Ch 5 Worksheets Key Name ___________________________ S. Stirling Page 1 of 20 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 U A D mADQ = 72.26mUAD = 86.28mQUA = 59.70mDQU = 141.77T R I mRIT = 60.21mTRI = 81.14mRT I = 38.6460 72 86 142 120 94 108 38 39 81 60 120 141 99
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Worksheet Chapter 5: Discovering and Proving Polygon ...€¦ · S. Stirling Page 1 of 20 Worksheet Chapter 5: Discovering and Proving Polygon Properties ... Finding one exterior
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Ch 5 Worksheets Key Name ___________________________
S. Stirling Page 1 of 20
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
Ch 5 Worksheets Key Name ___________________________
S. Stirling Page 2 of 20
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.
mDAB+mABC+mBCD+mCDA = 360.00
mDAB = 114
mABC = 77mBCD = 113
mCDA = 56
Quadrilateral ABCD
D
C
BA
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
mHAB+mEBC+mFCD+mGDA = 360.00
mHAB = 67
mEBC = 103
mFCD = 84
mGDA = 106
D
C
B
A
HG
F
E
mFAB+mGBC+mHCD+mIDE+mJEA = 360
mJEA = 61
mIDE = 56
mHCD = 104
mGBC = 80
mFAB = 59A
G
H
I
J
B
C
D
EF
B
I
J
K
M
G
C
D
E
F
A
H
mMFA = 72
mKEF = 54
mJDE = 33
mICD = 73
mHBC = 66
mGAB = 63
mGAB+mHBC+mICD+mJDE+mKEF+mMFA = 360
67
103
84
106
80
104
59 61
56
63
73
66
33
54
72
77
113 56
114
mIEF+mEFG+mFGH+mGHI+mHIE = 540.00
mIEF = 71
mEFG = 156
mFGH = 43
mGHI = 157
mHIE = 112
Pentagon EFGHI
I
H
G
F
E 43 71
157
156
112
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
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
Ch 5 Worksheets Key Name ___________________________
S. Stirling Page 3 of 20
Page 262-263 5.2 Investigation: Is there an Exterior Angle Sum?
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
Ch 5 Worksheets Key Name ___________________________