worksheet chapter 5: discovering and proving polygon ... · s. stirling page 1 of 24 worksheet...
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Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 1 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 2 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 3 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 4 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 5 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 6 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 7 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 8 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 9 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 10 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
You will need your book to read the directions and answer some of these questions. Show how you are
finding your answers!
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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 11 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 12 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 13 of 24
Investigation 2: Trapezoid Midsegment Properties.
Page 276-277 (alternate) Steps 1 – 8:
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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 14 of 24
5.4 EXERCISES Page 277-279 #1 – 7, 14
5.4 Page 279 Exercise #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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 15 of 24
Lesson 5.5 Investigations Properties of Parallelograms
Investigation: Four Parallelogram Properties
Page 281-282 (alternate) Steps 1 – 4:
(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.
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 16 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 17 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.
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 18 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 19 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 20 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 21 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 22 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
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 23 of 24
5.6 Page 297 Exercise #28
CH 5 REVIEW EXERCISES Page 304-305
You will need your book to read the directions and answer some of these questions. Show how you are
finding your answers!
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.
Ch 5 Worksheet L2 Key Name ___________________________
S. Stirling Page 24 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