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Unit 5 Polygons of 21

Geometry Unit 5 - Notes

Polygons

Syllabus Objective: 5.1 - The student will differentiate among polygons by their attributes. Review terms: 1) segment 2) vertex 3) collinear 4) intersect Polygon- a plane figure bounded by a closed path, composed of a finite sequence of straight line segments. These segments are called its sides, and the points where two edges meet are the polygon's vertices.

Polygon Type Number of Sides

Triangle 3 Quadrilateral 4

Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10

Hendecagon 11 Dodecagon 12

n-gon n

Polygon Parts

Side - one of the line segments that make up the polygon.

Vertex - point where two sides meet. Two or more of these points are called vertices.

Diagonal - a line connecting two vertices that isn't a side.

Interior Angle - Angle formed by two adjacent sides inside the polygon.

Exterior Angle - Angle formed by two adjacent sides outside the polygon.

http://en.wikipedia.org/wiki/Closed_curve�

http://en.wikipedia.org/wiki/Line_segment�

http://en.wikipedia.org/wiki/Line_segment�


Interior G H I

J K

L

E F

Interior B A

D C

Convex Polygon

- a polygon for which there is no line containing a side of the polygon that also contains a point in the interior of the polygon.

Polygon ABCD is a convex quadrilateral. Concave Polygon - a polygon for which there is

a line containing a side of the polygon that also contains a point in the interior of the polygon.

Polygon EFGHIJKL is a concave octagon.

I always draw a teddy bear hiding in the cave. I’m not a good artist, but they remember. Probably because I’m so bad!!

Examples: Determine which of the following figures is a polygon. If it is a polygon, state whether it is a convex or concave polygon.

“A” and “C” are polygons. “A” is concave and “C” is convex. Solutions:

“B” is not a polygon because some of its sides intersect more than two other sides. “D” is not a polygon because it has a side that is not a segment. “E” is not a polygon because it is not closed, two of its sides intersect only one other side.

E D C B A

*Take the opportunity to relate parts of the polygon name to real life objects and concepts. For instance, a pentagon has five sides. Show a picture of the Pentagon. The building is

called the Pentagon due to its shape. Show a picture of an octopus. It has eight legs just like an octagon has eight sides. A decade is ten years and a decagon has ten sides. A tricycle

has three wheels just like the triangle has three sides! A quadrilateral has four sides just like a four-wheeler is called a “quad” for short.


Equilateral

- a polygon with all sides congruent to each other.

Equiangular

- a polygon with all of its interior angles congruent to each other.

Regular - a polygon that is both equilateral and equiangular.

Examples:

Determine if the polygon is convex or concave; appear to be equiangular, equilateral, or regular and state the name of the polygon.

“A” is an equilateral, convex pentagon. It has five segments that are congruent to each other, but it is not equiangular. Therefore, it is not a regular polygon.

Solutions:

“B” is equiangular and equilateral. Thus, it is a regular, convex triangle. “C” is an equilateral, concave decagon. It is not equiangular, so it is not a regular decagon.

Syllabus Objectives: 5.2 - The student will classify polygons by their special properties. Diagonal

- a segment that joins two nonconsecutive vertices in a polygon.

The green (internal) segments in the three polygons above are examples of diagonals. Interior Angles of a Quadrilateral Theorem: The sum of the four angles of any quadrilateral

is 360°.

C B A


Quadrilateral

Isosceles Trapezoid

Kite Trapezoid

Square

Rhombus

Parallelogram

Rectangle

Types of Quadrilaterals:

Quadrilateral

– four sided polygon

When referring to the flow chart above, the quadrilateral at the top center is a four-sided polygon. The four polygons on the left side all have opposite sides parallel and congruent. Therefore, the parallelogram, rectangle, rhombus, and square are all parallelograms. All parallelograms have opposite sides congruent, opposite angles are congruent, the diagonals bisect each other, and consecutive angles are supplementary. The quadrilaterals on the right side are not parallelograms. The trapezoid is a quadrilateral that has one pair of opposite sides parallel. If the legs, or base angles, are congruent, then it is an isosceles trapezoid. {Link concept to isosceles triangles.} The diagonals of an isosceles trapezoid are congruent just like those of the rectangle; however they do not bisect each other. The diagonals of a kite are perpendicular like those in the rhombus; however they do not bisect each other. Only one pair of opposite angles is congruent to each other. The other pair of opposite angles is bisected by the diagonal that passes through them. ************************************************************************************ Parallelogram

– A quadrilateral with both pairs of opposite sides parallel. The parallelogram symbol is .

