WORKBOOK SAMPLER Chapter 7: Polygons - Nelson Education

WORKBOOK SAMPLER 

Chapter 7: Polygons

Nelson Mathematics 

for Apprenticeship 

and Workplace 12 

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

Nelson Mathematics for Apprenticeship and Workplace 12 

Table of Contents 

Chapter 1 

Getting Started 

Buying or Leasing a Vehicle 

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 Review 

Chapter Test 

This Sampler contains 


Chapter 2 Measuring Instruments 


2.1 Precision 

2.2 Precision and Calculations 

2.3 Solving a Measuring Puzzle 


2.4 Precision and Accuracy 

2.5 Uncertainty in Measurements 



Chapter 3 Statistics 


3.1 Mean 

3.2 Weighted Mean 

3.3 Median 

3.4 Mode 

3.5 Which Score is Higher 


3.6 Interpreting Data 

3.7 Percentiles 



NEL 

Chapter 7 Polygons 

1


Chapter 4 Linear Relations 


4.1 Describing Relations 

4.2 Interpreting Linear Relations 

4.3 Direct and Partial Relations 


4.4 Equations of Linear Relations 

4.5 Creating a Number Trick 

4.6 Scatter Plots 

4.7 Scatter Plots and Technology 



Chapter 5 Career Planning 


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 


6.1 Business Opportunities 

6.2 Business Expenses 

6.3 Planning for Taxes 

6.4 Sidewalk Sale Game 


6.5 Improving Profitability 

6.6 Break-Even Point 



2 

Apprenticeship and Workplace 12 

NEL




Polygons 

7.1 Triangles 

7.2 Quadrilaterals 

7.3 Creating Polygon Puzzles 


7.4 Regular Polygons 

7.5 Applications of Polygons 



Chapter 8 Transformations 


8.1 Translations 

8.2 Reflections 

8.3 Rotations 


8.4 Dilations 

8.5 Dilations and Technology 

8.6 Combining 2-D Transformations 

8.7 Solving a Transformation Puzzle 



Chapter 9 Trigonometry 


9.1 Exploring the Sine Law 

9.2 Solving Sine-Law Problems 

9.3 Reversing Triangle Puzzle 


9.4 Exploring the Cosine Law 

9.5 Solving Cosine-Law Problems 

9.6 Choosing the Sine Law or Cosine Law 



NEL 


3




Probability 

10.1 Experimental Probability 

10.2 Theoretical Probability 

10.3 Three-Cup Guessing Game 


10.4 Interpreting Odds 

10.5 Making Decisions 



Chapter 11 Owning a Home 


11.1 Qualifying for a Mortgage 

11.2 Closing Costs 

11.3 Mortgage Payments 


11.4 Managing Housing Costs 

11.5 Mortgages and Technology 

11.6 Solving Map Puzzles 



Glossary 

4 

Apprenticeship and Workplace 12 

NEL

Polygons 

7 

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 

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

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 

B. e.g., 

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

Different: My shape has three straight sides and three 

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

The sides and the angles of the hexagon are equal. 

NEL 

NEL 

Chapter 7 Polygons 161 


5

7 Getting 

You will need 

• a millimetre ruler 

• a protractor 

equilateral 

triangle 

a triangle with 

three equal sides 

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 

1 

2 

3 

isosceles 

triangle 


exactly two equal 

sides 

scalene triangle 

a triangle with no 

equal sides 

2. Use angles to classify the triangles 

in the picture of a crane. 

a) Which triangle is an acute 

triangle 3 

b) Which triangle is an obtuse 

triangle 1 

c) Which triangle is a right 

triangle 2 

3 

1 

2 

acute triangle 


each angle less 

than 908 

obtuse triangle 


one angle that is 

greater than 908 

right triangle 


one angle that is 

equal to 908 

3. Measure the side lengths and interior angles of the triangles 

below. Use millimetres for the side lengths. Record the 

measurements on the diagrams. 

34 mm 

34 mm 

1 

45° 

35 mm 35 mm 

2 

48 mm 

60° 

60° 

35 mm 

60° 

60 mm 

60 mm 

AW12SB 

33° 

33° 

0176519637 

45° 

3 

FN 36 mmC07-F02-AW12SB 

114° 36 mm 

CO 

CrowleArt Group 

Technical 

Pass 

3rd pass 

Approved 

Not Approved 

36 mm 

36 mm 

55 mm 

4. Use the triangles in Question 3. What do you notice about the 

measure of the angle opposite the longest side in each triangle 

33° 

4 

114° 

33° 

The largest angle is opposite the longest side. 

