worksheet chapter 5: discovering and proving polygon ... · discovering and proving polygon...

Ch 5 Worksheet L2 Key Name ___________________________

S. Stirling of 24

Worksheet Chapter 5:

Discovering and Proving Polygon Properties Lesson 5.1 Polygon Sum Conjecture & Lesson 5.2 Exterior Angles of a Polygon

Warm up:

Definition: Exterior angle is an angle that forms a linear pair with one of the interior angles

of a polygon.

Measure the interior angles of QUAD to the nearest degree and put the measures into the diagram.

Draw one exterior angle at each vertex of QUAD. Measure each exterior angle to the nearest degree and

put the measures into the diagram.

How could you have calculated the exterior angles if all you had was the interior angles?

Each interior angle forms a linear pair with an exterior angle (supplementary)

Are any of the angles equal? No

What is the sum of the interior angles? ≈ 360

What is the sum of the exterior angles? ≈ 360

Now repeat the above investigation for the triangle TRI at the right. Compare the different angle sums

with the angle sums for the quadrilateral.

Are any of the angles equal? No

What is the sum of the interior angles? 180

What is the sum of the exterior angles? 360

Do you see a possible pattern? (Various conclusions)

Q

UA

D

mADQ = 72.26

mUAD = 86.28

mQUA = 59.70

mDQU = 141.77

T

R

I

mRIT = 60.21

mTRI = 81.14

mRTI = 38.64

60

72

86

142

120

94

108

38

39

81

60

120

141

99


S. Stirling of 24

Page 258-259 5.1 Investigation: Is there a Polygon Sum Formula? Steps 1-2: Review your work from page 1 and examine the diagrams below.

Step 3-4: Complete the sum of the interior angles column and drawing diagonals on the next page.

Page 262-263 5.2 Investigation: Is there an Exterior Angle Sum?

Steps 1-5: Review your work from page 1 and examine the diagrams below. One exterior angle is drawn

at each vertex. Complete the sum of the exterior angles column on the next page.

mIEF+mEFG+mFGH+mGHI+mHIE = 540.00

mIEF = 71

mEFG = 156

mFGH = 43

mGHI = 157

mHIE = 112

Pentagon EFGHI

I

H

G

F

E

mOJK+mJKL+mKLM+mLMN+mMNO+mNOJ = 720.00

mOJK = 112

mJKL = 159

mKLM = 108

mLMN = 105

mMNO = 140

mNOJ = 96

Hexagon JKLMNO

ON

M

LK

J

96

105

108

112

140

159

mDAB+mABC+mBCD+mCDA = 360.00

mDAB = 114

mABC = 77mBCD = 113

mCDA = 56

Quadrilateral ABCD

D

C

BA

77

113

56

114

Quadrilateral interior angle sum = 360

Pentagon interior angle sum = 540

43

71

157

156

112

Hexagon interior angle sum = 720

Octagon interior angle sum = 1080

mWPQ = 119

mPQR = 130

mQRS = 154

mRST = 132

mSTU = 131

mTUV = 137

mUVW = 131

mVWP = 147

Octagon PQRSTUVW

W V

U

T

SR

Q

P

132

119

130

137

131

131

147

154

Quadrilateral exterior angle sum = 360

Pentagon exterior angle sum = 360

Hexagon exterior angle sum = 360

B

I

J

K

M

G

C

D

E

F

A

H

63

73

66

33

54

72

D

C

B

A

HG

F

E

67

103

84

106

A

G

H

I

J

B

C

D

EF

80

104

59

61

56


S. Stirling of 24

Page 262-263 5.2 Investigation: Equiangular Polygons

Steps 7-8: Use what you know about interior angle sums and exterior angle sums to calculate the measure

of each interior and each exterior angle of any equiangular polygon.

Try an example first. Use deductive reasoning.

Find the measure of an interior and an exterior angle of an equiangular pentagon. Show your calculations

below:

One interior angle = 540 ÷ 5 = 108

One exterior angle = 360 ÷ 5 = 72

What is the relationship between one interior

and one exterior angle?

