Polygons Booklet - Mennonite Brethren Collegiate Institute

Name: __________________________

The practice problems have been taken from a variety of sources including:

MathWorks 12

Math at Work 12

Apprenticeship and Workplace 12

Provincial exams

What You’ll Learn

Properties of various polygons

How to calculate for the sum of interior angle in a regular polygon

How to calculate for the measure of an interior angle in a regular polygon

Why It’s Important

Polygons are used by:

Professional tilers creating designs for flooring or backsplashes.

Engineers looking to keep soldiers safe.

Brick layers for pathways.

Key Formulas NOTE: Here n = the number of sides

Sum of the interior angles: 𝑆 = 180(𝑛 − 2)

Measure of an interior angles: 𝑀 =180 (𝑛−2)

𝑛

Measure of central angle: 𝐶 =360

𝑛

Grade 12 Essentials - Polygons

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Getting Started: Notes

Measuring Angles

To measure an angle:

- Put the vertical marker of the protractor at the

vertex (corner) of the angle you are measuring.

- Make sure that the 0̊ line is along one of the legs of

the angle.

- Follow out to the second leg and read the

measurement on the protractor. Depending on the

length of the line, it may be difficult to use the

outside measurements on the protractor.

Marking Sides and Angles

- Capital letters are used to mark vertices of a polygon.

- The line segment (side) of a polygon is denoted by the two vertices

(corners) it sits between.

- Congruent (equal) sides are marked with dashes.

- Congruent (equal) angles are marked with arcs.

Examples:

1. Congruency of Angles 2. Congruency of Sides

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Getting Started: Practice

1. Measure each of the following angles.

a)

b)

2. Record the side lengths and angles for ∆ABC.

AB = A=

AC= B=

BC= C=

A

B C

ABC=

A D

C

B ACB=

BCD=

NOTE: is the symbol for angles. The letter that is listed in the middle is the angle

that is being measured.

Ex. ABC = 60˚ means that angle B = 60˚. (We could also call this B)


4


5


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Triangles: Notes

Definitions

Triangle:

Vertex (pl. Vertices):

Classification

Triangles can be classified by their…

Angles

1. Right:

2. Acute:

3. Obtuse:

Side Lengths

1. Equilateral:

2. Isosceles:

3. Scalene:


7

Triangles: Practice

1. Circle the types of triangles that have each property.

NOTE: There may be more than one right answer.

a) Some sides are equal.

Equilateral Isosceles Scalene

b) No interior angles are equal.


c) The sum of the interior angles is 180˚.

Acute Right Equilateral

2. Circle the types of triangles that do not have each property.

a) All angles are less than 90˚.

Acute Right Obtuse

b) Some angles are equal.


c) There are at least 2 equal sides.


3. Use the angle measures to calculate the unknown angles in each triangle.

NOTE: The following are not to scale. Use calculations rather than a

protractor to solve.

The Esplanade Riel Bridge


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4. Use the following diagram to answer the questions below.

a) What is the measure of M?

b) Classify ∆MNP by angle measure and by side length.

5. Recall our activity at the start of class …

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?

c) Is this a property for all triangles? Explain.


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Quadrilaterals: Notes

Which of the following shapes are polygons? Circle each one.

Definitions

Polygon:

Quadrilateral:

Types of Quadrilaterals

Rectangle:

Square:

Parallelogram:

Rhombus:

Trapezoid:

Kite:


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Irregular Quadrilateral:

Concave Quadrilateral:

NOTE: Some polygons are also convex, which means that there are no interior

angles which are greater than 180˚.


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Activity – Properties of Quadrilaterals

We can take a polygon and draw diagonals, which are line segments joining

vertices that are NOT NEXT TO EACH OTHER.

Example) In pairs, complete the following instructions and answer each question

using the given square.

What do we find?

- The diagonals are . That is, they are the equal.

- The diagonals on a square are . That is, they cross at a

90˚ angle.

- The diagonals each other. That is, they cut each other in half.

All regular polygons have certain properties when you draw diagonals. See the

following page for an overview of the properties.

1. Draw the diagonal AC. Measure its length.

2. Draw the diagonal BD. Measure its length. What do

you notice?

3. Label the point where the diagonals intersect as E.

Measure the lengths of AE, BE, CE, and DE. What do

you notice?

4. Measure DEA, AEB, BEC, and CED. What do

you notice?

5. What is the sum of the angles where the diagonals

intersect?

A

D C

B


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Quadrilaterals: Practice

1. Determine the missing measurements using the properties of

quadrilaterals.

2. State two properties that would prove that a quadrilateral is a

parallelogram.