65° 

22 mm 

5 

25° 

50 mm 

162 Apprenticeship and Workplace 12 NEL 

6 Apprenticeship and Workplace 12 NEL

5. Which triangles in Question 3 match each description 

a) equilateral triangle: 2 

b) scalene triangle: 5 

c) obtuse triangle: 3 and 4 

d) regular polygon: 2 

6. a) Which two triangles in Question 3 are congruent 3 and 4 

b) Two angles in triangle 5 are complementary. 

What are the measures of these angles 658 and 258 

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

a) I J 

b) B C 

c) Q 

regular polygon 

a closed shape 

with all sides 

equal and all 

angles equal 

complementary 

angles 

two angles whose 

sum is 908 

R 

H 

K 

side HI 5 side 

side 

IJ 

5 side 

KJ 

HK 

8. a) The diagram below shows a transversal crossing two 

parallel lines. Record the angle measures on the diagram. 

Do not measure the angles. 

D 

/ DBC 

5 / 

DCB 

P 

S 

QR and PS are 

parallel . 

transversal 

a line that 

intersects two or 

more lines 

20° 

160° 

160° 

20° 

20° 

160° 

160° 

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

diagram 

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

20° 

opposite angles 

non-adjacent 

angles that are 

formed by two 

intersecting lines 

c) What are the measures of two supplementary angles in 

the diagram 1608 and 208 

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. 

Dawn knows the measurements in the diagram. 

a) What type of triangle is nRST 

e.g., isosceles 

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

The measure of /RST is 1048. The sum of the angles 

R 

supplementary 

angles 

two angles whose 

sum is 1808 

30 ft 30 ft 

38° 

S 

38° 

T 

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

NEL 

NEL 



7

7.1 

Triangles 

Try These e.g., 

You will need 

• string 

• scissors 

• plain paper 



Try These 

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 1808. 

Cut three pieces of string that you can use to make a triangle. 

How many different triangles can you make 

1. e.g., 

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 

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

3 What are the angle measures 

e.g., 668, 748, and 408 

ReflecTinG 

Suppose 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. 

property 

a characteristic 

that is shared by 

all the members 

of a group 

4 Two triangles are different if they are not congruent. Are any 

different triangles possible with your side lengths no 

5 Compare your triangle with other students’ triangles. Could 

anyone make more than one triangle no 

Example 1 

The bamboo stems in this photograph create 

an isosceles triangle. An isosceles triangle 

has two equal sides called legs. . The interior 

angles opposite the legs are also equal. 

MPS 

Do all isosceles triangles have these 

properties 

1st pass 

Solution 

A. Find the midpoint of side AC. 

Label it M. Draw MB. 

A 

B 

M 

C 

B. What are the side lengths, in millimetres 

nABM 

: 

19 mm, 50 mm, and 47 mm 

nCBM: 19 mm, 50 mm, and 47 mm 



C. Is nABM 

congruent to nCBM How do you know 

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

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

shape. 

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. 

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

congruent triangles. So the corresponding angles are equal. 

D 

D. e.g., 

E 

20 mm 20 mm 

45° M 45° 

35 mm 

F 

Example 2 

Pavlo 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. 

Solution 

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

below. Record the angle measures on the diagram. 

Acute triangle: 

105° 

Obtuse triangle: 

e.g., 

90° 

120° 

150° 

75° 

35° 70° 110° 

152° 

48° 

28° 

132° 

20° 

145° 

160° 

NEL 

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 

yes 

C. Is the following a property of all triangles Explain. 

• Each exterior angle is 908 or greater. 

AW12SB 

0176519637 

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

than 908. 

FN 

CO 

Technical 

Pass 

Approved 

Not Approved 

D. What triangles have the property in Part C 

NEL 

acute triangles and right triangles 

C07-F17-AW12SB 


2nd pass 

ReflecTinG 

Why is showing 

that something 

is not a property 

easier than 

showing that it is 

a property 

Hint 

Use the triangular 

bracket above as 

an example of a 

right triangle. 



