Name
SECTION
6A
Date
Class
Ready to Go On? Skills Intervention
6-1 Properties and Attributes of Polygons
Find these vocabulary words in Lesson 6-1 and the Multilingual Glossary.
Vocabulary
side of a polygon
vertex of a polygon
diagonal
regular polygon
concave
convex
Identifying Polygons
Tell whether each figure is a polygon. If it is a polygon, name it by the
number of its sides.
Is it a plane figure?
Is the figure closed ?
Is the figure formed by three or more segments?
Do the segments intersect only at their endpoints?
Are any of the segments with common endpoints collinear?
Is the figure a polygon?
How many sides does it have?
Name the polygon by the number of sides it has.
Finding Interior Angle Measures and Sums in Polygons
A. Find the sum of the interior angle measures of a convex 15-gon.
The sum of the interior angle measures of a convex polygon with
n sides is (n )180. What does the n represent?
Substitute 15 for n in the formula. (
2)180
Simplify the expression.
B. Find the measure of each interior angle of a regular 15-gon.
What is a regular polygon?
From Part A, the sum of the interior angle measures of a 15-gon is
.
How do you find the measure of each interior angle of a regular polygon?
To find the measure of each interior angle of a regular 15-gon,
divide 2340 by
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.
2340
______
76
Holt Geometry
Name
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Date
Class
Ready to Go On? Problem Solving Intervention
6-1 Properties and Attributes of Polygons
The Polygon Exterior Angle Sum Theorem states that the sum
of the exterior angle measures, one angle at each vertex, of a
convex polygon is 360.
A decorative garden in the shape of a pentagon is surrounded
by five railings. Find the measure of each exterior angle of the
garden.
14x °
A
8x °
E
B
12x °
C
17x °
9x °
D
Understand the Problem
1. What is a pentagon?
2. What is an exterior angle?
Make a Plan
3. The Polygon Exterior Angle Sum Theorem states that the sum of the exterior angle
measures, one angle at each vertex, of a convex polygon is
.
4. How will you find the measure of each exterior angle?
Solve
5. Substitute the given angle measures into the Polygon Exterior Angle Sum Theorem.
17x 12x 360
6. Combine like terms and solve for x.
7. Substitute the value of x into each given angle measure.
Look Back
8. Find the sum of the angle measures you found in Exercise 7.
9. What is the sum of exterior angle measures of a convex polygon?
10. Is your answer in Exercise 8 the same as your answer from Exercise 9?
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77
Holt Geometry
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SECTION
6A
Date
Class
Ready to Go On? Skills Intervention
6-2 Properties of Parallelograms
Find this vocabulary word in Lesson 6-2 and the
Multilingual Glossary.
Vocabulary
parallelogram
Using Properties of Parallelograms to Find Measures
䊐PQRS is a parallelogram. PT ⴝ 53, PS ⴝ 76, and m⬔QRS ⴝ 75ⴗ.
Find each measure.
Q
A. RT
T
4x + 1
In a parallelogram, the diagonals
_
R
each other, so PT and PT Since PT 53, RT .
.
7x – 41
P
S
B. QR
_
, so QR In a parallelogram, opposite sides are
and QR . Since PS 76, QR ,
.
C. mRQP
In a parallelogram, consecutive angles are
.
mRQP mQRS Substitute 75 for mQRS. mRQP Solve to find mRQP. mRQP D. mRSP
angles are congruent. RSP In a parallelogram,
mRSP . What is mRQP ?
Substitute to find mRSP. mRSP E. mQPS
mQPS . Substitute to find mQPS. mQPS F. RS
_
In a parallelogram, opposite sides are congruent so RS , and RS .
Substitute the given values for RS and QP.
7x 41
Subtract 4x from both sides.
3x 41
42 Add 41 to both sides.
x
Divide both sides by 3.
Substitute your solution into 7x 41 and simplify to find RS. 7(
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78
) 41 Holt Geometry
Name
SECTION
6A
Date
Class
Ready to Go On? Problem Solving Intervention
6-2 Properties of Parallelograms
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
The parking spots in a parking lot are in the shape of a parallelogram.
If m⬔DAB ⫽ 65⬚ and AE ⫽ 13.3 ft, find m⬔CDA, m⬔CBA, EC, DC, and DA.
C
D
Understand the Problem
1. Why is it important that the problem states that the parking spots
A
E
11 ft
B
20 ft
are in the shape of parallelograms?
Make a Plan
2. What do you know about consecutive angles of a parallelogram?
3. What do you know about opposite angles of a parallelogram?
4. What do you know about opposite sides of a parallelogram?
5. What do you know about the diagonals of a parallelogram?
Solve
6. Find m⬔CDA. Explain how you determined your answer.
7. Find m⬔CBA. Explain how you determined your answer.
8. Find EC. Explain how you determined your answer.
9. Find DC and DA. Explain how you determined your answer.
Look Back
10. Do your answers to Exercises 6–9 satisfy all of the properties of parallelograms
from Exercises 2–5?
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79
Holt Geometry
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Class
Ready to Go On? Skills Intervention
6-3 Conditions for Parallelograms
Verifying Figures are Parallelograms
Show that ⵥJKLM is a parallelogram for x ⫽ 9 and y ⫽ 2.5.
J
For JKLM to be a parallelogram, both pairs of
when x 9 and y 2.5.
have to be
10y + 3
4x + 6
K
6x – 12
Step 1 Find JM and KL.
KL JM M 14y – 7 L
Substitute x 9 into each expression and simplify.
JM 4(
)6
JM KL 6(
) 12
KL _
_
Since JM KL, JM
KL.
Step 2 Find JK and ML.
ML JK Substitute y 2.5 into each expression and simplify.
JK 10(
)3
JK ML 14(
)7
ML _
_
Since JK ML, JK
ML.
sides of the quadrilateral are
Step 3 Both pairs of
, so
.
the quadrilateral is a
Applying Conditions for Parallelograms
Determine if the quadrilateral must be a parallelogram. Justify your answer.
For a quadrilateral to be a parallelogram you must be able to prove that at least one
sides are parallel and
pair of
.
The arrows on the opposite sides indicate that the sides are
.
The tic marks on the opposite sides indicate that the sides are
.
Since one pair of
sides of a quadrilateral are
, the quadrilateral is a
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and
.
80
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Quiz
6A
6-1 Properties and Attributes of Polygons
Tell whether each figure is a polygon. If it is a polygon, name it by the number
of its sides.
1.
2.
3.
4.
5. Find the sum of the interior angle measures of a convex 21-gon.
6. The surface of a stop sign is in the shape of a regular octagon. Find the
measure of each interior angle of the stop sign.
7. A decorative pool in the shape of a pentagon is
bordered by five rows of bushes, as shown. Find the
measure of each exterior angle of the pool.
A
15x °
25x °
B
E
25x °
42x °
D
13x °
C
8. Find the measure of each exterior angle of a heptagon.
6-2 Properties of Parallelograms
Tiles used to cover a floor are in the shape of a parallelogram. In
ⵥPQRS, PS ⫽ 3 in., TQ ⫽ 3.5 in., and m⬔RSP ⫽ 58⬚. Find each
measure.