Supplementary, 108 + 72 = 180

Equiangular Polygon Conjecture

Or 180 360 180 360n n

n n n

More practice:

One exterior angle = 360 ÷ 6 = 60

What is the relationship between one interior

and one exterior angle?

Supplementary

Use this relationship to find the measure of one interior angle. 180 – 60 = 120

Use the formula to find the measure of one interior angle. (6 2)180 720

1206 6

Same results? Yes

Which method is easier? Finding one exterior angle

first, because sum is always 360.

You can find the measure of

each interior angle of an

equiangular n-gon by using

either of these formulas:

( 2)180n

n

or

360180

n

You can find the measure of

each exterior angle of an

equiangular n-gon by using the

formula:

360

n

108 72

120 60


S. Stirling of 24

5.1 EXERCISES Page 259-261 #3 – 8, 11, 13, 14, 12, 16.

Show how you are finding your answers!

a = 360 – 90 – 76 – 72 = 122

90

110

112

(6 – 2)180 = 720

b = (720 – 448)/2 = 136

e = (5 – 2)180/5

= 540/5

= 108

180 – 108 = 72

f = 180 – 2 * 72

= 180 – 144 = 36

108 72

72

44 102

Triangle: d = 180 – 44 – 30 = 106

Quad: c = 360 – 252 = 108

Penta: g = (540 – 225)/3 = 105

Quad: h = 360 – 278 = 82

122

360 – 108 – 130

= 122

j = 720/6 = 120

k = 360 – 322 = 38

60 120

120

142

60

18 sides

9 2 180140

9

140 360 – 200 = 160

360180 160

n

36020

360 20

18

n

n

n

2 180 2700

2 15

17

n

n

n

360180 156

36024

15

n

n

n


S. Stirling of 24

5.1 Page 260 Exercise #12

5.1 Page 261 Exercise #16 You are building the window frame below. You will need to know the

measures the angles in order to cut the trapezoidal pieces. Show and explain how you would calculate the

measures of the angles of the trapezoids

a = 116, b = 64, c = 90, d = 82, e = 99, f = 88,

g = 150, h = 56, j = 106, k = 74, m = 136, n = 118, p = 99

The angles of the trapezoid

measure 67.5 and 112.5.

Each angle of the octagon:

(8 2)180135

8

Around a point:

360 – 135 = 225

225 2 = 112.5

Angles between the bases are

supplementary.

180 – 112.5 = 67.5

135

112.5 112.5

67.5


S. Stirling of 24

5.2 EXERCISES Page 263-264 #1 – 6, 8 – 10

You will need your book to read the directions and answer some of these questions. Show how you are

finding your answers!

1. Sum exterior

angles decagon.

360 2. One exterior angle

equiangular

pentagon:

36072

5

equiangular hexagon:

36060

6

3. Number of sides if one

exterior angle measures

24:

36024

360 24

15

n

n

n

4. Number of sides if sum of interior

angles measures 7380:

( 2)180 7380

( 2)180 7380

180 180

2 41

43

n

n

n

n

112

Exterior angle sum is 360.

a = 360 – 252 = 108

40

Exterior angle sum is 360.

360 – 112 – 43 - 69 = 136

136/3 = 45.333

108

Pentagon: (5 – 2)180/5 = 108

Octagon: (8 – 2)180/8 = 135

e = 180 – 108 = 72

f = 180 – 135 = 45

g = 360 – 108 – 135 = 117

h = 360 – 117 – 72 – 45 = 126

108 135

135


S. Stirling of 24

136

44

30

44

106

30

a = 180 – 18 = 162

g = 180 – 86 – 39 = 55

d = 39 Isos. triangle

c = 180 – 39 * 2 = 102

e = (360 – 102)/2 = 129

f = 90 – 39 = 51

Large Pentagon:

540 – 94 – 90 – 162 = 194

h = 194/2 = 97

b = 180 – 97 = 83

Quad:

k = 360 – 129 – 51 – 97 = 83

162

55

39

102

129 129

51

97 97

83

83

Triangle: a = 180 – 56 – 94 = 30

b = 30 ||, alt. int. angles =

Triangle: c = 180 – 44 – 30 = 106

d = 180 – 44 = 136


S. Stirling of 24

5.3 Investigations Kites Review kite vocabulary on page 2 of your notes! Use this vocabulary in your answers.

Investigation: Angles of a Kite

(a) Look at the angles of the kites below.

