Download 6.4 Rhombuses, Rectangles, and Squares

Page 1 of 6
6.4
Goal
Use properties of special
types of parallelograms.
Key Words
Rhombuses, Rectangles,
and Squares
In this lesson you will study three special types of parallelograms.
A rhombus is a parallelogram with
four congruent sides.
rhombus
• rhombus
• rectangle
• square
A rectangle is a parallelogram with
four right angles.
rectangle
A square is a parallelogram with
four congruent sides and four right angles.
EXAMPLE
1
square
Use Properties of Special Parallelograms
In the diagram, ABCD is a rectangle.
A
B
a. Find AD and AB.
5
b. Find maA, maB, maC, and maD.
D
C
8
Solution
a. By definition, a rectangle is a parallelogram, so ABCD is a
parallelogram. Because opposite sides of a parallelogram
are congruent, AD 5 BC 5 5 and AB 5 DC 5 8.
b. By definition, a rectangle has four right angles,
so maA 5 maB 5 maC 5 maD 5 908.
Use Properties of Special Parallelograms
P
1. In the diagram, PQRS is a rhombus.
Find QR, RS, and SP.
6
P
R
S
6.4
Rhombuses, Rectangles, and Squares
325
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Student Help
COROLLARIES
STUDY TIP
Rhombus Corollary
The corollaries allow
you to show that a
quadrilateral is a
rhombus, rectangle, or
square without first
showing that it is a
parallelogram.
Words
If a quadrilateral has four congruent
sides, then it is a rhombus.
Symbols
If AB
&* c BC
&* c CD
&* c AD
&*,
then ABCD is a rhombus.
A
B
D
C
Rectangle Corollary
Words
If a quadrilateral has four right angles,
then it is a rectangle.
Symbols
If maA 5 maB 5 maC 5 maD 5 908,
then ABCD is a rectangle.
A
B
D
C
Square Corollary
Words
If a quadrilateral has four congruent sides
and four right angles, then it is a square.
Symbols
IStudent Help
EXAMPLE
If AB
&* c BC
&* c CD
&* c AD
&* and
maA 5 maB 5 maC 5 maD 5 908,
then ABCD is a square.
2
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
A
B
D
C
Identify Special Quadrilaterals
Use the information in the diagram
to name the special quadrilateral.
3
5
Solution
The quadrilateral has four right angles. So, by the Rectangle Corollary,
the quadrilateral is a rectangle.
Because all of the sides are not the same length, you know that the
quadrilateral is not a square.
Identify Special Quadrilaterals
Use the information in the diagram to name the special quadrilateral.
2.
326
Chapter 6
Quadrilaterals
3.
4
4
4
4
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THEOREM 6.10
Words
Symbols
EXAMPLE
In rhombus ABCD, AC
&* ∏ BD
&*.
3
C
B
The diagonals of a rhombus
are perpendicular.
A
D
Use Diagonals of a Rhombus
ABCD is a rhombus.
Find the value of x.
B
608
C
x8
E
A
Student Help
LOOK BACK
To review the Corollary
to the Triangle Sum
Theorem, see p. 180.
D
Solution
By Theorem 6.10, the diagonals of a rhombus are perpendicular.
Therefore, aBEC is a right angle, so TBEC is a right triangle.
By the Corollary to the Triangle Sum Theorem, the acute angles of
a right triangle are complementary. So, x 5 90 2 60 5 30.
THEOREM 6.11
Words
The diagonals of a rectangle
are congruent.
Symbols
In rectangle ABCD, AC
&* c BD
&*.
A
B
D
C
Carpentry
EXAMPLE
4
Use Diagonals of a Rectangle
a. You nail four pieces of wood together
4 ft
to build a four-sided frame, as shown.
What is the shape of the frame?
b. The diagonals measure 7 ft 4 in. and
7 ft 2 in. Is the frame a rectangle?
6 ft
6 ft
Solution
DOORS If a screen door is
not rectangular, you can use a
piece of hardware called a
turnbuckle to shorten the
longer diagonal until the door
is rectangular.
a. The frame is a parallelogram because both
4 ft
pairs of opposite sides are congruent.
b. The frame is not a rectangle because the
diagonals are not congruent.