9. SQ
12. mQPS
P
S
10. TS
11. QR
13. mQRS
14. mRQP
15. Three vertices of parallelogram WXYZ are W(1, 3), X(6, 4) and Y(4, 1).
Find the coordinates of vertex Z.
Copyright © by Holt, Rinehart and Winston.
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Q
T
R
y
2
–2
–2
81
2 4 6
x
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Quiz continued
6A
DEFG is a parallelogram. Find each measure.
3y + 11
G
F
16. FG
(3x + 15)ⴗ
17. DE
18. m⬔F
(7x – 5)ⴗ
19. m⬔E
6-3
D
E
6y – 1
Conditions for Parallelograms
20. Show that HIJK is a parallelogram for x ⫽ 6.5 and y ⫽ 7.
7y + 10
I
J
8x – 9
6x + 4
H
21. Show that ABCD is a parallelogram for x ⫽ 11 and y ⫽ 8.
K
9y – 4
D
C
(9x – 41)ⴗ
(15y + 2)ⴗ
(5x + 3)ⴗ
A
B
Determine if each quadrilateral is a parallelogram. Justify your answer.
22.
23.
24.
25. Show that a quadrilateral with vertices J(⫺2, 1), K(3, 3), L(⫺1, ⫺4),
and M(⫺6, ⫺6) is a parallelogram.
y
2
–6 –4 –2
–2
2
4 x
–4
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82
Holt Geometry
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Class
Ready to Go On? Enrichment
6A
Polygons and Parallelograms
Answer each question.
1. The measure of an exterior angle of a regular
polygon is x , and the measure of an interior angle
is (10x 15). Name the polygon.
2. Find the measure of each angle in the polygon at right if:
2
mA (x 7x 1)
B
A
mB (13x 21)
mC (11x 4)
G
C
2
mD (x 5x 1)
F
2
mE (2x x 1)
D
E
mF (17x 12)
2
mG (x 3x 4)
3. The coordinates of three vertices of a parallelogram
are (1, 5), (4, 3) and (2, 2). Find the coordinates of
two other possible locations of the fourth vertex.
4. ABCD is a parallelogram. mA (7y x),
mB (2x 5), and mD (3y 12). Find the
measure of each angle.
D
C
A
B
y
5. The diagonals of a parallelogram intersect at (1, 1). Two
vertices are located at (6, 4) and (3, 1). Find the
coordinates of the other two vertices.
6
4
2
–6 –4 –2
–2
2
4 6 x
–4
–6
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83
Holt Geometry
Name
SECTION
6B
Date
Class
Ready to Go On? Skills Intervention
6-4 Properties of Special Parallelograms
Find these vocabulary words in Lesson 6-4 and the Multilingual Glossary.
Vocabulary
rectangle
rhombus
square
Using Properties of Rectangles to Find Measures
PQRS is a rectangle. PQ ⫽ 64 ft and PR ⫽ 70 ft. Find each measure.
P
A. SR
_
PR Rectangle
64
PQ So, SR Q
T
opposite sides Definition of segments
S
R
.
B. TQ
_
PR Rectangle
PR QS 1
TQ __ 2
1
TQ __ 2
Definition of segments
diagonals
Parallelogram
Substitute and simplify.
diagonals bisect each other
ft
Using Properties of Rhombuses to Find Measures D
CDEF is a rhombus. Find DE.
CD FC
10x + 9
C
G
15x – 22
Definition of a rhombus
Substitute the given values. E
10x 9 10x 9 22 15x 22 22
F
Add 22 to both sides.
15x
10x 10x 15x Subtract 10x from both sides.
5x
Divide both sides by 5.
x
DE Definition of a rhombus
DE Substitute 15x 22 for FC.
DE 15 (
) 22
DE Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Substitute 6.2 for x.
Simplify.
84
Holt Geometry
Name
SECTION
6B
Date
Class
Ready to Go On? Problem Solving Intervention
6-4 Properties of Special Parallelograms
If a parallelogram is a rectangle, then its diagonals are congruent.
A rectangular door is inset with glass that has decorative strips along the
diagonals. In rectangle FGHI, FG 24 in. and FH 65 in. Find each length.
F
G
A. HI
B. GI
J
C. JH
O
Understand the Problem
1. Why is it important to know the shape of the door?
I
H
2. What measurements of the rectangle are important to finding the lengths?
Make a Plan
3. How are opposite sides of a rectangle related?
4. How are the diagonals of a rectangle related?
Solve
5. If FG 24 in., what do you know about HI ?
6. Since FH 65 in., what do you know about GI?
7. Since FH 65 in., what do you know about JH ?
So, JH Look Back
Use the properties of rectangles to see if your answers are logical.
8. Are FG and HI opposite sides of a rectangle?
9. Are FH and GI diagonals of a rectangle?
Do they have equal length?
10. Do diagonals of a rectangle bisect each other?
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Do they have the same length?
85
So, is 2 JH FH ?
Holt Geometry
Name
Date
Class
Ready to Go On? Skills Intervention
SECTION
6B
6-5 Conditions for Special Parallelograms
P
Applying Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional
information is needed to make it valid.
_
_
_
_
Given: PS QR , ⬔SPR ⬔QRT. PR ⬜ QS
T
Q
S
Conclusion: PQRS is a rhombus.
Step 1 Determine if PQRS is a parallelogram.
R
⬔SPR ⬔QRT
Given
PS QR
Converse of
_
_
_
Theorem
_
PS QR
PQRS is a parallelogram. Quad with one pair of
parallelogram
sides that are
and
Step 2 Determine if PQRS is a rhombus.
_
_
PR ⬜ QS
PQRS is a rhombus.
Parallelogram with
diagonals
with
Step 3 Since PQRS is a
rhombus
diagonals, it is a
. Is the conclusion valid?
y
6
Identifying Special Parallelograms in the Coordinate Plane
Use the diagonals to determine whether the parallelogram with the
given vertices is a rectangle, rhombus, or square. Give all names
that apply.
4
2
–6 –4 –2
–2
2
4 6 x
–4
J(1, 4), K(6, 1), L(3, ⫺4), M(⫺2, ⫺1)
–6
Step 1 Graph JKLM on the grid at right.
Step 2 Use the Distance Formula to find JL and MK to determine if JKLM is a rectangle.
1 ⫺ 2 ⫹ 4 ⫺
2 ⫽ ⫽ 2 JL ⫽
2
2
⫺6 ⫹
⫺1 ⫽
⫽ 2
KM ⫽
Since JL
KM, JKLM is a
_
.
_
Step 3 Find the slopes of JL and KM to tell if JKLM is a rhombus.
_
_
⫺4 ⫺
⫺1
slope of KM ⫽ ________ ⫽ ____ ⫽ ___
slope of JL ⫽ ________ ⫽ ____ ⫽
3⫺
⫺6
_
_
__1 ⫽ ⫺1, JL KM . JKLM is a
Since .
4
Since JKLM is a
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and a
, JKLM is also a
86
.
Holt Geometry
Name
SECTION
6B
Date
Class
Ready to Go On? Skills Intervention
6-6 Properties of Kites and Trapezoids
Find these vocabulary words in Lesson 6-6 and the Multilingual Glossary.