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3. List all the quadrilaterals that could fit each description.

a) Has at least one set of parallel sides

b) Has four equal side lengths

c) Has two equal diagonals

4. Sketch and name a quadrilateral that fits each description.

a) The diagonals are equal, but the sides are not all equal.

b) The diagonals are equal, and all the sides are equal.

c) The diagonals are not equal, and no two sides are equal.

5. Using your knowledge of the properties of quadrilaterals, find the

measures of the missing angles. What kind of quadrilateral is this?

6. Solve for the indicated length or angle, and identify the type of

quadrilateral.

a) b)

a.


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Regular Polygons: Notes

Definitions

Regular Polygon:

Activity – Measure and Sum of Interior Angles

1. Draw a square. Then draw a diagonal between two non-adjacent corners

so the square is divided into triangles.

2. How many triangles were created?

3. What is the sum of the interior angles of a square?

4. What is the measure of each interior angles of a square?

5. For the regular pentagon below, repeat steps 2-5.

NOTE: You will need to draw more than one diagonal to divide each

shape into triangles.

Common Polygons

Name Number of Sides

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Example 1) Are the following shapes regular

polygons? Explain.

a) b)


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6. For the regular hexagon below, repeat steps 2-5. Again, you will need to

draw more than one diagonal to divide each shape into triangles.

7. Using your results from steps 1-7, complete the following table:

Figure Number of

Sides

Number of

Triangles

Sum of Interior

Angles

Measure of Each

Individual Angle

Equilateral

Triangle

Square

Regular

Pentagon

Regular

Hexagon

8. Use your chart from Question 8 to answer the following questions.

a. How does the number of triangles you can make in a polygon

relate to the number of sides?

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

c. What is the sum of all the angles measures in a 12-sided polygon?

d. What is the measure of each angle in a 12-sided polygon?


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Properties

We can use formulas to find the measure of an interior

angle, as well as the sum of the interior angles of a

regular polygon.

Sum of the interior angles:

Example 2) Find the sum of the interior angles of a hexagon.

Example 3) Working backwards: The sum of the interior angles of a polygon is

900˚. Determine the number of sides of the polygon.

Measure of an interior angle:

If we know that all of the angles in a hexagon sum to 720˚, how can we find one

angle?

Example 4) Find the measure of an interior angle in a square.

𝑀 =180 (𝑛−2)

𝑛 where n is the number of sides

𝑆 = 180(𝑛 − 2) where n is the number of sides


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Measure of the central angle:

We can also determine the measure of the central angles in a regular polygon.

The central angle is the angle made at the center of a polygon by any two

adjacent vertices of the polygon.

All central angles would add up to 360º (a full circle), so the measure of the

central angle is 360 divided by the number of sides

Example 5) What is the measure of the central angle in a hexagon?

Example 6) A regular polygon has central angles of 45º.

a) State the number of sides for this polygon.

b) State the name of this polygon.

𝐶 =360

𝑛 where n is the number of sides.


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Regular Polygons: Practice

1. Which are regular polygons? Check with a ruler and a protractor.

2. What is the measure of each interior angle in a regular octagon?

3. Given the following regular polygon:

a) Calculate the sum of the interior angles in the

polygon.

b) State the measure of each interior angle in the polygon.

4. The sum of the interior angles is 900º. Determine the number of sides of the

polygon.


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5. A regular hexagon has a side length of 10 metres.

a) State the measure of angle A, the central angle, in degrees.

b) State the measure of the given diagonal in metres.

6. Draw ALL diagonals in each regular polygon.

a) How many diagonals does each polygon have?

b) Decide whether each property is true or false using the above

polygons.

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

If the number of vertices is even, the diagonals that connect opposite

vertices intersect at the centre.

The number of diagonals you can draw from one vertex of a regular

polygon is (𝑛 − 3), where 𝑛 is the number of vertices.

7. Determine the number of diagonals in a regular octagon.

NOTE: Use the formula 𝐷 =𝑛(𝑛−3)

2


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Applications of Polygons

Airless Tire Promises Grace under Pressure for Soldiers

The Pentagon investigates the use of a new type of airless

tire designed to get troop-carrying Humvees through hot

spots without stopping

In Iraq and elsewhere, improvised explosive devices (IEDs)

pack a double-deadly whammy: They can kill when they

explode, and then they turn surviving soldiers into sitting

ducks when Humvee tires blow out. Conventional Humvee tires need a certain amount

of air pressure, but also may include so-called "run-flat" inserts that wrap around the

tire's rim to keep it from going completely flat when the tire's surface is ruptured. The U.S.

Army, however, is looking for an alternative that can keep its vehicles running faster and

farther than a run-flat donut after an attack.

To keep troops from being stranded and easily ambushed on the battlefield, the Army is

working with researchers to develop tires for their Humvees that can better withstand

roadside attacks. One such design comes from Resilient Technologies, LLC, based in

Wausau, Wisc., and the University of Wisconsin–Madison's Polymer Engineering Center.

