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Chapter 7: Polygons

Nelson Mathematics for Apprenticeship and Workplace 12

Nelson Mathematics for Apprenticeship and Workplace resources are comprehensive supplementary workbooks that are carefully designed to engage students in real-life contexts of mathematics.

Three components are available for Nelson Mathematics for Apprenticeship and Workplace 12:

Student Workbook• 300+ page workbook• Each lesson includes prompts, examples, and exercises scaffolded in

manageable steps• Predictable layout assists students with weak organizational skills• Written at an appropriate reading level for struggling students• Real-world connections embedded throughout• Supports 100% of the outcomes in the new curriculum

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For more information, visit www.nelson.com/wncpmath/apprenticeship

1


Table of Contents Chapter 1 Buying or Leasing a Vehicle Getting Started 1.1 Buying a New Vehicle 1.2 Buying a Used Vehicle 1.3 Operating Costs for a Vehicle 1.4 Who’s Buying What? Mid-Chapter Review 1.5 Leasing a Vehicle 1.6 Lease or Buy? 1.7 Vehicle Options and Technology Chapter ReviewChapter Test

Chapter 2 Measuring Instruments Getting Started2.1 Precision 2.2 Precision and Calculations 2.3 Solving a Measuring Puzzle Mid-Chapter Review 2.4 Precision and Accuracy 2.5 Uncertainty in Measurements Chapter Review Chapter Test

Chapter 3 Statistics Getting Started 3.1 Mean 3.2 Weighted Mean 3.3 Median 3.4 Mode 3.5 Which Score is Higher? Mid-Chapter Review 3.6 Interpreting Data 3.7 Percentiles Chapter Review Chapter Test

This Sampler contains Chapter 7

NEL Chapter 7 Polygons

2


Chapter 4 Linear Relations Getting Started 4.1 Describing Relations 4.2 Interpreting Linear Relations 4.3 Direct and Partial Relations Mid-Chapter Review4.4 Equations of Linear Relations 4.5 Creating a Number Trick 4.6 Scatter Plots 4.7 Scatter Plots and Technology Chapter Review Chapter Test

Chapter 5 Career Planning Getting Started 5.1 Exploring Career Options 5.2 Researching Your Career Choice 5.3 Planning for Training Costs 5.4 Writing a Resumé 5.5 Financing Your Lifestyle Chapter Project

Chapter 6 Operating a Small Business Getting Started 6.1 Business Opportunities 6.2 Business Expenses 6.3 Planning for Taxes 6.4 Sidewalk Sale Game Mid-Chapter Review6.5 Improving Profitability 6.6 Break-Even Point Chapter Review Chapter Test

Apprenticeship and Workplace 12 NEL

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Chapter 7 Polygons Getting Started 7.1 Triangles 7.2 Quadrilaterals 7.3 Creating Polygon Puzzles Mid-Chapter Review7.4 Regular Polygons 7.5 Applications of Polygons Chapter Review Chapter Test

Chapter 8 Transformations Getting Started 8.1 Translations 8.2 Reflections 8.3 Rotations Mid-Chapter Review8.4 Dilations 8.5 Dilations and Technology 8.6 Combining 2-D Transformations 8.7 Solving a Transformation Puzzle Chapter Review Chapter Test

Chapter 9 Trigonometry Getting Started 9.1 Exploring the Sine Law 9.2 Solving Sine-Law Problems 9.3 Reversing Triangle Puzzle Mid-Chapter Review9.4 Exploring the Cosine Law 9.5 Solving Cosine-Law Problems 9.6 Choosing the Sine Law or Cosine Law Chapter Review Chapter Test

NEL Chapter 7 Polygons

Apprenticeship and Workplace 12 NEL4


Chapter 10 Probability Getting Started 10.1 Experimental Probability 10.2 Theoretical Probability 10.3 Three-Cup Guessing Game Mid-Chapter Review10.4 Interpreting Odds 10.5 Making Decisions Chapter Review Chapter Test

Chapter 11 Owning a Home Getting Started 11.1 Qualifying for a Mortgage 11.2 Closing Costs 11.3 Mortgage Payments Mid-Chapter Review 11.4 Managing Housing Costs 11.5 Mortgages and Technology 11.6 Solving Map Puzzles Chapter Review Chapter Test

Glossary

7777777Polygons

Zahra is a beekeeper near Melfort. The cells in a honeycomb are hexagons. This makes it possible for the bees to pack a lot of honey into a small space. It also gives the honeycomb strength.

A. How can you tell if a shape is a hexagon?

B. Draw a 2-D shape that is not a hexagon. How is your shape the same as the hexagon drawn on the honeycomb? How is it different?

e.g., It has six straight sides and six vertices.e.g., It has six straight sides and six vertices.

e.g., Same: Both have straight sides. e.g., Same: Both have straight sides.

Different: My shape has three straight sides and three Different: My shape has three straight sides and three

vertices. The sides and the angles of my shape are not equal. vertices. The sides and the angles of my shape are not equal.

The sides and the angles of the hexagon are equal.The sides and the angles of the hexagon are equal.