Vocabulary
kite
trapezoid
base angle of a trapezoid
base of a trapezoid
leg of a trapezoid
isosceles trapezoid
midsegment of a trapezoid
Using Properties of Kites
In kite WXYZ, m⬔VWX ⫽ 85⬚ and
m⬔VWZ ⫽ 48⬚. Find each measure.
W
V
Z
A. mWXV
X
Y
mWVX ⫽
Kite
mVWX ⫹ mVXW ⫽
Acute angles of a right triangle are
⫹ mVXW ⫽ 90
mVXW ⫽
Diagonals are
.
Substitute 85 for mVWX.
Subtract 85 from both sides.
B. mXYZ
ZWX ⬵
Kite
mZWX ⫽
Definition of
mZWX ⫽ mVWZ ⫹ mVWX
Angle
mZWX ⫽
⫹
one pair of opposite
angles.
angles
Postulate
Substitute 48° for mVWZ and 85° for mVWX.
mZWX ⫽
Simplify.
mXYZ ⫽
Substitute mZYX for mZWX.
Using Properties of Isosceles Trapezoids
If CA ⫽ 53, DE ⫽ 20, find EB.
D
C
E
A
B
_
DB ⬵
Isosceles trapezoid
DB ⫽
Definition of congruent
DB ⫽
Substitute
DE ⫹ EB ⫽ DB
⫹ EB ⫽
EB ⫽
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diagonals congruent
for CA.
Addition Postulate
Substitute 20 for DE and 53 for DB.
Subtract 20 from both sides.
87
Holt Geometry
Name
SECTION
6B
Date
Class
Ready to Go On? Problem Solving Intervention
6-6 Properties of Kites and Trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The
midsegment of a trapezoid is the segment whose endpoints are the
midpoints of the legs.
The front of a decorative end table is in the shape of a trapezoid. The
bases are 37 cm and 54 cm long. The bottom of the top drawer extends
from the midpoint of each leg of the trapezoid. How long is the bottom
of the top drawer?
Understand the Problem
of a trapezoid is the segment whose endpoints are the
1. The
midpoints of the
.
to each base, and its length is
2. The midsegment of a trapezoid is
the sum of the lengths of the
.
Make a Plan
3. Apply the Midsegment Theorem, using
of the bases.
and
for the lengths
Solve
4. Substitute the lengths of bases into the Midsegment Theorem and simplify.
_________
2
____
2
5. The length of the bottom of the drawer is
.
Look Back
6. Work backwards from your answer to check your solution. Multiply your answer
in Exercise 6 by 2.
7. What is the sum of the bases of the trapezoid?
8. Do your answers in Exercises 7 and 8 match?
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88
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Quiz
6B
6-4 Properties of Special Parallelograms
A
The flag of Florida is a rectangle with stripes along the diagonals.
In rectangle ABCD, AD ⫽ 45 in. and BD ⫽ 52.5 in. Find each length.
1. ED
2. AC
3. BC
4. EC
D
E
B
C
12x – 8
M
JKLM is a rhombus. Find each measure.
L
N
5. MJ
8x + 5
6. Find mMJN and mLMJ if mMNJ ⫽ (6d ⫺ 12)⬚ and
mNKJ ⫽ (4d ⫹1)⬚.
_ _
J
_
7. Given: ADCE is a rhombus with diagonal ED . CB ⬵ AB .
Prove: BCD ⬵ BAD.
Statements
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
_ _
D
B
W
X
Z
Y
_
8. Given: WY XZ , WZ ⬵ WX
Conclusion: WXYZ is a rhombus.
_
E
A
6-5 Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional
information is needed to make it valid.
_ _
C
Reasons
1.
_
K
_ _
_
9. Given: WX 储 ZY , WZ 储 XY , WZ ZY
Conclusion: WXYZ is a rectangle.
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89
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Quiz continued
6B
y
Use the diagonals to determine whether a parallelogram with the
given vertices is a rectangle, a rhombus, or a square. Give all the
names that apply.
6
4
2
10. H(3, 5), I(⫺1, 2), J(⫺3, ⫺4), K(1, ⫺1)
–6 –4 –2
–2
4 6 x
–4
11. P(2, 4), Q(4, ⫺2), R(⫺2, ⫺4), S(⫺4, 2)
–6
P
_
12. Given: 䉭MON is equilateral. M is the midpoint of LN . LMOP
is a parallelogram.
Prove: LMOP is a rhombus.
Statements
O
L
M
N
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6-6 Properties of Kites and Trapezoids
In kite CDEF, m⬔CDF ⫽ 39⬚, and m⬔EFC ⫽ 25⬚. Find each measure.
13. m⬔CFG
14. m⬔GEF
15. m⬔DCG
16. m⬔DEF
17. Find m⬔Q.
18. AC ⫽ 91.7 and BE ⫽ 33.9. Find ED.
Q
2
C
D
G
F
E
D
R
C
E
P
54°
S
A
B
19. The face of a stone wall is in the shape of a trapezoid. The
bases of the wall are 132 in. and 64 in. A steel bar is attached
between the midpoint of each leg of the trapezoid. How long is
the bar?
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90
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Enrichment
6B
Other Special Quadrilaterals
1. Given: ABCD is a rectangle. E, F, G, and H are midpoints of
their respective sides. Write a paragraph proof to show that
EFGH is a rhombus.
H
A
E
G
B
2. The quadrilateral at right is a kite,_
not drawn
_ to scale.
_
RS
and
RQ
or
RS
Which
two
sides
are
congruent,
_
and ST ? Why?
91
F
R
C
2x + 35
S
5 – 3x
4x + 31
Q
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D
T
Holt Geometry
SECTION
Ready to Go On? Quiz continued
5-6
5B
Inequalities in Two Triangles
7. Compare AB and ST.
8. Compare m⬔XWY
and m⬔ZWY.
T
A
29 120°
C 33
W
100°
R 29
B
9. Find the range of values
for x.
64
88
S
ST ⬍ AB
Relationships in Triangles
For Exercises 1–2, tell whether a triangle can have vertices with the given
coordinates. Explain.
1. 䉭PQR has vertices P(3, 11), Q(1, 3) and R(5, 5).
X
88
33
5-7
Ready to Go On? Enrichment
SECTION
5B
40
Y
44
74
Z
40°
5x – 11
55°
2. 䉭JKL has vertices J(5, 1), K(4, 2) and L(11, 1).
x ⬎ 17
m⬔XWY ⬍ m⬔ZWY
The Pythagorean Theorem
Satisfies 䉭 Ineq.
x
10. Find the value of x. Give the answer in simplest radical form.
19
兹 442
Answer each question.
3. A right triangle has legs with lengths x and 3(x 1), and hypotenuse 4x – 3.
Find x and the lengths of each side.
11. Tell if the measures 8, 9, and 15 can be the side lengths of a
triangle. If so, classify the triangle as acute, obtuse, or right.
2
Yes, because JK ⫽ 兹 82 , KL ⫽ 兹 58 , and JL ⫽ 2兹65 .