9

Practice 

ReflecTinG 

Does it matter 

which side of 

a triangle you 

extend to make 

an exterior angle 

Explain. 

Hint 

Use the diagrams 

and definitions of 

different types of 

triangles in Getting 

Started. 

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 1808 

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

When you extend one side, you create 

form 

a straight line . Angles that form 

have a sum of 1808 . 

two angles that 

a straight line 

c) Is this a property for all triangles Explain. 

• The sum of the interior angle plus the exterior angle 

is 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. 

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 

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

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

property of all triangles 

e.g., The sum of the interior angles is 1808. 



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. 

75° 

105° 43° 

1 

32° 

148° 

137° 

30° 

60° 120° 

2 

30° 

150° 

150° 

35° 

145° 

20° 

3 

125° 

55° 

160° 

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

Triangle 

1 

2 

3 

Sum of 3 interior 

angles 

1808 

1808 

1808 

Sum of 3 exterior 

angles 

3608 

3608 

3608 

Sum of 3 interior 

angles 1 sum of 

3 exterior angles 

5408 

5408 

5408 

ReflecTinG 

Do you think that 

the sum of the 

interior angles and 

the exterior angles 

is the same for all 

triangles Explain. 

NEL 

NEL 

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 

yes 

b) Compare your results with a classmate’s results. 

Did your classmate get the same results 

yes 

c) Will adding a line that is parallel to the base always create 

a smaller triangle with the same angles Explain. 

AW12SB 

0176519637 

FN 

CO 



60° 

60° 

60° 60° 

60° 

Hint 

One way to draw 

parallel lines is to 

draw along both 

sides of a ruler. 

6. a) e.g., 

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° 



11

7.2 

Quadrilaterals 

You will need 

• coloured pencils 



Try These 

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 

Brady installs stained-glass windows in Victoria. How can you 

describe the quadrilaterals in this window 

e.g., 

square 

trapezoid 

parallelogram 

rhombus 

rectangle 

kite 

triangle (not a 

quadrilateral) 

1 Label an example of each convex quadrilateral in Brady’s 

window. Draw an arrow to the quadrilateral. 

A rectangle has four 908 angles. 

Convex quadrilaterals 

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. 

AW12SB 

0176519637 

2 Some polygons have more than one name. What are three other 

names for a square 

rectangle, parallelogram, and rhombus 

3 Label a polygon in the window that is not a quadrilateral. How 

do you know that it is not a quadrilateral 

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

This triangle has three sides and three vertices. 

168 FN Apprenticeship and C07-F26-AW12SB 

Workplace NEL 

12 Apprenticeship and Workplace 12 

CO 


Technical 

NEL 

Pass 

3rd pass

Example 1 

Elena 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 

Solution 

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

each diagonal in each quadrilateral. 

diagonal 

a line segment 

joining opposite 

vertices 

rectangle parallelogram kite trapezoid 

B. Name quadrilaterals with each property. 

square 

irregular 

concave AW12SB 

concave 

quadrilateral 

quadrilateral 0176519637 1 

rhombus 

quadrilateral 2 

FN 


CO 


Technical 

Property 

Pass 

2nd pass 

ApprovedQuadrilaterals 

Both diagonals make congruent triangles. Not Approved 

One diagonal makes congruent triangles. 

No diagonals make congruent triangles. 

rectangle, parallelogram, 

rhombus, and square 

kite and concave 


trapezoid, irregular 

quadrilateral, and concave 


irregular 

a quadrilateral 

with different 

side lengths and 

different angle 

measures 

concave 

a polygon with an 

interior angle that 

is greater than 1808 

ReflecTinG 

How do you know 

if each result is 

a property of all 

quadrilaterals 

that are the same 

type Use side 

lengths to explain. 

Example 2 

Tessa 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 

30 mm 

20 mm 36 mm 36 mm 

20 mm 

NEL 

Solution 

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

Record them on the diagrams. 

B. How can Tessa use the properties of a rectangle 

NEL 

If the window frame is a rectangle, the opposite sides 

are equal and the diagonals are equal . 

20 mm 

30 mm 

30 mm 

30 mm 

30 mm 

41 mm 

20 mm 



13

Hint 

Use diagrams 

from Example 1 

to help you with 

Question 3. 