B. e.g.,

Chapter 7 Polygons 161NEL

5NEL Chapter 7 Polygons

7777777 GettingGettingGettingGettingGettingGettingGettingGetting

1. A triangle is a polygon with three straight sides and three vertices. Use side lengths to classify the triangles in the picture of a crane below.

a) Which triangle is an equilateral triangle?

b) Which triangle is an isosceles triangle?

c) Which triangle is a scalene triangle?

2. Use angles to classify the triangles in the picture of a crane.

a) Which triangle is an acutetriangle?

b) Which triangle is an obtusetriangle?

c) Which triangle is a righttriangle?

3. Measure the side lengths and interior angles of the triangles below. Use millimetres for the side lengths. Record the measurements on the diagrams.

You will need• a millimetre ruler• a protractor

equilateraltriangle

a triangle with three equal sides

isoscelesisoscelestriangle

a triangle with exactly two equal sides

scalene triangle

a triangle with no equal sides

acute triangle

a triangle with each angle less than 908

obtuseobtuse triangletriangle

a triangle with one angle that is greater than 908

rightright triangletriangle

a triangle with one angle that is equal to 908

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4. Use the triangles in Question 3. What do you notice about the measure of the angle opposite the longest side in each triangle?

34 mm

34 mm

48 mm

4545°

4545°

1

35 mm 35 mm

35 mm

60°

60°

60°

2

60 mm

36 mm 36 mm

33°

114°

33°3

60 mm

36 mm

36 mm

33°

114°

33°

4

22 mm22 mm

65°

25°

55 mm

50 mm5

3

2

1

3

1

2

The largest angle is opposite the longest side.The largest angle is opposite the longest side.

162 Apprenticeship and Workplace 12 NEL


5. Which triangles in Question 3 match each description?

a) equilateral triangle:

b) scalene triangle:

c) obtuse triangle:

d) regular polygon:

6. a) Which two triangles in Question 3 are congruent?

b) Two angles in triangle 5 are complementary. What are the measures of these angles?

7. Use the marks on each shape. Fill in the blanks below.

a) I J

H K

side HI 5 side

side 5 side

b) B C

D

/ /

c) Q

P S

R

QR and PS are .

8. a) The diagram below shows a transversal crossing two parallel lines. Record the angle measures on the diagram. Do not measure the angles.

160°

160°20°

20°160°

160°20°

20°

b) What are the measures of two opposite angles in the diagram?

c) What are the measures of two supplementary angles in the diagram?

9. Dawn plans to install a ridge vent on a roof. This will cool the attic. The angle of the vent needs to equal /RST at the peak of the roof. RST at the peak of the roof. RSTDawn knows the measurements in the diagram.

a) What type of triangle is nRST? RST? RST

b) What is the measure of /RST? How do you know?RST? How do you know?RST

complementaryangles

two angles whose sum is 908

regular polygon

a closed shape with all sides equal and all angles equalangles equal

transversal

a line that intersects two or more lines

supplementaryangles

two angles whose sum is 1808

opposite angles

non-adjacent angles that are formed by two intersecting lines

30 ft 30 ft

38°38°R T

S

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3 and 4

2

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658 and 258

IJ

KJ

HKDBC DCB5DBC DCB5 /DBC DCB/

parallelparallel

e.g., 20e.g., 208 and 208 OR 1608 and 1608

1608 and 208

e.g., isoscelese.g., isosceles

The measure of /RST is 104RST is 104RST 8. The sum of the angles

in any triangle is 1808. 1808 2 388 2 388 5 1048



Triangles7777777.7.7.1111111You will need• string• scissors• plain paper• a millimetre ruler• a protractor

Try TheseMake a paper triangle. Draw a dot at each vertex. Cut the triangle so that Make a paper triangle. Draw a dot at each vertex. Cut the triangle so that each vertex is separate. Show that the sum of the angles is 180each vertex is separate. Show that the sum of the angles is 1808.

ReflecTinGSuppose that the sum of the lengths of the two shortest

sides is less than the length of the longest side. Can these three pieces of string make a triangle? Explain.

A C

B

M

Cut three pieces of string that you can use to make a triangle. How many different triangles can you make?

1 Place your string on paper to make a triangle. Mark the vertices with a pencil. Join the vertices.

2 What are the side lengths?

3 What are the angle measures?

4 Two triangles are different if they are not congruent. Are any different triangles possible with your side lengths?

5 Compare your triangle with other students’ triangles. Could anyone make more than one triangle?

Example 1The bamboo stems in this photograph create The bamboo stems in this photograph create an isosceles triangle. An isosceles triangle an isosceles triangle. An isosceles triangle has two equal sides called legs. The interior . The interior angles opposite the legs are also equal.

Do all isosceles triangles have these properties?

SolutionA. Find the midpoint of side AC.

Label it M. Draw MB.

B. What are the side lengths, in millimetres?

nABMnABMn :

nCBM:

property

a characteristic that is shared by all the members of a group

e.g., 128 mm, 175 mm, and 184 mme.g., 128 mm, 175 mm, and 184 mm

e.g., 66e.g., 668, 748, and 408

no

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no

1. e.g., 1. e.g.,

19 mm, 50 mm, and 47 mm

19 mm, 50 mm, and 47 mm



C. Is nABMnABMn congruent to ABM congruent to ABM nCBM? How do you know?

D. Kate said that this property is a property of all isosceles triangles. Do you agree with Kate? Explain. Include a diagram.• The angles opposite the equal legs are equal.

