSESSMENT FOCUS Question 5 Unit 3 Lesson 6
Can you build an object that has different front and back views?If your answer is yes, build the object and sketch all 5 views.If your answer is no, explain why you cannot build the object.
Number Strategies
Write 4 different numbersthat have a remainder of 3when divided by 5. How did you find thenumbers?
Numbers Every Day
3. Use Snap Cubes to build the object shown in each mat plan.a) b)
Draw the front, left, and right views of each object.
4. Look at the object below.Which orthographic view represents the left view of the object?Explain how you know.
5. Use Snap Cubes to build an object that has these orthographic views.Draw the left and back views of the object.
Explain how you built the object.
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93
Animator
Wor
ld of Work
An animator uses artistic talent and sophisticated graphics softwareto make movie scenes. But while the software may be sophisticated,basic geometry is at its core.
Every movement of an object within an animated scene involvesone or more transformations. The animator chooses direction andspeed, and the software performs the transformations to match.Animators continually adjust their instructions to make a scene morerealistic or exciting. New “routines” are stored and shared with theother animators working on the project. Although everythingappears three-dimensional, calculations are done using two-dimensional transformations with sizing changes.
Computer animators can recreate details thatwould be impossible to get on film alone.Sometimes, they can make the unbelievableseem real.
Unit 3 93
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L E S S O N
94 LESSON FOCUS Describe and represent various cross-sections of solids.
Cross-Sections of Solids
You will need Plasticine and dental floss.
➤ Use the Plasticine to make four cones.Place the cones with their bases on the table.Suppose you were to cut the cone in each of these ways:• horizontally• vertically through its vertex• slanting• vertically, but not through its vertexSketch the figure you predict you would see after each cut.
➤ Use the dental floss to cut the cones.Record your findings.
Show and Share
Compare your results with those of another pairof students. How are the results the same?How are they different?
Math LinkScience
Magnetic resonance imaging (MRI)is a scan that generates cross-sectional images of the brain orother organs or body structures.
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Unit 3 Lesson 7 95
A cross-section is the 2-D face produced when a cut is made through a 3-D object.
➤ These pictures show the cross-sections of a pentagonal prism.
➤ These pictures show the cross-sections of a cylinder.
➤ These pictures show the cross-sections of a square pyramid.
Parallel to the base Vertically Slanting
Vertically Horizontally Slanting
You will need Plasticine and dental floss for question 3.
1. Name 2 figures found on the cross-sections of each solid.a) cube b) triangular prismc) square pyramid d) tetrahedron
A tetrahedronis a triangular pyramid
with 4 congruent faces.
Parallel to the base Vertically through the vertex
Vertically not through the vertex Slanting
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2. Describe how a hexagonal prism could be cut to produce each cross-section.a) a hexagon b) a rectangle
3. Use Plasticine to make 4 triangular prisms.a) Sketch the figures you predict you would see on each cross-section.
The prism is cut:i) parallel to its base
ii) parallel to one of its rectangular facesiii) slanting toward its baseiv) slanting toward a rectangular face
b) Use dental floss to cut the prism to check your answers to part a. Record your findings.
4. A student said:
a) Is this statement true? Explain.b) Is this statement true for pyramids? How do you know?
5. Explain how a square pyramid could be cut to produce each cross-section.a) a triangle b) a rectangle c) a trapezoid
6. Which 3-D solids have a cross-section that is an isosceles triangle?Use words and pictures to explain.
7. Are circles and ovals cross-sections of any prisms or pyramids?Use words and pictures to explain.
96 ASSESSMENT FOCUS Question 4Unit 3 Lesson x
Which cross-sections are easier to visualizethan others? Use words and pictures to explain.
Cutting off a vertex from any prism always produces a
triangular cross-section.
Mental Math
Which is greater:25% of $30 or 20% of $35?
Numbers Every Day
Unit 3 Lesson 7
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97LESSON FOCUS Make generalizations about the planes of symmetry of 3-D solids.
L E S S O N
Planes of Symmetry
You will need Plasticine, dental floss, and a ruler.
➤ Use Plasticine to make a cube.➤ Use dental floss to cut the cube into
2 congruent parts.➤ How many different ways can you cut
the cube to make 2 congruent parts?➤ Sketch each cut on a labelled view of the cube.
Show and Share
Compare your findings with those of another pair of students.Did you find all the different ways to cut the cube? Explain.
➤ A figure may have one or more lines of symmetry.
➤ Three-dimensional objects may also have symmetry.A solid has plane symmetry if a plane can divide the solid into 2 parts so that one part is the mirror image of the other.
This rectangle has 2 lines of symmetry.
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➤ A rectangular prism has 3 planes of reflective symmetry, as shown below.The lines of symmetry of each rectangle lie on the planes of symmetry of the prism.
➤ A square pyramid has 4 planes of symmetry.
98 Unit 3 Lesson 8
Two vertical planes cut through Two vertical planes cut throughthe midpoints of opposite sides. the diagonals of the base.Each plane makes 2 congruent Each plane makes 2 congruenttriangular faces. triangular faces.
You will need Plasticine for questions 1 and 7,and Snap Cubes for question 3.