Practice 

1. Which convex quadrilaterals have each property 

a) two pairs of equal sides: 

and rhombus 

b) four right angles: 

c) equal diagonals: 

d) equal angles at opposite vertices: 

parallelogram, and rhombus 

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. 

Property 

Diagonals are 

perpendicular. 

Diagonals cross at their 

midpoints. 

rectangle, square, parallelogram, 

rectangle and square 

rectangle and square 

opposite sides that are parallel and equal 

A quadrilateral with 

this property 

e.g., kite 

e.g., rhombus 

rectangle, square, 

A quadrilateral without 

this property 

e.g., parallelogram 

e.g., kite 

. 

Hint 

Draw some 

quadrilaterals. 

Measure the 

exterior angles to 

test your answer 

for Question 4. 

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 1808 . 

• There are four vertices. The total sum of all the interior 

and exterior angles of a quadrilateral is 4 ( 1808 ) 5 7208 . 

• The sum of the interior angles is 3608 . 

• The sum of the exterior angles is 7208 2 3608 5 3608 . 

• The sum of the exterior angles of any quadrilateral 

is 3608 . 

b) Do all quadrilaterals have diagonals that are perpendicular 

Explain. 

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 

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. 

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 

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

angles OR equal diagonals 

c) Compare your isosceles trapezoid with isosceles 

trapezoids drawn by your classmates. What is one 

property that is shared by all the isosceles trapezoids 

AW12SB 

0176519637 

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

FN 

CO 

Technical 

Pass 

Approved 

Not Approved 


angles OR two different angles that add to 1808 OR two 


isosceles triangles and two congruent triangles that are 

formed where the diagonals meet 

2nd pass 

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 

parallelogram 

4(90°) 360° 

Hint 

Think about 

• side lengths 

• angle measures 

• diagonals 

7. e.g., 

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. 

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

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

8. e.g., 

10 in. 

10 in. 

NEL 

NEL 



15

7.3 

You will need 

• 12 toothpicks 

creating Polygon Puzzles 

The convex polygons below are made of squares and triangles. 

pentagon 

hexagon 

Hint 

Here are names of 

polygons: 

5 sides: pentagon 

6 sides: hexagon 

7 sides: heptagon 

8 sides: octagon 

9 sides: nonagon 

10 sides: decagon 

12 sides: 

dodecagon 

A. Use 12 toothpicks to make a different convex hexagon with 

squares and triangles. Draw your hexagon. 

e.g., 

B. Use 12 toothpicks to make four other convex shapes. 

Make each shape out of smaller polygons. Draw your shapes. 

e.g., 

square pentagon pentagon parallelogram 

C. e.g., 

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. 

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

pentagon. 



Mid-chapter 

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. 

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. 

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 1 4 

of the area of the original equilateral 

triangle. 

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

I joined the midpoints, I made four small congruent 

triangles. So each triangle has 1 4 

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. 

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

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

1. e.g., 

2. a) e.g., 

C07-F54 -AW12.ai 

C07-F55 -AW12.ai 

2. b) e.g., 

NEL 

3. Hayley is the lighting director for a theatre in Manitoba. 

AW12 

She wants to place a spotlight so that it shines on the centre 

0176519637 

of a rectangular stage. 

NEL 

How can Hayley use a property Company of rectangles MPS to find the 

centre of the stage Use the AW12 Technical rectangle at the right in your 

explanation. 

Pass 

1st pass 

e.g., Hayley can use string or 

Figure 

tape 

Number 

to mark 

C07-F55-AW12.ai 

the diagonals. The 

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

can mark the midpoint of each Pass side and join 1st each pass midpoint 

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

the stage. 

Figure Number 

0176519637 

Approved 

Company Not Approved 

Technical 

Approved 

Not Approved 

C07-F54-AW12.ai 

MPS 

C07-F57 -AW12.ai 

C07-F58 -AW12.ai 

AW12 


0176519637 


Figure Number C07-F57-AW12.ai 

C07-F59 -AW12.ai 

Company 

MPS 

17

7.4 

Regular Polygons 

Try These 

You will need 

• 6 toothpicks 

• a ruler 


• a compass 

Use six toothpicks. Create an irregular hexagon. 

How do you know that your hexagon is not regular 

Use a diagram to explain. 