Example 2Pavlo is a carpenter. He uses triangular brackets for shelving. The sides of each bracket extend past the vertices to create exterior angles. What types of triangles have this property?• Each exterior angle is 908 or greater.

SolutionA. Measure the interior and exterior angles in the acute triangle

below. Record the angle measures on the diagram.

Acute triangle: Obtuse triangle:

B. Draw an obtuse triangle in Part A. Extend one side at each vertex to create three exterior angles. Measure the interior and exterior angles. Record the measures. Are any exterior angles acute?

C. Is the following a property of all triangles? Explain.• Each exterior angle is 908 or greater.

D. What triangles have the property in Part C?

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150°

120°

90°

ReflecTinGWhy is showing that something

is not a property easier than

showing that it is a property?

35° 110°

145°

70°

75°

105°

Use the triangular bracket above as an example of a right triangle.

Hint

Yes. e.g., They are congruent because only one triangle is Yes. e.g., They are congruent because only one triangle is

possible with these sides. OR They are the same size and possible with these sides. OR They are the same size and

shape.shape.

e.g., Yes, I agree. If you draw a centre line, you get two e.g., Yes, I agree. If you draw a centre line, you get two

congruent triangles. So the corresponding angles are equal.congruent triangles. So the corresponding angles are equal.

yesyes

D FM45° 45°

35 mm

20 mm 20 mm

E

D. e.g.,

e.g.,

No. e.g., One exterior angle on the obtuse triangle is less No. e.g., One exterior angle on the obtuse triangle is less

than 908.

acute triangles and right trianglesacute triangles and right triangles

28°

160°

152°

132°20°48°



Practice 1. Use your triangles from Example 2.

a) The sum of the interior angle plus the exterior angle is the same at each vertex. What is this sum?

b) Why does it make sense that each vertex has the same sum?

When you extend one side, you create angles that form . Angles that form have a sum of .

c) Is this a property for all triangles? Explain.• The sum of the interior angle plus the exterior angle

is 1808.

2. Cables on the Esplanade Riel Bridge in Winnipeg illustrate many types of triangles.

Circle the types of triangles that have each property.

a) Some sides are equal.

equilateral triangle isosceles triangle scalene triangle

b) Some exterior angles are equal.


c) No interior angles are equal.


d) All three exterior angles are 908 or greater.

acute triangle obtuse triangle right triangle

e) Each exterior angle is equal to the sum of the interior angles at the other two vertices.

acute triangle obtuse triangle right triangle

3. a) What is one property of isosceles triangles that is not a property of all triangles?

b) What is one property of isosceles triangles that is a property of all triangles?

ReflecTinGDoes it matter which side of a triangle you

extend to make an exterior angle?

Explain.

Use the diagrams and definitions of different types of triangles in Getting Started.

Hint

1808

twoa straight linea straight line a straight linea straight line

1808

Yes. e.g., You always create an exterior angle by extending

a side. The interior angle and exterior angle will always

form a straight line.

e.g., Isosceles triangles have exactly two equal sides.

e.g., The sum of the interior angles is 1808.166 Apprenticeship and Workplace 12 NEL


4. Use the angle measures to calculate the unknown angles in each triangle. Include interior angles and exterior angles. Record the measurements on the diagrams.

137°

75°

148°

43°105°

1

32°

60°

30°

30°

150°

150°

2

120°

125°

145°

160°

35°3

55°

20°

5. Use the triangles in Question 4. Complete this chart.

TriangleSum of 3 interior

anglesSum of 3 exterior

angles

Sum of 3 interiorangles 1 sum of3 exterior angles

1

2

3

6. Marcel’s crew builds A-frame cabins in Tofino.• The balcony is parallel to the base of a cabin. • The front of this cabin is an equilateral triangle. • The section above the balcony is also an

equilateral triangle.

Marcel wonders about this question.• Does drawing a line parallel to the base of any

triangle create a second triangle with angles that are equal to those in the original triangle?

a) Test Marcel’s idea. • Draw a triangle. Draw a line through your triangle so

that the line is parallel to the base.

Are the angles in the small triangle equal to the angles in the large triangle?

b) Compare your results with a classmate’s results. Did your classmate get the same results?

c) Will adding a line that is parallel to the base always create a smaller triangle with the same angles? Explain.

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60°

60° 60°

60°60°

ReflecTinGDo you think that

the sum of the interior angles and the exterior angles is the same for all triangles? Explain.

One way to draw parallel lines is to draw along both sides of a ruler.

Hint

180818081808

360836083608

540854085408

yesyes

yesyes

Yes. e.g., One angle is shared by both triangles. The other two angles are corresponding

angles, formed by transversals that meet the parallel lines at the same angle. So each

angle in the small triangle has a matching equal angle in the large triangle.

65°

25°

25°

6. a) e.g.,



Brady installs stained-glass windows in Victoria. How can you describe the quadrilaterals in this window?