Are any of the angles of the kite congruent? Which ones? Yes the non-vertex angles.

Are any of the angles of the kite supplementary? No

51

109

109

90

D

Q

A

U

90

90

104

75

B

FE

C

67135

91

67

H

G

K

O

(b) Now we will look at the angles formed when we draw the diagonals into the kites.

Are any of the angles formed congruent? Yes, the angles formed by the diagonal and vertex angles.

Are any of the angles formed supplementary? No

Are any of the diagonals angle bisectors? Yes, the one diagonal bisects the vertex angles.

(The non-vertex angles are not bisected!!)

How are the diagonals related to each other (based on the angle measures formed)?

The diagonals are perpendicular. (They form 90 angles.)

64

64

9049

26

41

26

49

41

D

Q

A

U

39

51

51

39

39

39

90 51

51

B

FE

C

2367

47

4343

67

23

47

90

H

G

K

O


S. Stirling of 24

Investigation: Segments in a Kite

(c) Now we will look at the segments formed when we draw the diagonals into the kites.

Are any of the segments of the kite congruent? Do you have any segment bisectors?

Yes the diagonal that connects the non-vertex angles is being bisected by the diagonal that connects the

vertex angles.

1.71.7

3.5

1.9

P

Q

D

A

U

5.3 Investigations Trapezoids Investigation: Angles of a Trapezoid

(a) Look at the trapezoids below. Are there any special relationships between the angles in a trapezoid?

(Besides the fact that the sum of the interior angles is 360.) Are there any unique relationships

between angles in an isosceles trapezoid?

65

132

48

115 C

B

A

D

Are any of the angles formed congruent? Yes, the base angles of the isosceles trapezoids are congruent.

Are any of the angles formed supplementary? Why?

Yes, the consecutive angles between the bases are supplementary.

The parallel bases always form pairs of same-side interior angles which are supplementary.

2.3

2.3 1.8

2.8Q

F

B

E

C

2.5

2.5

1.1

2.7

R

G

HK

O

30

74

106

150

D

GE

F57

57

123

123

KP

N

H

134

46

134

46

T

Q

S

R


S. Stirling of 24

Investigation: Diagonals of a Trapezoid

(b) Look at the trapezoids below. Do the diagonals of a trapezoid (or an isosceles trapezoid) have any

special relationships?

Do the diagonals bisect any angles? No

Do the diagonals bisect any segments? No

Are any of the segments formed congruent?

5.3 EXERCISES Page 271-273 #1 – 8, 11 – 13



EG = 1.23 in.

DF = 2.13 in.

EG

D

F

BC = 1.29 in.

AD = 2.41 in.

A

B

C

D TR = 1.83 in.

QS = 1.83 in.

T

R

S

Q

HN = 1.56 in.

PK = 1.56 in.

K

H

N

P

Yes, the diagonals are congruent

in an isosceles trapezoid.

Perimeter:

20 * 2 + 12 * 2 = 64

12

20

Non-vertex angles =. y = 146

x = 360 – 47 – 146 * 2 = 21

146

Isos. so base angles =. y = 128

Consecutive angles supplementary.

x = 180 – 128 = 52

128

52

21


S. Stirling of 24

Perimeter:

85 = 37 +18 + 2x

85 = 55 + 2x

30 = 2x

15 = x

29

x

90

Small Right triangle:

x = 180 – 90 – 18 = 72 Large Right triangle:

y = 180 – 90 – 29 = 61

Perimeter:

164 = y + 2(y +12) + (y – 12)

164 = y + 2 y + 24 + y – 12

164 = 4 y + 12

152 = 4 y

38 = y

81

x = 180 – 81

= 99

99

9

30 30

45

30

w = 180 – 2 * 30

= 120

3.0 cm

1.6 cm

48 90

y = 180 – 90 – 48

= 42 42

Vertex angle

11. One possible

answer:

E I

12. One possible answer:

Other base ZI

Base angles:

and Q U

and Z I

13. One possible answer:

Other base OW

Base angles:

and S H , and O W

SW HO

15 15


S. Stirling of 24

Lesson 5.4 Investigations Properties of Midsegments

Investigation 1: Triangle Midsegment Properties

Page 275-276 (alternate) Steps 1 – 3:

Draw the midsegments. You should have 4 small triangles.