1. Use Plasticine to make a square prism.Use dental floss to cut the prism along a plane of symmetry.a) Sketch the cross-section.b) How many planes of symmetry does the prism have?
2. How many planes of symmetry does each object have?a) b) c)
3. Use 10 Snap Cubes.Make an object that has:a) exactly one plane of symmetryb) exactly two planes of symmetryc) more than two planes of symmetry
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4. a) Compare the numbers of planes of symmetry of a square prism and a rectangular prism.Which prism has more planes of symmetry?Explain.
b) Does a cube have more or fewer planes of symmetry than each prism in part a?Explain.
5. Look at this vase.How many planes of symmetry does the vase have?Explain how you know.
6. How are the planes of symmetry of a cone and a cylinder related? Explain.
7. Use Plasticine and dental floss when you need to.a) How many planes of symmetry does a rectangular pyramid have?b) How many lines of symmetry does a rectangle have?c) Repeat parts a and b for a pyramid with a regular pentagon
as its base.d) Use the results of parts a, b, and c above, question 1, and question 2c.
What conclusions can you make about the planes of symmetry of a pyramid and the lines of symmetry of its base?
ASSESSMENT FOCUS Question 4
Do all prisms have at least one plane of symmetry?Use words and pictures to explain.
Number Strategies
Find the common factors ofthe numbers in each pair.
40, 7245, 6350, 8055, 132
Numbers Every Day
Unit 3 Lesson 8 99
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100
1. Owen says he can make an angle smaller by making the arms shorter.Do you agree? Why or why not?
2. a) Use a protractor to draw a 40° angle.b) Do not use a protractor.
Draw an angle that is 90° greater.c) Use a protractor to check the angle in part b.
3. Fold a piece of paper to make a 45° angle.What other angle have you made at the same time? Explain.
4. A quadrilateral has angles measuring 60°, 50°, and 120°.What is the measure of the 4th angle?How do you know?
5. A parallelogram has one 55° angle.Sketch the parallelogram.Explain how you can use the properties of a parallelogram to find the measures of the other angles.
6. Sketch a kite and a rectangle.List the properties of a kite that are the same as those of a rectangle.
7. Use square dot paper.Make a quadrilateral that has:a) three acute anglesb) exactly one right anglec) diagonals that bisect each otherName each quadrilateral if you can.
8. Consider the statement,“If a figure is a rhombus, it is also a square.”Is this statement true or false?How do you know?
Show What You Know
34
3
1
2
LESSON
Unit 3
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6
7
8
LESSON
9. Use 8 Snap Cubes to create a solid.Use grid paper to draw the top, front, back, left, and right views.Label each view.
10. Use Snap Cubes to build an object that has the views shown.Draw the back and left views.
11. How many different solids can you name that have these cross-sections?a) square b) rectangle c) circleSketch each solid you name and show the cross-section.
12. Use Plasticine to make 4 trapezoidal prisms.a) Sketch the figure you predict you would
see on each cross-section.The prism is cut:
i) parallel to its baseii) parallel to one of its rectangular faces
iii) slanting toward its baseiv) slanting toward a rectangular face
b) Use dental floss to cut the prisms to check your answers to part a. Record your findings.
13. a) Use Plasticine to make a tetrahedron.Use dental floss to find how many planes of symmetry a tetrahedron has.
b) Which figures make up the cross-sections of the planes of symmetry?
Learning GoalsU N I T
✓
✓
✓
✓
✓
✓
✓
estimate, measure, and drawangles using a protractormake and applygeneralizations about the sumof angles in triangles andquadrilateralsmake and applygeneralizations about diagonalproperties of quadrilateralssort quadrilaterals accordingto propertiesmake and interpretorthographic drawingscreate cross-sections fromsolidsmake generalizations aboutthe planes of symmetry of solids
Unit 3 101
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It’s a Slice
102 Unit 3
You will need:• Plasticine• dental floss
Use the Plasticine to make a prism, a pyramid, a cone,a cylinder, and a sphere.
Part 1
Try to make each cross-section by cutting the solids in different ways.• a hexagon• an octagon • a parallelogram that is not a rectangle• a circle • a square • an equilateral triangle • a rectangle that is not a square • a triangle that
is not equilateral • a pentagon
Record which figures you were able to create and how you created them.
Which figures were impossible to make?Explain why.
What conclusions might you make about the kinds of polygons that can be made from the cross-sections of a prism? A pyramid?
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Your work should showillustrations of your cross-sections with figures namedexplanations for the cross-sections that arenot possibleconclusions about the cross-sections that are possible
✓
✓
✓
Check List
Part 2
Try to make cross-sections with two or more of these properties by cutting the solids.• a triangle with at least two 60° angles• a figure with 4 lines of symmetry• a figure with perpendicular diagonals• a figure with diagonals that bisect each other• a figure whose angle sum is 360°• a figure with one pair of parallel sides• a figure with 2 lines of symmetry
Sketch and name the types of polygons you create.Identify the properties from the list above.
How are the properties of quadrilaterals related to the cross-sections and planes of symmetry of solids?Use diagrams in your explanation.
Unit 3 103
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