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square

rectangletrapezoid

rhombus

parallelogramkite

triangle (not aquadrilateral)

1 Label an example of each convex quadrilateral in Brady’s window. Draw an arrow to the quadrilateral.

Convex quadrilaterals

A rectangle has four 908 angles.

A square is a rectangle with four equal sides.

A parallelogram has opposite sides that are parallel and equal.

A rhombus is a parallelogram with four equal sides.

A trapezoid has only one pair of parallel sides.

A kite has two pairs of equal sides that are not opposite sides. If all four sides are equal, the quadrilateral is a rhombus.

2 Some polygons have more than one name. What are three other names for a square?

3 Label a polygon in the window that is not a quadrilateral. How do you know that it is not a quadrilateral?

Quadrilaterals7.27.27.27.27.27.27.27.27.27.27.27.27.27.27.27.27.27.2You will need• coloured pencils• a millimetre ruler • a protractor

Try These

Circle the polygons.Circle the polygons.

quadrilateral

a polygon with four straight sides and four vertices

convex

a polygon with no interior angles that are greater than 1808

e.g., A quadrilateral has four sides and four vertices. e.g., A quadrilateral has four sides and four vertices.

This triangle has three sides and three vertices.iangle has three sides and three vertices.

rectangle, parallelogram, and rhombusrectangle, parallelogram, and rhombus

e.g.,



Example 1Elena is a pastry chef in Fort Qu’Appelle. She cut these square pastries along a diagonal. This makes two congruent triangles. What other quadrilaterals have this property?

SolutionA. Draw diagonals to form triangles. Use a different colour for

each diagonal in each quadrilateral.

squarerectangle parallelogram kite trapezoid

irregularquadrilateral

rhombus

concavequadrilateral 1

concavequadrilateral 2

B. Name quadrilaterals with each property.

Property Quadrilaterals

Both diagonals make congruent triangles.

One diagonal makes congruent triangles.

No diagonals make congruent triangles.

Example 2Tessa is a carpenter in Whitehorse. She needs to check that a window frame is a rectangle. She only has a tape measure. How can she use the properties of a rectangle to check?

SolutionA. Measure the side lengths and diagonals in each quadrilateral.

Record them on the diagrams.

B. How can Tessa use the properties of a rectangle?

If the window frame is a rectangle, the opposite sides are and the diagonals are .

30 mm

30 mm

20 mm20 mm 36 mm36 mm36 mm

30 mm

30 mm

20 mm

30 mm

41 mm

20 mm

ReflecTinGHow do you know

if each result is a property of all quadrilaterals

that are the same type? Use side

lengths to explain.

diagonal

a line segment joining opposite vertices

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irregular

a quadrilateral with different side lengths and different angle different angle measures

concave

a polygon with an interior angle that is greater than 1808

rectangle, parallelogram, rhombus, and square

kite and concave quadrilateral 1

trapezoid, irregular quadrilateral, and concave quadrilateral 2

equalequal equalequal



Practice 1. Which convex quadrilaterals have each property?

a) two pairs of equal sides:

b) four right angles:

c) equal diagonals:

d) equal angles at opposite vertices:

2. How can you use properties to show that a rectangle is a parallelogram? A rectangle is a parallelogram because it has

.

3. The diagonals in a square are perpendicular. The diagonals cross at their midpoints. Record the names of other types of quadrilaterals to complete the chart.

PropertyA quadrilateral with

this propertyA quadrilateral without

this property

Diagonals are perpendicular.

Diagonals cross at their midpoints.

4. a) Show that the exterior angles of all quadrilaterals have the same sum.• At each vertex, the measure of the

interior angle plus the measure of the exterior angle equals .

• There are vertices. The total sum of all the interior and exterior angles of a quadrilateral is (( )) 5 .

• The sum of the interior angles is .• The sum of the exterior angles is 2 5 .• The sum of the exterior angles of any quadrilateral

is .

b) Do all quadrilaterals have diagonals that are perpendicular? Explain.

Draw some quadrilaterals. Measure the exterior angles to test your answer for Question 4.

Hint

Use diagrams from Example 1 to help you with Question 3.

Hint

rectangle, square, parallelogram, rectangle, square, parallelogram,

and rhombus

rectangle and squarerectangle and square

rectangle and squarerectangle and square

rectangle, square, rectangle, square,

parallelogram, and rhombusparallelogram, and rhombus

e.g., kite e.g., parallelogram

e.g., rhombus e.g., kitee.g., rhombus e.g., kite

1808

four1808 72084

3608

7208 3608 3608

3608

opposite sides that are parallel and equalopposite sides that are parallel and equal

No. e.g., The parallelogram, trapezoid, and irregular quadrilateral

in Example 1 do not have diagonals that are perpendicular.



5. Jay makes picture frames. The interior angles of the square picture frame have a sum of 3608. How can you draw a diagonal to show that all quadrilaterals have this property?

6. An isosceles trapezoid is cut from an isosceles triangle. It has two equal sides and two parallel sides. The parallel sides are called bases.

a) Use the isosceles triangle at the right. Draw a line parallel to the base to make an isosceles triangle. Measure to check that there are two equal side lengths in the trapezoid.

b) What are two properties of your isosceles trapezoid?

c) Compare your isosceles trapezoid with isosceles trapezoids drawn by your classmates. What is one property that is shared by all the isosceles trapezoids?