Use tracing paper and copy one of the 4 triangles.

Compare all 4 triangles by sliding the copy of one small

triangle over the three others.

Are there any relationships?

Steps 4 – 5: (Review Corresponding Angles Conjecture for parallel lines. The F shape!)

Draw one midsegment in the triangle below.

(Each person in your group should draw a different midsegment.)

Measure all of the angles. Using the Converse of the Corresponding

Angles Conjecture, what conclusions can you make about a midsegment

and the large triangle’s third side?

Compare the length of the midsegment to

the large triangle’s third side. How do they relate?

E

D

F

A

B

C

M

N

O

H

K

G

Yes, all 4 triangles are congruent.

ADF DBE FEC EFD

The midsegment is parallel to the 3rd side.

Various examples:

MN HG because corresponding angles are

congruent.

The midsegment is half the length of the 3rd

side. Various examples:

1

2MN HG

24

24

7 cm


S. Stirling of 24

Investigation 2: Trapezoid Midsegment Properties.


Draw the midsegment. Find all angle measures. Compare the angle measures.

Are the corresponding angles congruent? What can you conclude about the midsegment and the bases?

Measure the lengths of the midsegment and the bases. Is there a relationship between the three segments?

110

70

110

70

144

144

36

36

F

E

A

B

C

D

R

Z

T

P

R

Z

T

P

2.4

3.8

1

F

E

A

B

C

D

The midsegment is parallel to the bases.

EF AB CD because corresponding angles are congruent.

The midsegment is half the length of the sum of the bases.

Or The midsegment is the average of the bases.

1

2EF AB CD or

2

AB CDEF


S. Stirling of 24

5.4 EXERCISES Page 277-279 #1 – 7, 14


a = 54, b = 72, c = 108, d = 72, e = 162, f = 18, g = 81,

h = 49.5, i = 130.5, k = 49.5, m = 162, n = 99

65

8 10

Perimeter TOP =

8 + 2*10 = 28

10

Triangle 3 midsegments.

Trapezoid 1 midsegment.

60

140

Corresponding

angles of

congruent

triangles.

6 8

9

Corresponding sides of

congruent triangles.

Perimeter = 6 + 8 + 9 = 23

73

m = 180 – 51

= 129

1

36 48 422

p

1

24 132

48 13

35

q

q

q

||, same side int.. angles supp.

y = 180 – 40

= 140

||, corr. angles = . x = 60


S. Stirling of 24

Lesson 5.5 Investigations Properties of Parallelograms

Investigation: Four Parallelogram Properties


(a) What about the angles in a parallelogram?

Look at the opposite angles of the parallelogram.

Any relationships?

Look at the measures of each pair of consecutive

angles in each parallelogram. Any relationships?

How can you use the two conjectures to find the measures of a

parallelogram with only one angle measure given?

(b) How about the sides of a parallelogram? Any relationships?

52

12852

128

C

A D

B

116

116

64

64

Q

P

R

K

133

133

47

47

G

F

H

E

38

U

S

V

T

The opposite angles of a

parallelogram are congruent.

The consecutive angles of a

parallelogram are supplementary.

(because they are same-side

interior angles to parallel lines)

180 38 142m T consecutive angles of a

parallelogram are supplementary

142m T m V and 38m S m U

opposite angles of a parallelogram are congruent.

1.15

1.15

2.78

2.78C

A D

B

1.37

1.37

1.42

1.42

Q

P

R

K

0.84

0.84

2.222.22

G

F

H

E

The opposite sides of a

parallelogram are congruent.