9
2
No, because PQ ⫽ QR ⫽ 4兹 5 , and PR ⫽ 8兹 5 . Does not satisfy the 䉭 Ineq.
x ⫽ 7; The lengths of the sides are 7, 24, and 25.
2
The sides form an acute triangle because 8 ⫹ 9 ⬍ 15 .
12. A park developer want to put a bike trail from one corner of a
rectangular park to the opposite corner. What will be the length
of the trail? Round to the nearest yard.
tra
ike
4. The figure at the right is drawn to scale. Compare BC and AD.
Which segment is longer? Explain your answer.
il
450 yd
Since ⬔BCA is acute and ⬔ACD is its
B
C
supplement, ⬔ACD must be obtuse.
700 yd
832 yd
A
B
D
Therefore AD ⬎ BC.
5-8
_
Applying Special Right Triangles
14 in.
12 in.
60°
15.
45°
m⬔RPS ⬎ m⬔QPS and PS ⫽ PS; By the Hinge
60°
12兹 3
45°
11
x ⫽ 11兹 2
SECTION
6A
x ⫽ 20兹 2
74
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Holt Geometry
Ready to Go On? Skills Intervention
side of a polygon
vertex of a polygon
diagonal
regular polygon
concave
convex
Yes
Is the figure formed by three or more segments?
Yes
Do the segments intersect only at their endpoints?
Yes
076-091_CH06_RTGO_GEO_12738.indd 76
5/25/06 4:30:32 PM
14x °
A
8x °
E
B
17x °
12x °
C
9x °
D
A polygon with five sides
An angle formed by one side of a polygon
4. How will you find the measure of each exterior angle?
360ⴗ
.
Solve for x, and substitute
the value of x into each angle measure.
Solve
5. Substitute the given angle measures into the Polygon Exterior Angle Sum Theorem.
8x
17x 9x
12x 6. Combine like terms and solve for x.
2)180
14x
360
60x ⴝ 360ⴗ, x ⴝ 6
7. Substitute the value of x into each given angle measure. (Exterior angles named by the
vertex angle) m⬔A ⴝ 48ⴗ, m⬔B ⴝ 84ⴗ, m⬔C ⴝ 72ⴗ, m⬔D ⴝ 54ⴗ, m⬔E ⴝ 102ⴗ
A polygon that is both equiangular and equilateral.
Look Back
8. Find the sum of the angle measures you found in Exercise 7.
2340
______
15
360ⴗ
9. What is the sum of exterior angle measures of a convex polygon?
To find the measure of each interior angle of a regular 15-gon,
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
6-1 Properties and Attributes of Polygons
measures, one angle at each vertex, of a convex polygon is
Dodecagon
Divide the sum of the interior angle measures by the number of sides.
.
Ready to Go On? Problem Solving Intervention
3. The Polygon Exterior Angle Sum Theorem states that the sum of the exterior angle
12
From Part A, the sum of the interior angle measures of a 15-gon is 2340ⴗ .
How do you find the measure of each interior angle of a regular polygon?
15
Holt Geometry
Make a Plan
No
B. Find the measure of each interior angle of a regular 15-gon.
divide 2340 by
75
and the extension of a consecutive side.
2340ⴗ
What is a regular polygon?
N
1. What is a pentagon?
The number of sides in the polygon
Simplify the expression.
P
Q
Understand the Problem
)180. What does the n represent?
Substitute 15 for n in the formula. (
RP
2兹 3
A decorative garden in the shape of a pentagon is surrounded
by five railings. Find the measure of each exterior angle of the
garden.
A. Find the sum of the interior angle measures of a convex 15-gon.
The sum of the interior angle measures of a convex polygon with
15
2兹 6
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Finding Interior Angle Measures and Sums in Polygons
2
MN
2. What is an exterior angle?
How many sides does it have?
Name the polygon by the number of sides it has.
n sides is (n M
Yes
Are any of the segments with common endpoints collinear?
Is the figure a polygon?
R
The Polygon Exterior Angle Sum Theorem states that the sum
of the exterior angle measures, one angle at each vertex, of a
convex polygon is 360.
Identifying Polygons
Tell whether each figure is a polygon. If it is a polygon, name it by the
number of its sides.
Is it a plane figure?
兹6
SECTION
Vocabulary
Yes
057-075_CH05_RTGO_GEO_12738.indd
5/25/06 4:30:31
75
PM
Find these vocabulary words in Lesson 6-1 and the Multilingual Glossary.
Is the figure closed ?
NQ
6A
6-1 Properties and Attributes of Polygons
057-075_CH05_RTGO_GEO_12738.indd 74
_
40
x ⫽ 12, y ⫽ 24
S
_
x
y
60°
x
28°
R
6. 䉭MNP is an equilateral triangle. RM ⬵ RP . MQ 3兹2 .
Find the following:
x
Q
53°
Theorem, RS ⬎ QS.
16.
x
P
RS ⬎ QS; By Alt Int ⬔’s, m⬔QPS ⫽ 28⬚, so
Find the values of the variables. Give your answers in simplest radical form.
14.
_
5. In the figure at the right, m⬔RPS 53. PQ 储 RS , and
PQ PR. Compare QS and RS. Explain your answer.
13. A decorative platter is an equilateral triangle with side lengths of
14 inches. What is the height of the platter? Round to the nearest inch.
360ⴗ
10. Is your answer in Exercise 8 the same as your answer from Exercise 9?
Yes
156ⴗ
76
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Holt Geometry
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All rights reserved.
209
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5/25/06 4:31:26
77
PM
77
Holt Geometry
Holt Geometry
12/15/05 9:43:36 AM
Ready to Go On? Skills Intervention
SECTION
6A
SECTION
6A
6-2 Properties of Parallelograms
Find this vocabulary word in Lesson 6-2 and the
Multilingual Glossary.
The parking spots in a parking lot are in the shape of a parallelogram.
If m⬔DAB ⫽ 65⬚ and AE ⫽ 13.3 ft, find m⬔CDA, m⬔CBA, EC, DC, and DA.
parallelogram
Using Properties of Parallelograms to Find Measures
䊐PQRS is a parallelogram. PT ⴝ 53, PS ⴝ 76, and m⬔QRS ⴝ 75ⴗ.
Find each measure.
Q
A. RT
bisect
RT and PT ⫽ RT .
53 .
_
_
each other, so PT Since PT ⫽ 53, RT ⫽
T
are in the shape of parallelograms?
7x – 41
P
20 ft
A
So you can find the
11 ft
B
1. Why is it important that the problem states that the parking spots
R
4x + 1
C
E
D
Understand the Problem
missing lengths using the properties of parallelograms.
S
Make a Plan
B. QR
congruent
PS . Since PS ⫽ 76, QR ⫽ 76 .
In a parallelogram, opposite sides are
and QR ⫽
6-2 Properties of Parallelograms
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Vocabulary
In a parallelogram, the diagonals
Ready to Go On? Problem Solving Intervention
_
, so QR PS
2. What do you know about consecutive angles of a parallelogram?
,
Consecutive angles of a parallelogram are supplementary.
3. What do you know about opposite angles of a parallelogram?
C. m⬔RQP
Opposite angles of a parallelogram are congruent.
In a parallelogram, consecutive angles are
m⬔RQP ⫹ m⬔QRS ⫽
supplementary .