With a four-year, $18-million grant from the Pentagon, Resilient is working to create a

"non-pneumatic tire" (NPT) technology, called that because it doesn't require air.

The NPT looks like a circle of honeycombs bordered by a thick black tread. "There's a lot

of space for shrapnel to pass through," says Ed Hall, Resilient Technologies's director of

business affairs. "Even if you remove 30 percent of the webs, the tire will still work."

And for those of you wondering why all tires aren't simply made out of solid rubber,

some construction vehicles use them on sites with debris that can easily shred a

pneumatic tire, but solid tires give an incredibly rough ride, generate a lot of heat, and

might be even worse if a piece came off during an explosion, because it could not

easily be repaired. The NPT's honeycomb structure is designed to support the load

placed on the tire, dissipate heat and offset some of these issues.

http://www.resilienttech.com/


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Applications: Article Analysis

Based on the article “Airless Tire Promises Grace under Pressure for Soldiers”, fill

out the following chart.

Fact-Based Article Analysis

Title and topic of article

Summarize the main ideas in your own

words.

Draw a diagram to represent the main

idea of the article (i.e. an airless tire)

List, in point form, at least three facts in

the article.

Write one question you have from the

article.

This article is important because …

Polygons are often used in construction, commercial, industrial or

artistic applications. Come up with one other real world application

for polygons.


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Chapter Review

Definitions: You will not be asked to specifically define a term on the test.

However, this is vocabulary heavy unit. Make sure you know all of the different

terms used so that you understand the questions being asked. See the Polygon

Unit Vocabulary at the end of this booklet.

Short Answer/Problem Solving:

1. What type of triangle has two equal sides and all angles are less than 90°?

2. Is A congruent with B? What type of triangles are they? Explain your

reasoning.

3. What type of triangle is XYZ? Explain using XYW and YZW.

4. Calculate all unknown interior and exterior angles using the given angles.

5. Calculate all unknown interior and exterior angles using the given angles.


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6. The sum of the interior angles of a polygon is 540 ̊. Determine the number of sides

of the polygon.

7. Determine the unknown measurements using the properties of quadrilaterals.

a) Isosceles trapezoid

TW = 8

UV =

TV=12

UW=

TXW = 60°

UXV =

b) Square

OP = 8

OQ = 10

PQ =

PS =

PSO =

OSQ =

8. Sketch a rhombus and label ALL of the congruent parts.

9. Which of these quadrilaterals do not have equal angles at opposite vertices?

10.


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10. Use properties of quadrilaterals to show that a square is always a parallelogram,

but a parallelogram is not always a square.

11. Is this shape a regular polygon? Explain.

12. Is this shape a regular polygon? Explain.

13. What is the relationship between the number of triangles that can be formed

within a regular polygon and the sum of all angle measures? Explain.


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14. Determine the sum of the angles in each regular polygon. Then, state the

measure of each interior angle.

a) Pentagon

b) Octagon

c) 10-sided figure

d) 11-sided figure

15. Louis wants to put a hole in the centre of his patio table for a large sun umbrella.

The table is shaped like a regular decagon with ten equal sides. How can Louis

determine the location of the centre? Explain with an illustration.


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16. Given a regular hexagon with centre C.

a) Determine the measure of the central angle of the

hexagon.

b) Determine the length of side a. Justify your answer.

17. Determine (by illustration of calculation) the total number of diagonals in a

regular six-sided polygon.

NOTE: The following question will be asked on the test!

18. Polygons are often used in construction, commercial, industrial, or artistic

applications.

a) Demonstrate one use of the various properties of polygons in the real

world by performing the following two steps:

State a specific example where the various properties of polygons

are used.

Support your example with a written explanation of how various

properties of polygons are used.

b) Sketch a reasonable neat picture or diagram (not necessarily to scale)

that supports your example in Part A.


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Polygon Unit Vocabulary

Term Definition Diagram

polygon a closed shape made up of

straight lines

equilateral triangle a triangle with three equal

sides

isosceles triangle a triangle with exactly two

equal sides

scalene triangle a triangle with no equal

sides

acute triangle a triangle with each angle

less than 90o

obtuse triangle a triangle with one angle

that is greater than 90o

right triangle a triangle with one angle

that is equal to 90o


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regular polygon

a closed shape with all

sides equal and all angles

equal

congruent the same size and shape

(they are equal)

complementary angles two angles whose sum is

90o

transversal a line that intersects two or

more lines

opposite angles

non adjacent angles that

are formed by two

intersecting lines

supplementary angles two angles who sum is 180 o

quadrilateral

convex polygon


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Quadrilaterals: Triangles:

concave polygon

diagonal

bisect

perpendicular

Polygons Booklet - Mennonite Brethren Collegiate Institute

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