S. Stirling of 24

(c) Now consider the diagonals of a parallelogram. What can you conclude about the point of

intersection of the diagonals? How do the diagonals relate?

(d) How do the diagonals relate to the angles of the parallelogram? Do either of the diagonals bisect any

of the angles of the parallelogram? Do the diagonals intersect at a 90? Could they?

1.2

1.2

0.75

0.75

O

Q

K

R

P

1.2

1.2

0.8

0.8

N

G

EH

F

1

1.8

1.8

1

M

CB

DA

The intersection is the midpoint of the diagonals.

The diagonals are bisectors of each other.

The diagonals aren’t equal!

None of the angles are being bisected.

None of the diagonals intersect at a 90, but they

could.

Because of the parallel sides, you can get many

pairs of congruent alternate interior angles.

Since FG EH , 2 6 and 3 7

Since FE GH , 1 5 and 4 8

8

76

5

2

41

3

GF

HE


S. Stirling of 24

5.5 EXERCISES Page 283-285 #1 – 6, 7 – 8, 15 – 17.

Show how you are finding your answers!

34 cm

a = 180 – 48

= 132

27 cm 48

14 cm

16 cm

18

18

24

24

21

21

17

23

63

78

Perim

= 18 + 24 + 21

= 63

x – 3 = 17

x = 20

20 + 3 = 23

Perim = 2*17 + 2*23 = 80

L

1. Duplicate angle L.

2. Copy LA

3. Copy TL = AS

4. Make opposite sides =. 1. Copied L.

2. Measure & draw LA.

3 Make A = 130,

since consecutive

angles supp.

4. Measure & draw AS

= LT. Opp. Sides =.

5. Draw TS.


S. Stirling of 24

8. Construct parallelogram DROP. Given side DR and diagonals DO and PR .

D O

P R

D R

D O

78

Kite: non-vertex angles =. 102 102

Trap: 180 – 78 = 102

Isos, base angles =.

x = 360 – 102 – 154 = 104

y = 360 – 102 – 160 = 98

1. Make diagonals bisect

each other.

2. Construct intersection

of the opposite sides

Penta: 540 ÷ 5 = 108

90 69

108

Hexa:

720 ÷ 6 = 120

120

d = 360 – 90 – 120 – 108 = 42

e = (180 – 42)/2 = 69

42

90

(8 – 2)180 = 1080

1080 ÷ 8 = 135

135

check tiling:

90 + 2*135 = 360


S. Stirling of 24

5.6 Investigations Properties of Special Parallelograms

(alternate) Investigation 2: Do Rhombus Diagonals Have Special Properties?

(a) The diagonals and the sides of the rhombus form two angles at each vertex. Compare each pair of

angles in the rhombi below. What do you observe? Equal angles. Each diagonal bisects the opposite

pairs of angles.

Now examine the measures of the angles formed by the intersection of the two diagonals. Are the

diagonals perpendicular? Yes Does that happen for every parallelogram? NO

(b) How do the diagonals relate to each other? Any segments equal? Are the diagonals equal?

Since a rhombus is a parallelogram, the diagonals bisect each other. They are not equal.

(c) What are all of the properties of a rhombus?

6060

6060

30

3030

30

90 N

G

E

F

H

90

69

69

69

69 21

21

21

21

M

CB

A D

0.85

1.5

1.72

1.72

0.85

1.5

N

G

F

E

H

1.83

1.71

0.651.71

0.65

1.83 M

C

A

B

D

Since a rhombus is a parallelogram, I know…

opposite angles are congruent

consecutive angles are supplementary

opposite sides are congruent

the diagonals bisect each other

Since a rhombus is a quadrilateral, I know…

the sum of the interior angles is 360

The properties specific to a

rhombus are…

all sides are congruent

opposite angles are

bisected by the diagonals

diagonals are

perpendicular


S. Stirling of 24

(alternate) Investigation 3: Do Rectangle Diagonals Have Special Properties?

Why is each angle of a rectangle 90? Because the sum is 360 and 360 4 equal angles = 90.