4. What do you know about opposite sides of a parallelogram?
180ⴗ
75ⴗ
Substitute 75⬚ for m⬔QRS. m⬔RQP ⫹
Solve to find m⬔RQP. m⬔RQP ⫽
⫽
Opposite sides of a parallelogram are congruent.
180ⴗ
5. What do you know about the diagonals of a parallelogram?
105ⴗ
The diagrams of a parallelogram bisect each other.
D. m⬔RSP
opposite
angles are congruent. ⬔RSP ⬔RQP
m⬔RQP . What is m⬔RQP ? 105ⴗ
Substitute to find m⬔RSP. m⬔RSP ⫽ 105ⴗ
In a parallelogram,
Solve
m⬔RSP ⫽
6. Find m⬔CDA. Explain how you determined your answer.
E. m⬔QPS
7. Find m⬔CBA. Explain how you determined your answer.
m⬔QRS
m⬔QPS ⫽
. Substitute to find m⬔QPS. m⬔QPS ⫽
75ⴗ
F. RS
8.
_
QP , and RS ⫽ QP .
4x ⴙ 1
⫽ 7x ⫺ 41
Substitute the given values for RS and QP.
1 ⫽ 3x ⫺ 41
Subtract 4x from both sides.
Add 41 to both sides.
42 ⫽ 3x
Divide both sides by 3.
x ⫽ 14
Substitute your solution into 7x ⫺ 41 and simplify to find RS. 7( 14 ) ⫺ 41 ⫽ 57
9. Find DC and DA. Explain how you determined your answer. DC
Opposite sides of a parallelogram are congruent so they have equal length.
Look Back
10. Do your answers to Exercises 6–9 satisfy all of the properties of parallelograms
from Exercises 2–5?
78
Holt Geometry
Ready to Go On? Skills Intervention
SECTION
6A
79
Holt Geometry
Ready to Go On? Quiz
6A
076-091_CH06_RTGO_GEO_12738.indd
12/15/05 9:43:37
79
AM
Verifying Figures are Parallelograms
Show that ⵥJKLM is a parallelogram for x ⫽ 9 and y ⫽ 2.5.
have to be
Yes
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
SECTION
6-3 Conditions for Parallelograms
076-091_CH06_RTGO_GEO_12738.indd 78
For ⵥJKLM to be a parallelogram, both pairs of
m⬔CBA ⫽ 115⬚ because
⬔CBA and ⬔CDA are opposite angles of the parallelogram.
Find EC. Explain how you determined your answer. EC ⫽ 13.3; Diagonals of a
parallelogram bisect each other so C is the midpoint of AC.
⫽ 20 ft and DA ⫽ 11 ft;
In a parallelogram, opposite sides are congruent so RS Copyright © by Holt, Rinehart and Winston.
All rights reserved.
m⬔CDA ⫽ 115⬚ because
⬔CDA and ⬔DAB are supplementary.
J
opposite sides
congruent when x ⫽ 9 and y ⫽ 2.5.
10y + 3
4x + 6
5/25/06 4:31:27 PM
6-1 Properties and Attributes of Polygons
Tell whether each figure is a polygon. If it is a polygon, name it by the number
of its sides.
K
6x – 12
1.
Yes, hexagon
2.
Not a polygon
3.
Yes, quadrilateral
4.
Not a quadrilateral
Step 1 Find JM and KL.
JM ⫽
4x ⫹ 6
KL ⫽
6x ⫺ 12
M 14y – 7 L
Substitute x ⫽ 9 into each expression and simplify.
9
42
JM ⫽ 4(
JM ⫽
)⫹6
9
42
KL ⫽ 6(
_
Since JM ⫽ KL, JM
⬵
KL ⫽
) ⫺ 12
_
KL.
Step 2 Find JK and ML.
JK ⫽
10y ⫹ 3
ML ⫽
14y ⫺ 7
3420⬚
Substitute y ⫽ 2.5 into each expression and simplify.
5. Find the sum of the interior angle measures of a convex 21-gon.
JK ⫽ 10( 2.5 ) ⫹ 3
6. The surface of a stop sign is in the shape of a regular octagon. Find the
ML ⫽ 14( 2.5 ) ⫺ 7
ML ⫽ 28
_
⬵ ML.
Step 3 Both pairs of opposite sides of the quadrilateral are congruent , so
the quadrilateral is a parallelogram .
JK ⫽
28
measure of each interior angle of the stop sign.
_
Since JK ⫽ ML, JK
x ⫽ 3, measures of exterior angles (named
using 1 letter) m⬔A ⫽ 45⬚, m⬔B ⫽ 75⬚,
Applying Conditions for Parallelograms
Determine if the quadrilateral must be a parallelogram. Justify your answer.
m⬔C ⫽ 75⬚, m⬔D ⫽ 39⬚, m⬔E ⫽ 126⬚
For a quadrilateral to be a parallelogram you must be able to prove that at least one
pair of
opposite
sides are parallel and
8. Find the measure of each exterior angle of a heptagon.
congruent .
.
The tic marks on the opposite sides indicate that the sides are
congruent .
Since one pair of
9. SQ
opposite
sides of a quadrilateral are
parallel
and
7 in.
12. m⬔QPS
congruent , the quadrilateral is a parallelogram .
122⬚
10. TS
3.5 in.
122⬚
13. m⬔QRS
A
15x °
B
25x °
42x °
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
076-091_CH06_RTGO_GEO_12738.indd 80
80
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Holt Geometry
Copyright © by Holt, Rinehart and Winston.
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210
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81
AM
(⫺3, ⫺2)
81
13x °
D
C
360 ⬚ ⬇ 51.43⬚
____
7 P
Q
T
S
11. QR
R
3 in.
58⬚
14. m⬔RQP
15. Three vertices of parallelogram WXYZ are W(⫺1, 3), X(6, 4) and Y(4, ⫺1).
Find the coordinates of vertex Z.
25x °
E
6-2 Properties of Parallelograms
Tiles used to cover a floor are in the shape of a parallelogram. In
ⵥPQRS, PS ⫽ 3 in., TQ ⫽ 3.5 in., and m⬔RSP ⫽ 58⬚. Find each
measure.
The arrows on the opposite sides indicate that the sides are
parallel
135⬚
7. A decorative pool in the shape of a pentagon is
bordered by five rows of bushes, as shown. Find the
measure of each exterior angle of the pool.
y
2
–2
–2
2 4 6
x
Holt Geometry
Holt Geometry
12/15/05 9:43:38 AM
Ready to Go On? Quiz continued
SECTION
SECTION
6A
DEFG is a parallelogram. Find each measure.
17. DE
18. m⬔F
19. m⬔E
3y + 11
G
23
23
66⬚
114⬚
16. FG
6-3
Ready to Go On? Enrichment
6A
Polygons and Parallelograms
Answer each question.
F
(3x + 15)ⴗ
(7x – 5)ⴗ
D
1. The measure of an exterior angle of a regular
polygon is x ⬚, and the measure of an interior angle
is (10x ⫹ 15)⬚. Name the polygon.
