Download Lesson 5.3 Kite and Trapezoid Properties

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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
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?
Are any of the angles equal?
What is the sum of the interior angles?
What is the sum of the exterior angles?
Now repeat the above investigation for the triangle TRI at the right. Compare the different angle sums
with the angle sums for the quadrilateral. Do you see a pattern?
Are any of the angles equal?
What is the sum of the interior angles?
What is the sum of the exterior angles?
R
D
Q
I
T
U
A
S. Stirling
Page 1 of 10
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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 3-5: Complete the sum of the interior angles column and drawing diagonals on the next page.
Hexagon JKLMNO
Quadrilateral ABCD
mBCD = 113
mABC = 77
mDAB = 114
mCDA = 56
mLMN = 105
mMNO = 140
mNOJ = 96
mOJK = 112
mJKL = 159
mKLM = 108
mOJK+mJKL+mKLM+mLMN+mMNO+mNOJ = 720.00
mDAB+mABC+mBCD+mCDA = 360.00
B
A
L
K
J
M
C
N
D
O
Oc tagon PQRSTUVW
mWPQ = 119
mPQR = 130
mQRS = 154
Pentagon EFGHI
mGHI = 157
mIEF = 71
mEFG = 156 mHIE = 112
mFGH = 43
mIEF+mEFG+mFGH+mGHI+mHIE = 540.00
mRST = 132 mUVW = 131
mST U = 131 mVWP = 147
mTUV = 137
S
R
Q
F
T
G
E
U
P
H
I
V
W
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. Complete the sum of the
exterior angles column on the next page.
F
C
E
B
I
mHAB = 67
D
mEBC = 103
C
H
mFCD = 84
A
mGDA = 106
G
H
D
mHAB+mEBC+mFCD+mGDA = 360.00
B
J
E
C
G
G
A
D
K
F
mFAB = 59
E
M
A
m GAB+m HBC+m ICD+m JDE+m KEF+m MFA = 360
mGBC = 80
mHCD = 104
mIDE = 56
mJEA = 61
J
mFAB+mGBC+mHCD+mIDE+mJEA = 360
S. Stirling
m JDE = 33 
m KEF = 54 
m MFA = 72 
I
B
F
m GAB = 63 
m HBC = 66 
m ICD = 73 
H
Page 2 of 10
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(Investigation 5.1 Step 5: Draw all possible diagonals from one vertex, which divides each polygon into triangles.
Use these to develop a formula for the Polygon Sum Conjecture.
Quadrilateral
Pentagon
Hexagon
Diagonal forms 2 triangles, so
Diagonals form __ triangles:
Diagonals form __ triangles:
____ (180) = _______
____ (180) = _______
____ (180) = _______
Octagon
Decagon
Diagonals form __ triangles:
Diagonals form __ triangles:
____ (180) = _______
____ (180) = _______
Investigation 5.1 and 5.2 Summary
Use the results from Lesson 5.1 and 5.2 to fill in the table at the left. (The last column of the table should be
completed after 5.2 Investigation.)
Number of
sides of a
polygon
Sum of
measures of
interior angles
Sum of measures
of exterior angles
(one at a vertex)
3
4
5
6
7
8
9
10
11
12
13
n
180
360
S. Stirling
Polygon Sum Conjecture.
The sum of the measures of the n angles of an ngon is
Exterior Angle Sum Conjecture.
The sum of the measures of a set of exterior angles
of an n-gon is
Page 3 of 10
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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:
Find the measure of an interior and an exterior angle of an equiangular pentagon. Show your calculations
below:
One interior angle =
One exterior angle =
What is the relationship between one interior
and one exterior angle?
Equiangular Polygon Conjecture
You can find the measure of
each interior angle of an
equiangular n-gon by using
either of these formulas:
You can find the measure of
each exterior angle of an
equiangular n-gon by using the
formula:
One exterior angle =
What is the relationship between one interior
and one exterior angle?
Use this relationship to find the measure of one interior angle.
Use the formula to find the measure of one interior angle.
Same results?
S. Stirling
Page 4 of 10
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Lesson 5.3 Kite and Trapezoid Properties
Definition of kite
Name ___________________________
Label vocab in the drawing.:
K
E
I
Measure then compare the opposite angles of the kite.
Which pair will be congruent?
I
T
vertex angles (of a kite) The angles
of congruent sides.
mDIA = 90
nonvertex angles (of a kite) The two
T
angles between consecutive noncongruent
The diagonals of the kite (c ompared to eachother):
sides of a kite.
DI = 1.57 cm IB = 1.57 cm
Kites
K
between
thekite:
pairs
Angle formed by the diagonals
of the
E
IA = 1.68 cm CI = 4.57 cm
Kites
Kite Angles Conjecture
(Dodiagonals
proof onof Ch
WS page 2.)
Angle formed by the
the5kite:
The angles of the kite:
mDIA = 90
The
angles of a kite mDCB
are = 38 mCBA = 118
The diagonals of the kite (c ompared mBAD
to eachother):
= 86 mADC = 118
Kites
DI = 1.57
cm IB
= 1.57
cm
Additional measures:
Label
the
diagram
with the measures to help you write the conjectures.