7. When you join the midpoints of all four sides of a quadrilateral, you always get the same type of polygon. Test some quadrilaterals. What is the polygon?

8. Darcy is cutting glass for a window that is a rhombus. She knows that the sides are 10 in. long.

Does she have all the information she needs to cut the glass? Explain. Include a diagram.

Think about • side lengths• angle measures• diagonals

Hint

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4(90°) � 360°

10 in.10 in.

e.g., I looked at one diagonal in each quadrilateral

in Example 1. You get two triangles. The sum of the

interior angles of each triangle is 1808. The sum of the

interior angles of both triangles is 2(1808) 5 3608.

e.g., one pair of equal sides and two pairs of equal

angles OR equal diagonals

e.g., one pair of equal sides OR two pairs of congruent

angles OR two different angles that add to 1808 OR two

isosceles triangles and two congruent triangles that are

formed where the diagonals meet

No. e.g., She also needs to know the angle measures. There are

many rhombuses such as a square with the same side lengths.

7. e.g.,

parallelogramparallelogram

8. e.g.,



The convex polygons below are made of squares and triangles.

pentagon hexagon

A. Use 12 toothpicks to make a different convex hexagon with squares and triangles. Draw your hexagon.

B. Use 12 toothpicks to make four other convex shapes. Make each shape out of smaller polygons. Draw your shapes.

C. Create a puzzle about a convex polygon made of smaller polygons. • Tell how many toothpicks to use. • Name the shape. • Draw the solution.

Trade puzzles with a partner. Solve each other’s puzzles.

creating Polygon Puzzles7.37.37.37.37.37.37.37.37.37.37.37.37.3You will need• 12 toothpicks

Here are names of polygons:

5 sides: pentagon6 sides: hexagon7 sides: heptagon8 sides: octagon9 sides: nonagon

10 sides: decagon12 sides:

dodecagon

Hint

e.g.,

e.g., Use 14 toothpicks to make a pentagon. Draw your

pentagon.

CrowleArt Group

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CrowleArt Group

CrowleArt Group

square pentagon pentagon parallelogram

e.g.,

C. e.g.,



1. Ty says that if you know all the angle measures in a triangle, you will know whether the sides are equal. Do you agree? Explain. Use diagrams.

2. Is each property true or false? Use diagrams to explain your answers.

a) The midpoints of the three sides of an equilateral triangle are joined to form four small triangles. The area of each small triangle is 14 of the area of the original equilateral triangle.

b) The diagonals of a convex quadrilateral always create two pairs of equal angles where they intersect.

3. Hayley is the lighting director for a theatre in Manitoba. She wants to place a spotlight so that it shines on the centre of a rectangular stage.

How can Hayley use a property of rectangles to find the centre of the stage? Use the rectangle at the right in your explanation.

Mid-chapter

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Yes. e.g., If three angles are equal, then the triangle is an

equilateral triangle with three equal sides. If two angles are

equal, the triangle is an isosceles triangle with two equal sides.

True. e.g., I tested some equilateral triangles. When

I joined the midpoints, I made four small congruent

triangles. So each triangle has 14 of the area of the

original equilateral triangle.

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True. e.g., The diagonals are straight lines. Where two straight

lines cross, they always create two pairs of equal angles.

e.g., Hayley can use string or tape to mark the diagonals. The

diagonals will cross at the centre of the stage. OR Hayley

can mark the midpoint of each side and join each midpoint

to the opposite midpoint. The lines will cross in the centre of

the stage.

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1. e.g.,

2. b) e.g.,

2. a) e.g.,


07_AW12_Ch07.indd 173 1/9/12 10:42 PM


Pavithra uses regular polygons to make wooden trays. Which regular polygons are in this tray?

1 Find a triangle in this tray. Is it a regular polygon? Explain.

2 Name all the regular polygons in this tray.

Example 1Olivia designs and sells cloth potholders. To make the design at the right, she sewed a light-coloured hexagon on a dark-coloured hexagon. The dark hexagon is slightly larger. The light hexagon is a smaller similar polygon.

How can Olivia use a tracing of the dark hexagon to make a model of the light hexagon?

SolutionA. Mark the midpoint of each side. Draw

straight lines to join each midpoint to the next midpoint.

B. How do you know the shapes are similar?

Regular Polygons7.47.47.47.47.47.47.47.47.47.47.47.47.4Use six toothpicks. Create an irregular hexagon. Use six toothpicks. Create an irregular hexagon. How do you know that your hexagon is not regular? How do you know that your hexagon is not regular? Use a diagram to explain.Use a diagram to explain.

Try These

similar polygons

polygons that are congruent, or enlargements or reductions of each other; the ratios of corresponding sides are equal and the corresponding angles are equal

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You will need• 6 toothpicks• a ruler• a protractor• a compass

No. e.g., Only two sides are equal.No. e.g., Only two sides are equal.

e.g., The interior angles are not equal.e.g., The interior angles are not equal.e.g., The interior angles are not equal.e.g., The interior angles are not equal.

square, octagon, pentagon, and hexagonsquare, octagon, pentagon, and hexagon

e.g.,

irregular

e.g., The original shape has six equal sides and equal interior e.g., The original shape has six equal sides and equal interior

1208 angles. The new shape also has these properties, but its angles. The new shape also has these properties, but its

side lengths are smaller.side lengths are smaller.174 Apprenticeship and Workplace 12 NEL


C. Test these regular polygons. Does the midpoint reducing method always work to make a similar smaller polygon?

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D. Will all regular polygons have this property? Explain.• Joining the midpoints of each side of a regular polygon

creates a similar regular polygon.