6.4
Rhombuses, Rectangles, and Squares
327
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Use Diagonals
Find the value of x.
4. rhombus ABCD
5. rectangle EFGH
F
B
6. square JKLM
G
K
L
x8
x8
A
C
12
x8
x
E
H
D
J
M
6.4 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. What is the name for a parallelogram with four congruent sides?
List all of the properties that must be true for the quadrilateral.
2. Parallelogram
A. All sides are congruent.
3. Rectangle
B. All angles are congruent.
4. Rhombus
C. The diagonals are congruent.
5. Square
D. Opposite angles are congruent.
P
6. PQRS is a rectangle. The length of
&* is 12. Find PR and PT.
QS
R
T
P
S
Practice and Applications
Extra Practice
Using Properties Find the measures.
See p. 686.
7. rhombus ABCD
E
B
A
Example 1:
Example 2:
Example 3:
Example 4:
328
Exs. 7–9
Exs. 10–12
Ex. 22
Ex. 13
Chapter 6
Quadrilaterals
F
9. square WXYZ
Z
C
4
Homework Help
8. rectangle EFGH
D
W
3
H
G
Y
X
AB 5 __?__
maE 5 __?__8
maW 5 __?__8
BC 5 __?__
maF 5 __?__8
YZ 5 __?__
AD 5 __?__
maG 5 __?__8
XY 5 __?__
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Careers
Using Corollaries Use the information in the diagram to name the
special quadrilateral.
10.
11.
12.
5
5
5
5
13. Making a Chair If you
FURNITURE DESIGNERS
use geometry, trigonometry,
and artistic skills to create
designs for furniture.
Career Links
measure the diagonals of the
chair frame as shown and find
that they are congruent, can
you conclude that the frame is
rectangular? If not, what other
information do you need?
Explain your reasoning.
Sorting Quadrilaterals In Exercises 14–17, list each quadrilateral for
which the statement is true.
CLASSZONE.COM
parallelogram
square
rhombus
rectangle
14. It has four right angles.
15. Opposite sides are congruent.
16. Diagonals bisect each other.
17. Diagonals are perpendicular.
Use Properties of Quadrilaterals
EXAMPLE
PQRS is a rectangle.
Find the value of x.
P
P
2x
x15
S
R
Solution
PR 5 SQ
Diagonals of a rectangle are congruent.
2x 5 x 1 5
Substitute 2x for PR and x 1 5 for SQ.
x55
Subtract x from each side.
Using Algebra Find the value of x.
18. rhombus KLMN
K
19. square ABCD
A
L
5x 8
B
20. rectangle EFGH
F
3x
E
N
x12
M
D
G
6x
J
3x 1 9
H
C
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Rhombuses, Rectangles, and Squares
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21. Logical Reasoning In g JKLM, aJ is a right angle. Explain why
g JKLM is a rectangle.
22. Using Theorems Find the value of
R
S
x in rhombus QRST.
x
5
P
4
P
T
Challenge GHJK is a square with diagonals intersecting at L. Given
that GH 5 2 and GL 5 Ï2
w, complete the statement.
Ï2
24. maKLJ 5 __?__
L
25. maHJG 5 __?__
26. Perimeter of THJK 5 __?__
Standardized Test
Practice
H
2
G
23. HK 5 __?__
K
J
27. Multiple Choice In g KLMN, KL 5 LM. What is maN?
A
X
C
X
308
B 458
X
D Cannot be determined
X
908
28. Multiple Choice In rhombus ABCD, AB 5 7x 2 3 and CD 5 25.
What is the value of x?
F
X
H
X
Mixed Review
3
A
G 4
X
J 25
X
7
D
25
7x 2 3
B
C
Finding Angle Measures Find the measure of the numbered angle.
(Lesson 3.4)
29.
30.
31.
2
3
358
1
Algebra Skills
1208
Finding Ratios Find the ratio of the length to the width for the
rectangle. Write the ratio in simplest form. (Skills Review, p. 660)
32.
33.
34.
7 in.
5 cm
9 cm
330
Chapter 6
Quadrilaterals
12 m
14 in.
18 m