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

e.g., 

irregular 

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. 

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

2 Name all the regular polygons in this tray. 

square, octagon, pentagon, and hexagon 

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 

Example 1 

Olivia 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. AW12SB 

0176519637 

How can Olivia use FN a tracing of C07-F62-AW12SB the dark hexagon 

to make a model of CO the light hexagon 


Technical 

Pass 

2nd pass 

Solution 

Approved 

A. Mark the midpoint Not Approved of each side. Draw 

straight lines to join each midpoint 

to the next midpoint. 

B. How do you know the shapes are similar 

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

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

side lengths are smaller. 



C. Test these regular polygons. Does the midpoint reducing 

method always work to make a similar smaller polygon 

ReflecTinG 

Gabriel drew a 

square. He says 

that all squares 

are similar to the 

square he drew. 

Do you agree 

Explain. 

yes yes yes 

D. Will all regular polygons have this property Explain. 

• Joining the midpoints of each side of a regular polygon 

creates a similar regular polygon. 

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 

C07-F65-AW12.ai 

C07-F67-AW12.ai 

C07-F66-AW12.ai 

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

does not change, so both polygons have equal angles. 

Example 2 

Craig 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 

AW12 

0176519637 

-F65-AW12.ai 

Figure Number 

S 

Company 

Technical 

pass 

Pass 

Approved 

Not Approved 

Solution 

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

AW12 

0176519637 

Figure Number 

C07-F66-AW12.ai 

Company 

MPS 

Technical 

Pass 

1st pass (1808) 5 

Approved 

Not Approved 

B. Draw diagonals from one vertex to divide your pentagon into 

triangles. How many triangles did you make three 

C07-F67-AW12.ai 

MPS 

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

1st pass 

3 5408 

The sum is 5408 . 

D. What angle measure should Craig use 

5408 4 5 5 1088 

Craig should use 1088 . 

A. e.g., 

ReflecTinG 

Can you always 

divide a shape 

into triangles to 

determine its angle 

measures Is 

this a property of 

regular polygons 

Explain. 

NEL 

NEL 

C07-F68-AW12.ai 


Chapter 7 Polygons 19

Practice 

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

in these two shapes. Are they regular polygons Explain. 

a) 

b) 

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

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

2. Draw diagonals from one vertex of each shape to divide the 

shape into triangles. Use the triangles to complete the chart. 

Property 

Number of triangles 

Hexagon 

C07-F69-AW12.ai 

4 

Heptagon 

C07-F70-AW12.ai 

5 

Octagon 

6 

Sum of all the angle 

measures 

7208 

9008 

10808 

Measure of each interior 

angle 

1208 

128.571 …8 

1358 

Measure of each exterior 

angle 

608 

51.428 …8 

458 

Sum of the measures of 

all the exterior angles 

3608 

C07-F71 -AW12.ai 

3608 

C07-F72 -AW12.ai 

3608 

C07-F73 -AW12.ai 

2 

519637 

e Number 

any 

nical 

ved 

pproved 

ReflecTinG 

As the number of 

sides in a polygon 

increases, C07-F69-AW12.ai the 

polygon MPS looks 

more and more 

like a circle. What 

1st pass 

happens to the 

interior angles 

3. Use your chart from Question 2. 

a) What happens to the number of triangles when you add 

one side C07-F70-AW12.ai 

to a polygon It increases by 1. 

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

10 

1st pass 

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

polygon 1508 

d) How does the number of triangles you can make in any 

AW12 

0176519637 

Figure Number 

Company 

Technical 

Pass 

Approved 

Not Approved 

AW12 

AW12 

polygon AW12 relate to the number of sides 

0176519637 

0176519637 

0176519637 

Figure Number C07-F71-AW12.ai 

Figure Number number C07-F72-AW12.ai 

Figure of Number triangles 5 C07-F73-AW12.ai number of sides 2 2 

Company 

MPS 

Company 

e) What 

MPS Company is the measure of MPSeach angle in a regular decagon 

Technical 

Technical 

Technical 

Pass 

1st pass 

Pass 

(1808)(10 1st pass Pass 2 2) 4 10 51st pass 1448 

Approved 

Approved 

Approved 

176 Apprenticeship 

Not Approved 

Not Approved and Workplace 12 NEL 

Not Approved 


AW12SB 

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

diagonals does each polygon have 

4 sides 5 sides 6 sides 7 sides 

2 diagonals 5 diagonals 9 diagonals 14 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. false 