Example 2Craig designs and makes signs in Regina. A customer wants a sign that is a regular pentagon.

How can Craig determine the angle measures for the sign?

SolutionA. Sketch a convex pentagon. It can be regular or irregular.

B. Draw diagonals from one vertex to divide your pentagon into triangles. How many triangles did you make?

C. What is the sum of all the interior angles of your pentagon?

(1808) 5

The sum is .

D. What angle measure should Craig use?

4 5 5

Craig should use .

ReflecTinGGabriel drew a

square. He says that all squares

are similar to the square he drew. Do you agree?

Explain.

ReflecTinGCan you always divide a shape into triangles to

determine its angle measures? Is

this a property of regular polygons?

Explain.

yes yes yes

Yes. e.g., The number of midpoints matches the number of

vertices, so the polygons are the same type. The distance

between the midpoints does not change, so both polygons

have equal sides. The angle from one midpoint to the next

does not change, so both polygons have equal angles.

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A. e.g.,

three

3 5408

5408

5408 1088

1088


07_AW12_Ch07.indd 175 1/9/12 10:42 PM


Practice 1. Billy engraves names on ID bracelets. The bracelets are made

in these two shapes. Are they regular polygons? Explain.

a)

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b)

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2. Draw diagonals from one vertex of each shape to divide the shape into triangles. Use the triangles to complete the chart.

ReflecTinGAs the number of sides in a polygon

increases, the polygon looks more and more

like a circle. What happens to the interior angles?

Property

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Hexagon

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Heptagon

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Octagon

Number of triangles

Sum of all the angle measures

Measure of each interior angle

Measure of each exterior angle

Sum of the measures of all the exterior angles

3. Use your chart from Question 2.

a) What happens to the number of triangles when you add one side to a polygon?

b) How many triangles can you make in a 12-sided polygon?

c) What size are the interior angles of a 12-sided regular polygon?

d) How does the number of triangles you can make in any polygon relate to the number of sides?

e) What is the measure of each angle in a regular decagon?

No. e.g., The interior angles are not equal. No. e.g., The sides are not equal.

4

7208

1208

608

3608

5

9008

128.571 …8

51.428 …8

3608

6

10808

1358

458

3608

It increases by 1.

10

1508

number of triangles 5 number of sides 2 2

(1808)(10 2 2) 4 10 5 1448


07_AW12_Ch07.indd 176 1/9/12 10:42 PM


4. Draw all the diagonals in each regular polygon. How many diagonals does each polygon have?

4 sides 5 sides 6 sides 7 sides

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diagonals diagonals diagonals diagonals

5. Use the polygons in Question 4. Test the following properties of regular polygons. Decide whether each property is true or false.

a) If the number of vertices is odd, the number of diagonals is odd.

b) If the number of vertices is even, the diagonals that connect opposite vertices intersect at the centre.

c) The number of diagonals you can draw from one vertex of a regular polygon is n 2 3, where n is the number of vertices.

6. Regular octagons are often used for the Chinese New Year. The number 8 is associated with wealth and good luck.

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The perspective was deleted.

a) Draw diagonals to join pairs of opposite vertices in this octagon. What is the measure of each angle where the diagonals meet?

b) What type of triangle do the diagonals make?

c) Place one end of a compass at the centre. Place the other end at a vertex. Draw a circle. What happens?

ReflecTinGWhich properties

of regular octagons are shared by all

regular polygons?

ReflecTinGUse one part in

Question 5 where you wrote “false” for the answer. Explain why it

is false.

2 5 9 14

false

true

true

3608 4 8 5 458

isosceles

The circle touches all the vertices.


07_AW12_Ch07.indd 177 1/9/12 10:42 PM


Angela creates designs with floor tiles. To cover a floor with no gaps, the tiles must fit together so that the angle measures have a sum of 3608. Create a design that Angela could use.

1 Draw a design that could be made with floor tiles. Use at least two different polygons. The polygons must fit together so that the angle measures have a sum of 3608. Record angle measures with a sum of 3608.

ExampleJordan is a machinist in Yellowknife. This chart shows a method he uses to space bolt holes at equal distances around a circle.

How can Jordan use a regular pentagon to space five holes at equal distances around a wheel?

Applications of Polygons7.57.57.57.57.57.57.57.57.57.57.57.57.5Look around your classroom.Look around your classroom.i) Where do you see a polygon? Name this polygon. do you see a polygon? Name this polygon.

ii) Describe some Describe some properties of this polygon.

Try TheseYou will need• a millimetre ruler• a protractor• a compass

To get this many holes around a circle …

… multiply the diameter of the circle by the number below. The result is the side length of a regular polygon with vertices at the locations of the holes.