Theorem: If a quadrilateral is a parallelogram, then its opposite sides are congruent. {Students often confuse this with the definition of parallelogram. The definition (above) does not indicate congruence of sides.}

Properties of Being a Parallelogram

Theorem: If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Theorem: If a quadrilateral is a parallelogram, then its diagonals bisect each other.


************************************************************************************ Rectangle

– A parallelogram with four right angles.

The rectangle is equiangular. Its diagonals not only bisect each other but they are also congruent to each other.

Properties of Being a Rectangle

************************************************************************************ Rhombus

– A parallelogram with four congruent sides.

The rhombus is equilateral. Its diagonals not only bisect each other but they intersect perpendicularly. The diagonals of a rhombus bisect a pair of opposite angles.

Properties of Being a Rhombus

************************************************************************************ Square

– A parallelogram with four congruent sides and four right angles.

The square is a regular quadrilateral. It has all the properties of the quadrilateral, parallelogram, rectangle, and rhombus.

Properties of Being a Square

************************************************************************************ Trapezoid

– A quadrilateral with exactly one pair of parallel sides, called bases. The nonparallel sides are legs.

Midsegment of a trapezoid

– a segment that connects the midpoints of the legs of a trapezoid.

The midsegment of a trapezoid is half the sum of the bases. {Link concept to the midsegment of a triangle, imagine a trapezoid with one base length equal to 0. What would be the best name for that figure? A triangle.} Midsegment Theorem for Trapezoids

************************************************************************************

– The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

Kite

– a quadrilateral that has two pairs of consecutive congruent sides, but in which opposite sides are not congruent.

In a kite, the diagonals are perpendicular, one pair of opposite angles are congruent and the other pair are bisected by the diagonal. ************************************************************************************


31

24

R S

TW

*Below is a table that shows the different types of quadrilaterals and the properties that they have.

Given: RSTW

Prove: RST TWR≅∆ ∆

Statements Reasons

1) RSTW is a ogram 1) Given 2) RS WT 2) Def. of ogram

3) 21 ≅ ∠∠ 3) 2 lines cut by trans., alt. int. 's∠ ≅

4) RT RT≅ 4) Reflexive 5) RW ST 5) Def. of ogram

6) 43≅ ∠∠ 6) 2 lines cut by trans., alt. int. 's∠ ≅ 7) RST TWR≅∆ ∆ 7) ASA

Property Quad ║ogram Rectangle Rhombus Square Trapezoid Isos. Trap. Kite 4 sides X X X X X X X X

Interior angle sum 360°. X X X X X X X X

Both pairs of opp. sides . X X X X

Exactly 1 pair of opp. sides . X X

Both pairs of opp. sides ≅ . X X X X

Exactly 1 pair of opp. sides ≅ . X

All sides are ≅ . X X All angles are ≅ . X X Both pairs of opp.

angles are ≅ . X X X X

Exactly 1 pair of opp. angles is. X

Diagonals are ≅ . X X X Diagonals are ⊥ . X X X Diagonals bisect

each other. X X X X


Knowing the diagonal separates a parallelogram into 2 congruent triangles suggests some more relationships.

o The opposite sides of a parallelogram are congruent. o The opposite angles of a parallelogram are congruent.

Both of those can be proven by adding CPCTC to the last proof. Many students mistakenly think the opposite sides of a parallelogram are equal by definition. That’s not true, the definition states the opposites sides are parallel. This proof allows us to show they are also equal or congruent.

The Problem: **Inner Rectangle**

Let ABCD be a convex quadrilateral and let the angle bisectors of the angles at A, B, C, D form the sides of a second quadrilateral PQRS. What can you say about the original quadrilateral ABCD , if PQRS is to be a rectangle?

Solution: ABCD must be a parallelogram. For suppose PA , PB are the bisectors of the angles at A and B. Then

90m APB∠ = ° if and only if 90m PAB m PBA∠ + ∠ = ° ,that is, if and only if ( )2 2 2 90 180m PAB m PBA m DAB m ABC∠ + ∠ = ° → ∠ + ∠ = ° , which in turn happens if

and only if the lines DA

and BC

are parallel. Similarly at the other vertices. Syllabus Objective: 5.3 - The student will find the sum of the measures of the interior angles of a polygon. Review the Triangle Sum Theorem. {Show students how to derive

the formula from constructing diagonals in a polygon from a single vertex and applying the Triangle Sum Theorem.}

Polygon Interior Angle Theorem: The sum of the measures of the interior angles of a convex n-gon is (n - 2)(180°).


Corollary to Polygon Interior Angle Theorem: The measure of each interior angle of a regular

n-gon is ( )2 180− °nn

.