b) If the number of vertices is even, the diagonals that 

connect opposite vertices intersect at the centre. true 

C07-F74 -AW12.ai C07-F75 -AW12.ai 

C07-F77-AW12.ai 

c) The number of diagonals you C07-F76-AW12.ai 

can draw from one vertex 

of a regular polygon is n 2 3, where n is the number of 

vertices. true 

ReflecTinG 

Use one part in 

Question 5 where 

you wrote “false” 

for the answer. 

Explain why it 

is false. 

6. Regular octagons are often used for the Chinese New Year. 

The number 8 is associated with wealth and good luck. 

2 

19637 

74-AW12.ai Number 

any 

ical 

ss 

ved 

pproved 

AW12 

AW12 

0176519637 

0176519637 

C07-F75-AW12.ai 

Figure Number 

Figure Number 

C07-F76-AW12.ai 

C07-F77-AW12.ai 

MPS Company 

Company 

MPS 

MPS 

Technical 

Technical 

1st Pass pass a) Draw diagonals Pass 

1st pass 

1st pass to join pairs of opposite vertices in this 

Approved 

Approved 

octagon. What is the measure of each angle where the 

Not Approved 

Not Approveddiagonals meet 3608 4 8 5 458 

b) What type of triangle The perspective do the diagonals was deleted. make isosceles 

c) Place one end of a compass at the centre. Place the other 

end at a vertex. Draw a circle. What happens 

The circle touches all the vertices. 

ReflecTinG 

Which properties 

of regular 

octagons are 

shared by all 

regular polygons 

NEL 

NEL 



21

7.5 

Applications of Polygons 

You will need 



• a compass 

Try These 

Look around your classroom. 

i) Where do you see a polygon Name this polygon. 

e.g., The top of 

my desk is a rectangle. 

ii) Describe some properties of this polygon. 

e.g., It has two pairs 

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

angles are right angles. 

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. 

360° 

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. 

e.g., 

120° 

90° 

60°45° 45° 

120° 

60° 45° 45° 

75° 90° 

120° 90° 

75° 

Example 

Jordan 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 

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 



Solution 

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

your circle 

e.g., 

24 mm 

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. 

( e.g., 40 mm)(0.5878) 5 e.g., 23.512 mm 

The side length is 

e.g., 24 

e.g., 40 

mm. 

ReflecTinG 

The 

measurements 

in this example 

are given in 

millimetres. When 

would a machinist 

have to measure 

much more 

precisely 

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 

regular octagon 

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

millimetre 

(60 cm)(0.3827) 5 22.962 cm 

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

Hint 

Use the charts 

inside the back 

cover for converting 

units. 

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. 

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

b) What is one regular polygon that could not make a tiling 

design Explain. 

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. 

NEL 

NEL 



23

T 

12.0 ft 

S 

22.0 ft 

28.0 ft 

U 6.7 ft 

V 

9.0 ft 

W 

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 WT 

in the diagram of the wall. 

b) What is the area of each triangle 

Hint 

For nWUV, use 

9.0 ft as the base. 

Subtract 22.0 ft 

from 28.0 ft to get 

the height. 

Triangle Dimensions Area 

nWST Base 5 

Height 5 

nWTU Base 5 

Height 5 

nWUV Base 5 

Height 5 

28.0 ft 

12.0 ft 

22.0 ft 

12.0 ft 

9.0 ft 

6.0 ft 

Area 5 1 2 (base)(height) 

5 1 (28.0 ft)(12.0 ft) 

2 

5 

168.0 sq ft 

1 

2 (base)(height) 5 1 (22.0 ft)(12.0 ft) 

2 

5 132.0 sq ft 

1 

2 (base)(height) 5 1 (9.0 ft)(6.0 ft) 

2 

5 27.0 sq ft 

c) What is the total area of pentagon STUVW 

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. 

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

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

He will need to buy 7 L. 