3 0.8660

4 0.7071

5 0.5878

6 0.5000

8 0.3827

10 0.3080

12 0.2588

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360°

e.g., The top of e.g., The top of

my desk is a rectangle.my desk is a rectangle.my desk is a rectangle.my desk is a rectangle.e.g., It has two pairse.g., It has two pairs

of equal sides. The equal sides are parallel. All four interior of equal sides. The equal sides are parallel. All four interior of equal sides. The equal sides are parallel. All four interior of equal sides. The equal sides are parallel. All four interior

angles are right angles.angles are right angles.angles are right angles.angles are right angles.

e.g.,

120°

120°60°60°45°°

45°90°

45°45°

90°

90°

75°

120°75°



SolutionA. Use a compass to construct a circle. What is the diameter of

your circle?

The diameter of the circle is mm.

B. Use the chart. Multiply the diameter by the number for five holes. The product is the side length of a regular pentagon.

( mm)(0.5878) 5 mm

The side length is mm.

Practice 1. A machinist wants to space eight bolt holes at equal distances

around a wheel. The diameter of the wheel is 60 cm.

a) What regular polygon can the machinist use to locate the holes?

b) How long will the sides of the polygon be, to the nearest millimetre?

2. Elaine is looking at floor-tile designs. Each design is made by repeating a regular polygon. The polygon in each design is different.

a) What is one regular polygon that could be used to make a floor-tile design? Explain.

b) What is one regular polygon that could not make a tiling design? Explain.

ReflecTinGThe

measurements in this example

are given in millimetres. When would a machinist have to measure

much more precisely?

Use the charts inside the back cover for converting units.

Hint

24 mme.g.,

e.g., 40e.g., 40

e.g., 40e.g., 40

regular octagonregular octagon

(60 cm)(0.3827) 5 22.962 cm

The sides of the polygon will be about 230 mm long.

e.g., A square; the interior angle is 908, and (4)(908) 5 3608.

e.g., A regular octagon; the interior angle is 1358. Regular octagons do not fit

together so that the sum of the angle measures is 3608.

e.g., 23.512e.g., 23.512

e.g., 24e.g., 24



3. Cameron is a painter in Cambridge Bay. He knows that 1 L of paint covers about 50 sq ft. How many litres of paint will Cameron need to buy for this attic wall?

a) Separate a polygon into triangles by drawing diagonals. Draw diagonals WU and WU and WU WT in the diagram of the wall. WT in the diagram of the wall. WT

b) What is the area of each triangle?

Triangle DimensionsTriangle Dimensions Area

nWST Base 5

Height 5Area 5

12

(base)(height)

512

(28.0 ft)(12.0 ft)

5

nWTU Base 5

Height 5

nWUV Base 5

Height 5

c) What is the total area of pentagon STUVW?STUVW?STUVW

d) How many litres of paint will Cameron need to buy?

4. Alex makes signs in Moose Jaw. The owner of a pie company wants a slice of pie on a circular sign. He wants each vertex of the triangle to be on the circle.

a) Find the midpoint of each side of the triangle at the left. Use a protractor. Draw a perpendicular line through each midpoint. Extend the perpendicular lines so they meet.

b) Place a compass where the lines meet. Use this as the centre. Draw the circle for the sign.

5. Triangles are useful for building bridges because they provide support. What are three other places you see triangles in construction or industry?

4. 4. 4.

12.0

ft

28.0 ft

T U

S W

V

9.0

ft

6.7 ft22.0 ft

For nWUV, use 9.0 ft as the base. Subtract 22.0 ft from 28.0 ft to get the height.

Hint

e.g., roofs of houses, hydro towers, cranes

168.0 sq ft 1 132.0 sq ft 1 27.0 sq ft 5 327.0 sq ft

The total area is 327.0 sq ft.

327 sq ft 4 50 sq ft/L 5 6.54 L

He will need to buy 7 L.

1

2(base)(height) 5

1

2(9.0 ft)(6.0 ft)

5 27.0 sq ft

1

2(base)(height) 5

1

2(22.0 ft)(12.0 ft)

5 132.0 sq ft

28.0 ft

168.0 sq ft168.0 sq ft

22.0 ft

9.0 ft

12.0 ft

6.0 ft

12.0 ft



1. Circle the polygons.

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2. Name the quadrilaterals in Question 1.

3. Name the regular polygons in Question 1.

4. Sketch a polygon that has each property below. Classify each polygon.

a) Each interior angle is 608.

b) There are five equal sides.

5. Look at the polygons in the photographs at the right. One of the polygons has this property:• You can draw a line through the polygon so that the line

is parallel to the base to create a smaller polygon. The original polygon and the smaller polygon are similar.

Which polygon has this property? Explain. Include drawings.

chapter

square, parallelogram, rectangle, concave quadrilateral, and trapezoid

square and regular pentagon

equilateral triangle

e.g., regular pentagon

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shortersidelengths

same sidelength asoriginalrectangle

60°

60°

60°

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4. a) e.g.,

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4. b) e.g.,

e.g.,

The triangle has this property. e.g., Drawing a line through the triangle gives a smaller

triangle with the same angles. The triangles are similar. When you draw a line through

the rectangle, you get smaller rectangles. One side is equal to a side in the original

rectangle; the other side is not. The rectangles are not similar.