Angle formed by the diagonals of the kite:
The diagonals of the kite and how they divide the angles of the kite:
IA = 1.68 cm CI = 4.57 cm
D
mDIA = 90
The vertex angles:
The angles of the kite:
The diagonals of the kite (c ompared to eachother):
mDCB = 38
DI = 1.57 cm IB = 1.57 cm
mBAD = 86
IA = 1.68 cm CI = 4.57 cm
The nonvertex angles:
mDCI = 19
mCDI = 71
mCBA = 118
mICB = 19
mADC = 118
mIDA = 47
mBAI = 43
mIBA = 47
mIAD = 43
mIBC
The diagonals of the kite and how they divide the angles of the
kite: = 71
The angles of the kite:
The vertex angles:
The nonvertex angles:
mDCB = 38
mBAD = 86
mCBA = 118
mDCI
= 19
mADC
= 118
mCDI = 71
mICB = 19
mIDA = 47
A
I
The diagonals of the kite
and how
angles of the kite:
mBAI
= 43they divide the
mIBA = 47
The vertex angles:
B
mIAD = 43
The nonvertex angles:mIBC = 71
mDCI = 19
mCDI = 71
mICB = 19
mIDA = 47
C
Kite
Angle
Conjecture
mBAI
= 43 Bisector
mIBA = 47
mIAD
= 43 anglesmIBC
71 are
The
vertex
of a =kite
Kite Diagonals Conjecture
The diagonals of a kite are
(Do proof on Ch 5 WS page 2.)
by a
(Do proof on Ch 5 WS page 3.)
Kite Diagonal Bisector Conjecture (Do proof on Ch 5 WS page 3.)
The diagonal connecting the vertex angles of a kite is the
of the other diagonal.
S. Stirling
Page 5 of 10
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Definition of trapezoid
Label vocab in
the drawing:
Definition of isosceles trapezoid
C
B
Measure the angles of the trapezoids below. Label the diagram
with the measures to help you write the conjectures.
B
D
mDC A = 59
D
mCAB = 121
mBDC = 139
mABD = 41
A
bases (of a trapezoid) The two
parallel sides.
base angles (of a trapezoid) A pair
of angles with a base of the
trapezoid as a common side.
legs are the two nonparallel sides.
C
mABC = 143
mDAB = 37
mBCD = 143
mCDA = 37
B
A
D
A
R
C
Trapezoid Consecutive Angles Conjecture
The consecutive angles between the bases of a trapezoid are
A
P
T
Isosceles Trapezoid [Base Angles] Conjecture
The base angles of an isosceles trapezoid are
I
S
C
Measure diagonals of the trapezoids below.
Isosceles Trapezoid Diagonals Conjecture
The diagonals of an isosceles trapezoid are
S. Stirling
(Do proof on Ch 5 WS page 4.)
Page 6 of 10
O
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Lesson 5.4 Properties of Midsegments
Page 275-276 Investigation 1: Triangle Midsegment Properties
Midsegment (of a triangle) is the line segment connecting the midpoints of the two sides.
Steps 1 – 3:
C
A
Three Midsegments Conjecture
The three midsegments of a triangle divide it into
B
Steps 4 – 5: (Review Corresponding Angles Conjecture for parallel lines. The F shape!) Also label the
drawing below.
R
Triangle Midsegment Conjecture
A midsegment of a triangle is
third
side and
the length of
to the
I
T
Page 276-277 Investigation 2: Trapezoid Midsegment Properties.
Midsegment (of a trapezoid) is the line segment connecting the midpoints of the two
nonparallel sides.
P
Steps 1 – 8: If you do not have tracing paper, you
may measure instead. Label angles with
measures!!
A
N
M
R
T
Trapezoid Midsegment Conjecture
The midsegment of a trapezoid is
to the bases and
is equal in length to
Side lengths:
S. Stirling
Page 7 of 10
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Lesson 5.5 Properties of Parallelograms
Name ___________________________
Page 281-282 Investigation: Four Parallelogram Properties
Definition of a Parallelogram: A quadrilateral with two pairs of opposite sides parallel.
Steps 1 – 4: Angles!
Parallelogram Opposite Angles
Conjecture (Do proof on Ch 5 WS page 5.)
The opposite angles of a
parallelogram
L
M
J
K
Parallelogram Consecutive
Angles Conjecture
The consecutive angles of a
parallelogram
So to find all angles of a parallelogram:
Steps 5 – 7: Side and diagonal lengths!
P
A
Parallelogram Opposite Sides
Conjecture (Do proof on Ch 5 WS page 5.)
The opposite sides of a parallelogram
Parallelogram Diagonals Conjecture
(Do proof on Ch 5 WS page 6.)
The diagonals of a parallelogram
L
S. Stirling
R
Page 8 of 10
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5.6 Properties of Special Parallelograms
Name ___________________________
Definition of a Rhombus: A quadrilateral with all sides congruent.
Page 291 Investigation 1: What Can You Draw with the Double-Edged Straightedge?
Steps 1 – 3: Complete in the space below. Use your ruler!
Double-Edged Straightedge
Conjecture
If two parallel lines are intersected by a
second pair of parallel lines that are the
same distance apart as the first pair,
then the parallelogram formed is a
Page 292 Investigation 2: Do Rhombus Diagonals Have Special Properties?
Steps 1 – 3:
H
R
O
M
Rhombus Diagonals Conjecture
The diagonals of a rhombus are
(Do proof on Ch 5 WS page 7.)
Rhombus Diagonals Angles Conjecture
The
of a rhombus
S. Stirling
, and they
(Do proof on Ch 5 WS page 6.)
the angles of the rhombus.
Page 9 of 10
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Definition of a Rectangle: A quadrilateral with all angles congruent.
Page 293-4 Investigation 3: Do Rectangle Diagonals Have Special Properties?
Steps 1 – 2:
R
E
Rectangle Diagonals Conjecture
The diagonals of a rectangle are
and
T
C
Definition of a Square: A quadrilateral with all angles and sides congruent
Square Diagonals Conjecture
The diagonals of a square are
A
U
S
Q
S. Stirling
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