Table of Polygon values:

Example:

Find the measure of each interior angle.

First, the sum of the interior angles is found to be (3)180 = 540°. x – 8 + 3x – 11 + x + 8 + x + 2x + 7 = 540 8x – 4 = 540 8x = 544 x = 68

60 , 193 , 76 , 68 , and 143m U m V m W m Y m Z∠ = ° ∠ = ° ∠ = ° ∠ = ° ∠ = °Solutions:

.

Polygon Type Number of Sides (n)

Number of ∆s Inside the Polygon

Sum of the Measures of the Interior Angles

Measure of each Interior Angle

(regular polygon)

Triangle 3 1 1(180°) = 180° 3 1(180°)

Quadrilateral 4 2 2(180°) = 360° 4 2(180°)

Pentagon 5 3 3(180°) = 540° 5 3(180°)

Hexagon 6 4 4(180°) = 720° 6 4(180°)

Heptagon 7 5 5(180°) = 900° 7 5(180°)

Octagon 8 6 6(180°) = 1080° 8 6(180°)

Nonagon 9 7 7(180°) = 1260° 9 7(180°)

Decagon 10 8 8(180°) = 1440° 10 8(180°)

Hendecagon 11 9 9(180°) = 1620° 11 9(180°)

Dodecagon 12 10 10(180°) = 1800° 12 10(180°)

n-gon n n - 2 (n - 2)(180°) n (n - 2)(180°)

(3 x - 11) °

( x + 8) °

x ° (2 x + 7) °

( x - 8) °

Z Y

W

V

U


95°

b a

11

8

y

x

Syllabus Objective: 5.4 - The student will solve problems involving properties of special quadrilaterals. Review the special properties of all quadrilaterals and special angles created when parallel lines are cut by a transversal.

Example:

Find the value of each variable in a parallelogram.

x = ? y = ? a = ? b = ? Solutions: x = 11 y = 8

a = 85° b= 95° Example: Find the angle measurements of trapezoid ABCD. AB CD

{Review AIA, AEA, CIA, and CA.}

∠ = ∠ =___?___ ____?___m DAB m CBA Solutions:

∠ =m DAB

135° ∠ =m CBA

120°

Example:

If KLMN is a parallelogram, find the value of x and y.

Opp. sides ≅ , so 9x = 36 → x = 4. Diag. bisect each other, so 3y – 7 = y + 15 → 2y = 22 → y = 11

**The Sliced Cube** The picture shows a cube of side length 2 units. M and N are the mid-points of sides DE and BG.

60° 45°

B

C D

A

9x36

3y-

y+1J

MN

LK


The Problem: What sort of shape is ANHM ? Calculate the lengths of AH and MN. Any comment about your previous answer now? Find the area of the shape ANHM. Solution: This concerns the ‘equal-diagonal’ property of a square. Clearly all four sides of the required shape are equal in length, so the choice is between a square (the obvious choice !) and a rhombus. We can use Pythagoras to find that AH is 2 3 units in length, while MN is 2 2 units in length. ANHM cannot be a square if it has unequal diagonals, so it must be a rhombus. Finally, the area of a rhombus is half the product of its diagonals 2 6= units².

Syllabus Objective: 5.5 - The student will write proofs involving special quadrilaterals.

Example: Write a two-column proof. Given: FGHJ Prove: ≅ ≅∠ ∠ ∠ ∠,F H J G

Statements Reasons 1) FGHJ 1) Given 2) ,FG JH FJ GH 2) Def. of Parallelogram

3) and are supplementary,and are supplementary,and are supplementary

F JJ HH G

∠ ∠∠ ∠∠ ∠

3) ║ lines cut by a transversal, CIA

are supp.

4) ≅ ≅∠ ∠ ∠ ∠,F H J G 4) ≅ supp. Th. Syllabus Objective: 5.6 - The student will solve problems involving properties of polygons. Review the Types of Quadrilaterals chart from p. 5 of these notes.

Reasoning with Quadrilaterals

J H

GF

The flow chart on page 3 of these notes can help students answer these. Movement from the bottom towards the top will always result in a true statement.

Movement from the top towards the bottom will only sometimes be true. Finally, movement sideways across the chart is never true.


Examples: Decide if the statement is Always, Sometimes, or Never true. a) A rhombus is a kite. a) Never b) A parallelogram is a rectangle. b) Sometimes c) A square is a quadrilateral. c) Always

Syllabus Objective: 5.7 - The student will perform constructions involving polygons. (supplemental material required)

Example: Construct an equilateral triangle. Mark a center, O, and draw a circle with a given radius. Draw the diameter, AB . Using A as the center and AO as the radius, draw an arc that intersects the circle at C and D. Connect B, C, and D to form a triangle. Example: Construct a square. Mark a center, O, and draw a circle with a given radius. Draw the diameter, AB . Draw another diameter, CD , which is the perpendicular bisector of AB . Connect A, D, B, and C to form a square.