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 

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



chapter 

1. Circle the polygons. 

2. Name the quadrilaterals in Question 1. 

square, parallelogram, rectangle, concave quadrilateral, and trapezoid 

3. Name the regular polygons in Question 1. 

square and regular pentagon 

C07-F86-AW12.ai 

4. Sketch a polygon that has each property below. 

Classify each polygon. 

a) Each interior angle is 608. equilateral triangle 

b) There are five equal sides. e.g., regular pentagon 

4. a) e.g., 

60° 

60° 60° 

4. b) e.g., 

AW12 

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. 

0176519637 

Figure Number 

e.g., 

Company 

Technical 

Pass 

Approved 

Not Approved 

Which polygon has this property Explain. Include drawings. 

C07-F86-AW12.ai 

shared 

angle 

MPS 

1st pass 

same angles in 

both triangles 

same side 

length as 

original 

rectangle 

shorter 

side 

C07-F87-AW12.ai 

lengths 

C07-F89-AW12.ai 

NEL 

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. 

NEL 



25 

AW12 

AW12 

C07-F91-AW12.ai

6. a) e.g., 

6. b) e.g., 

6. c) e.g., 

P 

60° 

C07-F92-AW12.ai 

120° 

Q 

90° 

C07-F93-AW12.ai 

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. 

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. 

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

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

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 

50 ft 

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

measure also increases. 

e.g., Area nQRS 

5 1 (50 ft)(50 ft) 

2 

2.ai 

15 ft S 50 ft R 

C07-F95-AW12.ai 

C07-F94-AW12.ai 

C07-F96-AW12.ai 

8. e.g., 

5 1250 sq ft 

Area nPQS 

5 1 (50 ft)(15 ft) 

2 

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. 

W12.ai 

-AW12.ai 

ai 

i 

C07-F98-AW12.ai 

b) What property of polygons did you use to solve the 

problem in Part a) 

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

8. Kylie designs and sews quilts. Describe a property of 

polygons that she might use in a quilt. Include a diagram with 

your description. 

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

polygon, you get a smaller similar polygon. 

2.ai 



C07-F100-AW12.ai

Chapter 

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

a) What are the side lengths of 

your triangle, in millimetres 

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

b) What are the angle measures 

of your triangle 

e.g., 1308, 308, and 208 

c) What are two names for the 

type of triangle you made 

e.g., obtuse triangle and 

scalene triangle 

d) What is one property that 

your triangle shares with all 

triangles e.g., The sum of 

the interior angles is 1808. 

Whitehorse 

Victoria 

Yellowknife 

Fort 

McMurray 

Edmonton 

Regina 

0 500 1000 km 

Winnipeg 

Baker 

Lake 

Churchill 

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

some triangles 

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

Toronto 

N 

Iqaluit 

Fredericton 

Québec 

City 

Ottawa 

St. John’s 

Charlottetown 

Halifax 

Moncton 

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

and Fort McMurray. 

a) What are the side lengths, in millimetres 

18 mm, AW12SB 18 mm, 11 mm, and 11 mm 

0176519637 

b) What are the angle measures 908 

FN 

C07-F101-AW12SB 

c) What are two names for the quadrilateral you made 

CO 


rectangle Technical and parallelogram 

d) What is Pass one property 2nd that pass this quadrilateral shares with 

another Approved type of quadrilateral Use diagrams. 

Not Approved 

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. 

2. d) e.g., 

NEL 

NEL 



C07-F102-AW12.ai 

27

2. e) e.g., 

e) What is one property that other quadrilaterals have, but 

this quadrilateral does not have Use diagrams. 

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., 

C07-F104-AW12.ai 

C07-F105-AW12.ai 144° 

4. b) e.g., 

Angles are 

all 60°. 

C07-F106-AW12.ai 

Angles are 

all 144°. 

3. Claude says that a polygon is a regular polygon if all of its 

sides are equal. Do you agree Include a diagram. 

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. 

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

Include diagrams. 

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

opposite vertices, all the diagonals cross in the centre. 

b) Describe one property that some other regular polygons 

have, but a regular decagon does not have. Include 

diagrams. 

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

regular decagon does not. 

4-AW12.ai 

i 

5. e.g., 

C07-F112-AW12.ai 

106-AW12.ai 

C07-F107-AW12.ai 

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. 

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. 

ss 

12.ai 

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 

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 

C07-F108-AW12.ai 

will be equal. 

C07-F109-AW12.ai 



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