24-gon
E
6y – 1
2. Find the measure of each angle in the polygon at right if:
Conditions for Parallelograms
m⬔A ⫽ (x ⫹ 7x ⫺ 1)⬚
2
20. Show that HIJK is a parallelogram for x ⫽ 6.5 and y ⫽ 7.
7y + 10
I
8(6.5) ⫺ 9 ⫽ 6(6.5) ⫹ 4 ⫽ 43. 7(7) ⫹ 10 ⫽
8x – 9
9(7) ⫺ 4 ⫽ 59. Since both pairs of opposite
B
A
m⬔B ⫽ (13x ⫹ 21)⬚
J
m⬔C ⫽ (11x ⫺ 4)⬚
6x + 4
G
C
m⬔D ⫽ (x ⫹ 5x ⫹ 1)⬚
2
H
sides have equal length, HIJK is a parallelogram.
21. Show that ABCD is a parallelogram for x ⫽ 11 and y ⫽ 8.
D
9(11) ⫺ 41 ⫽ 5(11) ⫹ 3 ⫽ 58⬚. 15(8) ⫹ 2 ⫽ 122⬚.
K
9y – 4
D
E
2
m⬔G ⫽ (x ⫹ 3x ⫺ 4)⬚
(15y + 2)ⴗ
A
Since ⬔CBA is supplementary to both of its
2
m⬔F ⫽ (17x ⫺ 12)⬚
C
(9x – 41)ⴗ
(5x + 3)ⴗ
F
m⬔E ⫽ (2x ⫺ x ⫺ 1)⬚
m⬔A ⫽ 143⬚, m⬔B ⫽ 138⬚, m⬔C ⫽ 95⬚, m⬔D ⫽ 127⬚,
B
m⬔E ⫽ 152⬚, m⬔F ⫽ 141⬚, and m⬔G ⫽ 104⬚
consecutive angles, ABCD is a parallelogram.
3. The coordinates of three vertices of a parallelogram
are (1, 5), (4, 3) and (2, ⫺2). Find the coordinates of
two other possible locations of the fourth vertex.
Determine if each quadrilateral is a parallelogram. Justify your answer.
22.
23.
(0, ⫺1) and (3, 10)
24.
4. ABCD is a parallelogram. m⬔A ⫽ (7y ⫹ x)⬚,
m⬔B ⫽ (2x ⫺ 5)⬚, and m⬔D ⫽ (3y ⫺ 12)⬚. Find the
measure of each angle.
D
C
m⬔D ⫽ m⬔B ⫽ 39⬚, m⬔A ⫽ 141⬚
No, not enough
Yes, one pair of
opposite sides are ⬵.
information given
opposite sides and 25. Show that a quadrilateral with vertices J(⫺2, 1), K(3, 3), L(⫺1, ⫺4),
and M(⫺6, ⫺6) is a parallelogram.
2 and the slope of
The slope of JK and the slope of LM are both _
5
7. Since the quadrilateral has
KL and the slope of MJ are both _
4
A
B
y
Yes, both pairs of
5. The diagonals of a parallelogram intersect at (1, 1). Two
vertices are located at (⫺6, 4) and (⫺3, ⫺1). Find the
coordinates of the other two vertices.
(5, 3) and (8, ⫺2)
y
6
4
2
–6 –4 –2
–2
2
4 6 x
2
–4
–6 –4 –2
–2
2
4 x
–6
–4
two pair of opposite parallel sides, it is a parallelogram.
82
Copyright © by Holt, Rinehart and Winston.
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SECTION
6B
Holt Geometry
Ready to Go On? Skills Intervention
SECTION
6B
6-4 Properties of Special Parallelograms
076-091_CH06_RTGO_GEO_12738.indd 82
Holt Geometry
Ready to Go On? Problem Solving Intervention
6-4 Properties of Special Parallelograms
12/15/05 9:43:40 AM
If a parallelogram is a rectangle, then its diagonals are congruent.
A rectangular door is inset with glass that has decorative strips along the
diagonals. In rectangle FGHI, FG ⫽ 24 in. and FH ⫽ 65 in. Find each length.
Vocabulary
rhombus
83
076-091_CH06_RTGO_GEO_12738.indd
5/25/06 4:31:27
83
PM
Find these vocabulary words in Lesson 6-4 and the Multilingual Glossary.
rectangle
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
square
F
G
A. HI
B. GI
Using Properties of Rectangles to Find Measures
PQRS is a rectangle. PQ ⫽ 64 ft and PR ⫽ 70 ft. Find each measure.
P
A. SR
SR
PQ ⫽
SR
So, SR ⫽
Rectangle
⫽ 64
64
T
Definition of segments
S
_
PR Understand the Problem
H
.
The lengths of the sides and the length of the diagonals.
Rectangle
35
70
diagonals
Make a Plan
Definition of segments
Parallelogram
have equal measure.
Substitute and simplify.
4. How are the diagonals of a rectangle related?
CD ⫽ FC
10x ⫹ 9 ⫽
31
31
31
6.2
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
C
G
Solve
15x – 22
15x ⫺ 22
Substitute the given values. E
F
⫽ 15x ⫺
⫽ 5x
10x
HI ⫽ 24 in.
GI ⫽ 65 in.
5. If FG ⫽ 24 in., what do you know about HI ?
6. Since FH ⫽ 65 in., what do you know about GI?
It is one-half of FH.
7. Since FH ⫽ 65 in., what do you know about JH ?
Add 22 to both sides.
⫽ 15x
So, JH ⫽
Subtract 10x from both sides.
Divide both sides by 5.
1
1
JH ⫽ __FH ⫽ __(65) ⫽ 32.5
2
2
Look Back
⫽x
FC
DE ⫽ 15x ⫺ 22
DE ⫽ 15 ( 6.2 ) ⫺ 22
DE ⫽ 71
DE ⫽
10x + 9
Definition of a rhombus
10x ⫹ 9 ⫹ 22 ⫽ 15x ⫺ 22 ⫹ 22
⫹ 10x ⫹
They are congruent and
have equal length and they bisect each other.
ft
10x ⫹
They are congruent and
3. How are opposite sides of a rectangle related?
diagonals bisect each other
Using Properties of Rhombuses to Find Measures D
CDEF is a rhombus. Find DE.
07 -091_CH0 _ TGO_GEO_12738.
I
to use the properties of rectangles to help answer the question.
R
2. What measurements of the rectangle are important to finding the lengths?
QS
PR ⫽ QS ⫽
1
TQ ⫽ __ SQ
2
1
TQ ⫽ __ 70
2
⫽
O
1. Why is it important to know the shape of the door? Sample answer:
opposite sides _
B. TQ
⫺10x
Q
_
_
PR J
C. JH
Use the properties of rectangles to see if your answers are logical.
Definition of a rhombus
8. Are FG and HI opposite sides of a rectangle?
Substitute 6.2 for x.
9. Are FH and GI diagonals of a rectangle?
Simplify.
84
8
Copyright © by Holt,
Rinehart and Winston.
All rights reserved.
Yes
Do they have the same length?
Yes, both equal 24.
Substitute 15x ⫺ 22 for FC.