Syllabus Objective: 5.8 - The student will find the measure of interior, exterior, and central angles of a given regular polygon. Compare and contrast interior, exterior, and central angle measurements of polygons in general.

It is very important to stress that students read carefully to understand exactly what they are being asked to find; a total or a single angle

measurement. They must also be aware if the polygon is regular or not.

Show an example with a triangle and have students create and measure exterior angles. Total the three angles up and then do

another example with a rectangle. Students will be able to come to the conclusion that the sum of the exterior angles of a polygon is

360°.

Discuss how to find the measure of each central angle. Guide students into examples that show how the central angles create a full circle, so you

can divide 360° by how many central angles the polygon has.

O C

B

D

A

O

C

B

D

A


Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360° . Corollary to Polygon Exterior Angles Theorem: The measure of each exterior angle of a regular

n-gon is 360n

.

Central angle of a regular polygon – is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. The central angle of a regular polygon can be

found by 360n

.

What if students are told the measure of an interior angle and asked to find how many sides the polygon has?

Example: The measure of an interior angle of a regular polygon is 156°. Find the number of sides in the polygon.

156 = ( )2 180−nn

→ ( )= −156 2 180n n → = −156 180 360n n → − = −24 360n →

15n = . OR

If an interior angle of a regular polygon is 156°, then the exterior angle will measure 24°. Since the sum of the exterior angles will measure 360° and all exterior angles are equal, one can divide 360 by 24 to get an answer of 15 sides.

Syllabus Objective: 5.9 - The student will utilize the distance, slope, and midpoint formulas to classify a given quadrilateral. Review: 1) distance formula 2) slope formula 3) midpoint formula

Students must be told that although a graph may look like a particular quadrilateral, it should only give them a hint of the direction their work should follow. There MUST be mathematical proof to justify

their answer. Review the distance, slope, and midpoint formulas. Also, discuss how you know if two lines are parallel.

I always point out what a hard formula this one is!

Or… ( )180 interior angle° − °

It depends what you already know! There’s no reason to make it harder than necessary.


**What conditions are required to prove that a given quadrilateral is a parallelogram?**

Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

**Ok, I know I have a parallelogram. How do I get more specific?**

Theorem: A parallelogram is a rhombus if and only if its diagonals are perpendicular. Theorem: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite

angles. Theorem: A parallelogram is a rectangle if and only if its diagonals are congruent.


**Now I don’t have a parallelogram. Can I classify these quadrilaterals?** Theorem: If a trapezoid is isosceles, then each pair of base angles is congruent. Theorem: If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. Theorem: A trapezoid is isosceles if and only if its diagonals are congruent. Theorem: If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Example 12: Use the figure below to prove that quadrilateral ABCD is an isosceles trapezoid, given that .BC AD

Proof: It must be shown that two sides are parallel and the two legs are congruent. Given BC AD, it must only be proven that AB CD≅ but not parallel. Begin by identifying the ordered pair for each point on quadrilateral ABCD. A(-1,-1), B(-3,2), C(-1,4), and D(2,2). Next, use the distance formula to determine the length of AB and CD . Use the slope formula to show that the two segments have different slopes.

4

2

-

-

- 5 1

B

C

D

A


AB = 2 22 1 2 1( ) ( )x x y y− + − = 2 2( 3 ( 1)) (2 ( 1))− − − + − − = 4 9+ 13.=

CD = 2 2

2 1 2 1( ) ( )x x y y− + − = 2 2(2 ( 1)) (2 4)− − + − = 9 4+ 13.= Therefore, AB = CD, or AB ≅ CD !

Slope of AB = 2 1

2 1

y yx x

−=

−2 ( 1)3 ( 1)

− −− − −

3 .2

=−

Slope of CD = 2 1

2 1

y yx x

−=

−2 4

2 ( 1)−

− −2 .

3−

=

Since they have different slopes, the two segments will intersect. It has now been shown that there is exactly one pair of opposite sides parallel and that the legs, AB and CD , are congruent. It can now be determined that quadrilateral ABCD is an isosceles trapezoid.


This unit is designed to follow the Nevada State Standards for Geometry, CCSD syllabus and benchmark calendar. It loosely correlates to Chapter 6 of McDougal Littell Geometry © 2004, sections 6.1 – 6.6 & 11.1. The following questions were taken from the 1st

Multiple Choice

semester common assessment practice and operational exams for 2008-2009 and would apply to this unit.