(a) Look at the diagonals of rectangles and compare their lengths. They are congruent.

Recall that the diagonals also have the properties of a parallelogram’s diagonals. So what else do you

know about the diagonals of a rectangle? They bisect each other.

Are the diagonals perpendicular? Could they be? Only if the rectangle is a square.

(b) What happens if you combine a rectangle and a rhombus? Since a square is a parallelogram, a

rectangle and a rhombus, what properties does a square have? Find all of the measures of Square ABDC.

0.87

3.9

2

2

2

277M

C

A D

B1.2

1.2

1.2

57

1.97

1.2

1.27

N

G

H

E

F

2.3

1.6

M

D

C

B

ASince a square is a parallelogram, I know…

opposite angles are congruent

consecutive angles are supplementary

opposite sides are congruent

the diagonals bisect each other

Since a square is a quadrilateral, I know…

the sum of the interior angles is 360

Since a square is a rhombus, I know…

all sides are congruent

opposite angles are bisected by the diagonals

diagonals are perpendicular

Since a square is a rectangle, I know…

diagonals are congruent

Squares are composed of a

special type of right triangle,

a 45:45:90 triangle, or right

isosceles triangle. How

many can you find?

4 congruent to BAC

4 congruent to AMB

45

45 90

45

45


S. Stirling of 24

5.6 EXERCISES Page 294-297 #1 – 13, 17 – 19, 21, 28

On #17 – 19, you may draw and/or construct. Label measures and show arc marks.

y = 180 – 48 – 95 = 37

10

10 10

20

95 90

45

L V Make diagonals perpendicular

bisectors of each other and

equal in length.

O

Sometimes

Always

Always

Sometimes

Always

Sometimes

Always

Always

Always

Sometimes: only if the

parallelogram is a rectangle.

90

3.1 cm

3.1 cm 3.1 cm

3.1 cm


S. Stirling of 24

They check to make sure the

diagonals are equal. This will

guarantee a rectangle.

Use opposite angles are bisected by the diagonal.

Use opposite angles are equal.

1. bisect angle B.

2. duplicate each angle at points B and K.

Where the sides intersect form points A and E.

B K

A

E

Use diagonals of rectangle are

equal.

Use opposite sides of a

parallelogram are equal.

1. bisect PE, call midpoint A.

2. make a circle with diameter PE.

3. construct sides PI = ES.

4. join E to I and P to S

I

S

P E

37

37

37

37

6.6 cm

4.6 cm 4.6 cm

4.9 cm

4.9 cm

A


S. Stirling of 24


CH 5 REVIEW EXERCISES Page 304-305



a = 54, b = 36, c = 72, d = 108, e = 36, f = 144, g = 18,

h = 48, i = 48, k = 84

y = 180 – 140

= 40 80

90

x = 180 – 170 = 10

266 = 94 + 52 + 2x

266 = 146 + 2x

120 = 2x

60 = x

c = 180 – 116

= 64

64

116

20

26

36

Perim = 36+20+18+26 = 100

32 = ½ (x – 12 + x)

64 = 2x – 12

76 = 2x

38 = x

1. One exterior

angles of a regular

polygon.

360

n

5.

Measure from corner to corner. The diagonals of the rectangle

should be =. And if it is a rectangle, you will have right

angles.


S. Stirling of 24

Exercise #13

Kite Isosceles

trapezoid

Parallelogram Rhombus Rectangle Square

Opposite sides are

parallel

No One pair Yes Yes Yes Yes

Opposite sides are

congruent

No One pair Yes Yes Yes Yes

Opposite angles are

congruent

Non-Vertex No Yes Yes Yes Yes

Diagonals bisect each

other

No No Yes Yes Yes Yes

Diagonals are

perpendicular

Yes No No Yes No Yes

Diagonals are

congruent

No Yes No No Yes Yes

Exercise #15

a = 120, b = 60,

c = 60, d = 120,

e = 60, f = 30,

g = 108, m = 24,

p = 84

17 = ½ y

34 = y

34 = ½ (17 + z)

68 = 17 + z

51 = z

34

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