Yes
10. Do diagonals of a rectangle bisect each other?
Holt Geometry
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211
076-091_CH06_RTGO_GEO_12738.indd
12/15/05 9:
85 3: 0 AM
85
Do they have equal length?
Yes
So, is 2 ⭈ JH ⫽ FH ?
Yes
Yes
Holt Geometry
Holt Geometry
12/15/05 9:43:41 AM
Ready to Go On? Skills Intervention
SECTION
6B
6B
P
Applying Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional
information is needed to make it valid.
_
_
6-6 Properties of Kites and Trapezoids
Find these vocabulary words in Lesson 6-6 and the Multilingual Glossary.
Vocabulary
_
_
Ready to Go On? Skills Intervention
SECTION
6-5 Conditions for Special Parallelograms
Given: PS QR , SPR QRT. PR QS
T
Q
S
kite
trapezoid
base angle of a trapezoid
Conclusion: PQRS is a rhombus.
base of a trapezoid
leg of a trapezoid
isosceles trapezoid
midsegment of a trapezoid
Step 1 Determine if PQRS is a parallelogram.
Given
PS QR
Converse of
PS QR
Given
_
_
_
_
Alternate
Interior
Angles
Theorem
opp.
sides that are
and
PQRS is a rhombus.
Step 3 Since PQRS is a
rhombus . Is the conclusion valid? Yes
6
4
ZWX J
–4
J(1, 4), K(6, 1), L(3, 4), M(2, 1)
–6
4 6 x
2
L
Step 1 Graph JKLM on the grid at right.
_
Angle
48⬚ 85⬚
mZWX 133⬚
mXYZ 133⬚
Simplify.
DB _
rhombus , JKLM is also a square
SECTION
6B
B
diagonals congruent
Definition of congruent
segments
Substitute
53
Segment
for CA.
Addition Postulate
Substitute 20 for DE and 53 for DB.
Subtract 20 from both sides.
.
86
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
C
E
A
53
33
EB EB D
Isosceles trapezoid
DE EB DB
20
_
⫺8
⫺1 1 ⫺2
1
4 4
slope of KM ________ ____ ___
slope of JL ________ ____ ⫺4
2
⫺2 6 ⫺8
4
3 1
_
_
1
Since ⫺4 __ 1, JL ⬜ KM . JKLM is a rhombus .
4
and a
Substitute mZYX for mZWX.
CA
CA
53
DB Holt Geometry
Ready to Go On? Problem Solving Intervention
87
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SECTION
Holt Geometry
Ready to Go On? Quiz
6B
6-6 Properties of Kites and Trapezoids
076-091_CH06_RTGO_GEO_12738.indd 86
076-091_CH06_RTGO_GEO_12738.indd
5/25/06 4:31:28
87
PM
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The
midsegment of a trapezoid is the segment whose endpoints are the
midpoints of the legs.
5/25/06 4:31:29 PM
6-4 Properties of Special Parallelograms
A
The flag of Florida is a rectangle with stripes along the diagonals.
In rectangle ABCD, AD ⫽ 45 in. and BD ⫽ 52.5 in. Find each length.
The front of a decorative end table is in the shape of a trapezoid. The
bases are 37 cm and 54 cm long. The bottom of the top drawer extends
from the midpoint of each leg of the trapezoid. How long is the bottom
of the top drawer?
1. ED
3. BC
26.25 in.
45 in.
2. AC
4. EC
52.5 in.
26.25 in.
midsegment
midpoints of the
5. MJ
of a trapezoid is the segment whose endpoints are the
legs
2. The midsegment of a trapezoid is
one half
parallel
M
3. Apply the Midsegment Theorem, using
of the bases.
8x + 5
_ _
Statements
and
54 cm
for the lengths
J
1.
Given
2.
2.
Given
4.
4. Substitute the lengths of bases into the Midsegment Theorem and simplify.
37 54
_________
91
____
CB AB
_
_
CD
AD
_ _
_
3. Def.
DB DB
5.
SSS 䉭 6.
⬔BCD ⬔BAD
6.
CPCTC
_
076-091_CH06_RTGO_GEO_12738.indd 88
88
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X
Z
Y
_
_
_ _
_ _
_
9. Given: WX ZY , WZ XY , WZ ZY
Conclusion: WXYZ is a rectangle.
45.5 2 ⫽ 90
7. What is the sum of the bases of the trapezoid?
W
Not enough information. Need to know that WXYZ is a parallelogram.
6. Work backwards from your answer to check your solution. Multiply your answer
in Exercise 6 by 2.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
_ _
8. Given: WY XZ , WZ WX
Conclusion: WXYZ is a rhombus.
Look Back
B
Prop. Of 䉭DAB 䉭DCB
6-5 Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional
information is needed to make it valid.
45.5 cm .
D
of a rhombus
4. Refl.
2
45.5
C
E
A
5.
2
8. Do your answers in Exercises 7 and 8 match?
K
Reasons
a rhombus.
1. ADCE
_ is_
3.
Solve
5. The length of the bottom of the drawer is
L
N
7. Given: ADCE is a rhombus with diagonal ED . CB AB .
Prove: BCD BAD.
.
Make a Plan
37 cm
12x – 8
mMJN ⫽ 21⬚ and LMJ ⫽ 138⬚
to each base, and its length is
bases
the sum of the lengths of the
C
31
6. Find mMJN and mLMJ if mMNJ (6d 12) and
mNKJ (4d 1).
.
D
E
B
JKLM is a rhombus. Find each measure.
Understand the Problem
1. The
angles.
Substitute 48° for mVWZ and 85° for mVWX.
_
_
DB _
rectangle
congruent
congruent angles
Addition Postulate
one pair of opposite
mZWX mVWZ mVWX
Step 3 Find the slopes of JL and KM to tell if JKLM is a rhombus.
Since JKLM is a
Kite
Definition of
rectangle .
KM, JKLM is a
⬔XYZ
m⬔XYZ
mZWX ⫽
Subtract 85 from both sides.
Using Properties of Isosceles Trapezoids
If CA ⫽ 53, DE ⫽ 20, find EB.
Step 2 Use the Distance Formula to find JL and MK to determine if JKLM is a rectangle.
1 3 2 4 ⫺4 2 68 2 17
JL ⫺2 6 2 ⫺1 1 2 68 2 17
KM Since JL
Substitute 85 for mVWX.
5⬚
mZWX K
2
–6 –4M–2
–2
Diagonals are
B. mXYZ
y
Identifying Special Parallelograms in the Coordinate Plane
Use the diagonals to determine whether the parallelogram with the
given vertices is a rectangle, rhombus, or square. Give all names
that apply.
90⬚
mVXW Parallelogram with perpendicular diagonals
rhombus
parallelogram with perpendicular diagonals, it is a
perpendicular
Acute angles of a right triangle are complementary .
Kite
mVXW 90
Given
PR QS
90⬚
mVWX mVXW 85
X
Y
mWVX Step 2 Determine if PQRS is a rhombus.
_
V
Z
A. mWXV
PQRS is a parallelogram. Quad with one pair of
parallelogram
_
W
Using Properties of Kites
In kite WXYZ, m⬔VWX ⫽ 85⬚ and
m⬔VWZ ⫽ 48⬚. Find each measure.