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sharedangle

same angles in both triangles


07_AW12_Ch07.indd 181 1/9/12 10:43 PM


6. Is each property true or false? Draw a diagram on plain paper to show your answer.

a) If some interior angles of a polygon are equal, some exterior angles are equal.

b) If a polygon has equal angles, it also has parallel sides.

c) Increasing the number of sides in a regular polygon decreases the measure of the interior angles.

7. Ryan is a landscaper in Brandon. He is going to sod a lawn in the shape of quadrilateral PQRS at the left.

a) What is the area that Ryan will cover with sod?

b) What property of polygons did you use to solve the problem in Part a)?

8. Kylie designs and sews quilts. Describe a property of polygons that she might use in a quilt. Include a diagram with your description.

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50 ft15 ft

50 ftP

S R

Q

True. e.g., When you add the interior angle and exterior angle

at a vertex, the sum is always 1808. If two interior angles are

equal, then the corresponding exterior angles are equal.

False. e.g., An isosceles triangle has equal angles but no

parallel sides.

False. e.g., As the number of sides increases, the angle

measure also increases.

e.g., When you join the midpoints of the sides of a regular

polygon, you get a smaller similar polygon.

e.g., If you draw a diagonal in a quadrilateral, you get two triangles.

e.g., AreanQRS

5 1

2 (50 ft)(50 ft)

5 1250 sq ft

AreanPQS

5 1

2 (50 ft)(15 ft)

5 375 sq ft

1250 sq ft 1 375 sq ft 5 1625 sq ft

Ryan will cover an area of 1625 sq ft with sod.

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60°

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90°

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120°

6. a) e.g.,

6. b) e.g.,

6. c) e.g.,

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8. e.g.,


07_AW12_Ch07.indd 182 1/9/12 10:43 PM


1. Make a triangle by joining three cities on the map.

a) What are the side lengths of your triangle, in millimetres?

b) What are the angle measures of your triangle?

c) What are two names for the type of triangle you made?

d) What is one property that your triangle shares with all triangles?

e) What is one property that your triangle does not share with some triangles?

2. Draw a rectangle by joining Yellowknife, Baker Lake, Churchill, and Fort McMurray.

a) What are the side lengths, in millimetres?

b) What are the angle measures?

c) What are two names for the quadrilateral you made?

d) What is one property that this quadrilateral shares with another type of quadrilateral? Use diagrams.

Chapter

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Iqaluit

Edmonton

Halifax

Regina

Whitehorse

Winnipeg

Churchill

BakerLakeYellowknife

FortMcMurray

Charlottetown

Moncton

St. John’sFredericton

Toronto

QuébecCity

Ottawa

Victoria

N

0 1000 km500

e.g., 1308, 308, and 208

18 mm, 18 mm, 11 mm, and 11 mm

e.g., obtuse triangle and

scalene triangle

e.g., It does not have equal sides or equal angles.

e.g., 16 mm, 20 mm, and 34 mm

e.g., The sum of

the interior angles is 1808.

908

rectangle and parallelogram

e.g., It has four right angles, like a square. OR When you

draw a diagonal in a rectangle, you always get two

congruent triangles. A rhombus has the same property.

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2. d) e.g.,

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07_AW12_Ch07.indd 183 1/9/12 10:57 PM


e) What is one property that other quadrilaterals have, but this quadrilateral does not have? Use diagrams.

3. Claude says that a polygon is a regular polygon if all of its sides are equal. Do you agree? Include a diagram.

4. a) Describe and illustrate two properties of regular decagons. Include diagrams.

b) Describe one property that some other regular polygons have, but a regular decagon does not have. Include diagrams.

5. Andy is paving a walkway in Airdrie. He is using stones that are regular polygons, but they are different shapes.

Can he use the three shapes at the right to pave the walkway without leaving gaps between the stones? Explain.

6. Melissa is installing a square skylight. She does not have a protractor. How can she check that the hole she cut for the skylight is a square?

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No. e.g., A polygon can have equal sides and different

interior angles. To be regular, a polygon must have equal sides

and equal interior angles.

e.g., All the interior angles measure 1448. If you join

opposite vertices, all the diagonals cross in the centre.

e.g., An equilateral triangle has acute interior angles, but a

regular decagon does not.

Yes. e.g., The side lengths are equal.

The stones fit together so that the

sum of the angle measures is 3608: 608 1 1208 1 908 1 908 5 3608.

e.g., She can measure the four side lengths and the diagonals. If the hole

is square, the side lengths will be equal and the diagonals will be equal.

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e.g., A trapezoid has two different angles, but the angles

in my rectangle are all equal. OR If you extend the sides

of a trapezoid, you get a triangle. My rectangle does not

have this property.

3. e.g.,

4. a) e.g.,

4. b) e.g.,

5. e.g.,

2. e) e.g.,

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144°

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Angles are

all 60°.Angles are

all 144°.


07_AW12_Ch07.indd 184 1/9/12 10:43 PM


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WORKBOOK SAMPLERolygons - · PDF filefor Apprenticeship ... breeze, with hundreds of multiple choice, ... Chapter 2 Measuring Instruments Getting Started 2.1 Precision