# Practice Exam (08-09) Operational Exam (08-09) 19. Use the triangle below.

What is the value of x?

A. 29 B. 33 C. 44 D. 49

Use the triangle below.


A. 15 B. 20 C. 25 D. 70

36. How many sides does a nonagon have? A. 7 B. 9 C. 11 D. 19

How many sides does a heptagon have? A. 6 B. 7 C. 16 D. 17

37. Which figure is a polygon?

A.

B.

C.

D.

Which figure is a polygon?

A.

B.

C.

D.

( )3 10x + ° 70°

x°

( )3 3x + °

45°

x°


38. A hexagon is shown below.

What is the value of a?

A. 90 B. 100 C. 130 D. 150

A pentagon is shown below.

What is the value of a?

A. 100 B. 110 C. 120 D. 150

39. Use the figure below.


A. 70 B. 60 C. 50 D. 40

The figure below is a hexagon with an external angle.


A. 20 B. 60 C. 80 D. 160

40. Parallelogram ABCD is given below.


A. 2 B. 3 C. 6 D. 16

Rectangle ABCD is given below.


A. 2 B. 3

C. 142

D. 295

41. What is the measure of each exterior angle of a regular hexagon?

A. 60° B. 90° C. 120° D. 135°

What is the measure of each exterior angle of a regular octagon?

A. 144° B. 135° C. 45° D. 36°

C

B

D

A

10x – 4

4x + 8

26

C

B

D

A 31

11x + 9

x°

150°

130°

100°

x°

60°

40°

130°

a°

110°

100°

a°

100°

150°

6(x + 4)


42. Which statement is true about a kite? A. A kite has 4 congruent sides. B. A kite has 2 pairs of parallel sides. C. A kite has perpendicular diagonals. D. A kite has congruent diagonals.

Which statement is true about a trapezoid? A. A trapezoid has 1 acute angle and 3 obtuse

angles. B. A trapezoid has 2 pairs of parallel sides and

4 sides. C. A trapezoid has 4 right angles and 2 pairs of

parallel sides. D. A trapezoid has 4 sides and exactly 1 pair of

parallel sides. 43. Which statement below is true about an

isosceles trapezoid? A. Both pairs of opposite sides are parallel. B. Both pairs of opposite sides are congruent. C. One pair of opposite sides is congruent and

the other is parallel. D. One pair of opposite sides is both parallel

and congruent.

Which statement below is true? A. All rhombi are squares. B. All squares are rectangles. C. All quadrilaterals are parallelograms. D. All rectangles are rhombi.

44. In the figure below, KLM ABC∆ ≅ ∆ .

Which statement must be true?

A. 8cmAC = B. 6cmBC = C. 53m A∠ = ° D. 80m C∠ = °

In the figure below, ABC XYZ∆ ≅ ∆ .

Which statement must be true?

A. 45m X∠ = ° B. 45m Z∠ = ° C. 3cmYZ = D. 3cmXY =

45. Use the rhombus below.

What is m CDE∠ ?

A. 25° B. 65° C. 90° D. 115°

In trapezoid GHIJ below, GH JI .


A. 75° B. 85° C. 95° D. 105°

J

I

G

H

x°

( )10x + °

A

B

D

C

E

65°

C

A

B

4 cm

5 cm

45°

Z

Y

X

80°

M

K

L

8 cm

10 cm

47°

B

C

A

53°


46. A regular polygon has interior angles that measure 144°. How many sides does this polygon have?

A. 6 B. 8 C. 10 D. 12

A regular polygon has interior angles that measure 150° . How many sides does this polygon have?

A. 30 B. 15 C. 12 D. 10

47. Use the figure below.


A. 64 B. 74 C. 116 D. 126

Use the figure below.


A. 23 B. 37 C. 43 D. 57

86°

123°

x° 75°

x° 41°


Free Response Practice Exam (08-09)

3. Show that the quadrilateral QUAD, having vertices Q(–1,–1), U(3,2), A(6,–2), and D(2,–5); is a square.

Operational Exam (08-09) 3. Prove that the quadrilateral QUAD, having vertices Q(–1,0), U(–3,6), A(0,7), and D(2,1) is a rectangle.

(A blank coordinate grid is provided.)


Sample SAT Question(s): Taken from College Board online practice problems.\

In the figure above, a shaded polygon which has equal sides and equal angles is partially covered with a sheet of blank paper. If 80x y+ = , how many sides does the polygon have?

(A) Ten (B) Nine (C) Eight (D) Seven (E) Six

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