R
SPR QRT
90
Yes
Valid conclusion
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89
AM
89
Holt Geometry
Holt Geometry
5/25/06 4:31:30 PM
Ready to Go On? Quiz continued
SECTION
SECTION
6B
y
Use the diagonals to determine whether a parallelogram with the
given vertices is a rectangle, a rhombus, or a square. Give all the
names that apply.
Other Special Quadrilaterals
6
1. Given: ABCD is a rectangle. E, F, G, and H are midpoints of
their respective sides. Write a paragraph proof to show that
EFGH is a rhombus.
4
2
10. H(3, 5), I(1, 2), J(3, 4), K(1, 1)
–6 –4 –2
–2
HIJK is a rhombus.
2
4 6 x
–6
P
_
12. Given: 䉭MON is equilateral. M is the midpoint of LN . LMOP
is a parallelogram.
Prove: LMOP is a rhombus.
Statements
L
1.
Given
_ _
2. OM ⬵ MN
_ _
3. LM ⬵ MN
_ _
2.
Def. of equilateral 䉭
3.
Def. of midpt.
4.
LM ⬵ OM
4.
Trans. Prop. Of ⬵
5.
LMOP is a rhombus.
5. Parallelogram
cons. sides ⬵
w/ one pair
rhombus
6-6 Properties of Kites and Trapezoids
In kite CDEF, m⬔CDF ⫽ 39⬚, and m⬔EFC ⫽ 25⬚. Find each measure.
13. m⬔CFG
12.5⬚
14. m⬔GEF
77.5⬚
15. m⬔DCG
51⬚
16. m⬔DEF
128.5⬚
126⬚
Q
N
M
Reasons
䉭MON is equilateral.
LMOP is a parallelogram.
17. Find m⬔Q.
O
D
G
E
57.8
18. AC 91.7 and BE 33.9. Find ED.
D
R
C
E
54°
P
S
A
B
F
R
C
2x + 35
S
5 – 3x
Sample answer: If RS ⫽ ST, 4x ⫹ 31 ⫽ 2x ⫹ 35
and x ⫽ 2. Substituting x ⫽ 2 into the side
lengths, RS ⫽ ST ⫽ 39 and RQ ⫽ ⫺1, which is
impossible. If RS ⫽ RQ, 5 ⫺ 3x ⫽ 2x ⫹ 35 and
x ⫽ ⫺6. Substituting x ⫽ ⫺6 into the side
lengths,
_ _RQ ⫽ RS ⫽ 23 and ST ⫽ 7. Therefore,
RS ⬵ RQ .
F
G
B
2. The quadrilateral at right is a kite,_
not drawn
_ to scale.
_
Which
_two sides are congruent, RS and RQ or RS
and ST ? Why?
C
D
E
AD ⬵ BC and AB ⬵ DC. AD ⫽ BC and AB ⫽ DC
by the definition of congruent
_
_ segments.
_
_By the
definition of a midpoint, AH ⬵ HD and FB ⬵ FC
and AH ⫽ HD and FB ⫽ FC. By the Transitive
Property of Equality, AH ⫽ HD ⫽ FB ⫽ FC.
Similarly,
of a midpoint,
_ definition
_
_ _ by the
AE ⬵ EB and DG ⬵ CG, and AE ⫽ EB and
DG ⫽ CG. By the Transitive Property of Equality,
AE ⫽ EB ⫽ DG ⫽ CG. Since ABCD is a rectangle,
⬔A, ⬔B, ⬔C, and ⬔D are right angles.
䉭AHE
⬵ 䉭BFE
䉭CFG ⬵ 䉭DHG. By CPTCT,
_⬵_
_ _
HE ⬵ FE ⬵ FG ⬵ HG. EFGH is a rhombus
because it is a quadrilateral with four congruent
sides.
PQRS is a rectangle and a square.
H
A
Sample
_answer:
_ ABCD is a rectangle,
_
_ Since
–4
11. P(2, 4), Q(4, 2), R(2, 4), S(4, 2)
1.
Ready to Go On? Enrichment
6B
4x + 31
Q
T
19. The face of a stone wall is in the shape of a trapezoid. The
bases of the wall are 132 in. and 64 in. A steel bar is attached
between the midpoint of each leg of the trapezoid. How long is
98 in.
the bar?
90
Copyright © by Holt, Rinehart and Winston.
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Holt Geometry
Ready To Go On? Skills Intervention
SECTION
7A
SECTION
7A
7-1 Ratio and Proportion
076-091_CH06_RTGO_GEO_12738.indd 90
extremes
means
1320
3
_____
Simplify.
–4
2
–2
4
–2
2. What is the problem asking you to find?
3. Let x represent the width of the actual building. Write a proportion that
compares the ratios of the height to the width.
width of actual building
width of model building
_________________________
_________________________
height of model building height of actual building
1.5
x
4. Substitute the known values into the proportion: ______ ____
192
2.4
Apply the Cross Products Property.
55b
Solve
Simplify.
1.5
x
5. Solve the proportion. ______ ____
192
2.4
Divide both sides by 55.
55
b
1.5 (
Solve for b.
3 ) ( t ⫺ 7 )2
81 (t 7)2
81 ⫾9
t7
t
9
16
7
Simplify.
⫺2
⫺9
width
092-108_CH07_RTGO_GEO_12738.indd 92
120
x
)x
120 ft
.
1.5 __
5
120 5
___
⫽ , and ____ ⫽ __
2.4
Solve for t.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2.4
2.4x
7. Substitute your value for x into the proportion in Exercise 4. Check to see if both
ratios are equal when simplified.
Rewrite as two equations.
92
)(
288
of the actual building is
8. Are the ratios equal?
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
192
Look Back
Find the square root of both sides.
or t 7 or t 6. The
Apply the Cross Products Property.
2
t 7 t
The width of the actual building.
Make a Plan
–4
t7
27
B. _____ _____
t7
3
27(
The widths and heights
of a scale model building and the actual building it represents.
x
Substitute the given values.
1320 ____
55b
_______
55
24
1. What numbers are being compared in this problem?
t
2
1 ⫺2
___________
2 (3)
5/25/06 4:31:31 PM
Understand the Problem
y
4
y1
y_________
rise
2 slope ____
run x
2 x1
Solving Proportions
Solve each proportion.
15
55
A. ___ ___
88
b
88
) b( 55 )
15(
7-1 Ratio and Proportion
An architect builds a scale model of an office building. The width of the model is
1.5 ft and the height is 2.4 ft. The actual building is 192 ft tall. What is the width of
the building?
cross products
Writing Ratios
Write a ratio expressing the slope of line t.
5
Ready To Go On? Problem Solving Intervention
A proportion is an equation stating that two ratios are equal.
Vocabulary
proportion
Holt Geometry
076-091_CH06_RTGO_GEO_12738.indd
5/25/06 4:31:30
91
PM
Find these vocabulary words in Lesson 7-1 and the Multilingual Glossary.
ratio
91
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Holt Geometry
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213
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12/14/05 7:36:07
93
PM
Yes
8
192
8
Is your answer correct?
93
Yes
Holt Geometry
Holt Geometry
12/14/05 7:36:08 PM