Rhombi and Squares

1
Lesson 8-5

• Rhombi and Squares

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Transparency 8-5
5-Minute Check on Lesson 8-4
WXYZ is a rectangle. Find each value. 1. If ZX
6x 4 and WY 4x 14, find ZX. 2. If WY
26 and WR 3y 4, find y. 3. If m?WXY 6a² -
6, find a. RSTU is a rectangle. Find each
value. 4. m?VRS 5. m?RVU 6.
What are
the coordinates of W if WXYZ is a rectangle and
X(2,6), Y(4,3), and Z(1,1)?
X
W
R
Y
Z
R
S
V
38
U
T
Standardized Test Practice
(1,4)
(1,-4)
(-1,4)
(-1,-4)
A
C
B
D
Click the mouse button or press the Space Bar to
display the answers.
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Polygon Hierarchy
Polygons
Quadrilaterals
Parallelograms
Kites
Trapezoids
IsoscelesTrapezoids
Rhombi
Rectangles
Squares
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Objectives

• Recognize and apply the properties of rhombi
• All Parallelogram Properties
• All 4 Sides Congruent
• Diagonals bisect a pair of opposite ?s
• Diagonals form right angles with each other
• Recognize and apply the properties of squares
• All Parallelogram Properties
• All Rectangle Properties
• All Rhombus Properties
• Diagonals divide into 4 congruent ?s (45-45-90)

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Vocabulary

• Rhombus quadrilateral with all four sides
  congruent
• Square a quadrilateral that is both a rhombus
  and a rectangle

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Rhombi and Squares
A
B
Rhombus CharacteristicsAll Parallelogram
Properties All 4 Sides Congruent Diagonals bisect
a pair of opposite ?s Diagonals form right
angles with each other
C
D
A
B
Square Characteristics All Parallelogram
Properties All Rectangle Properties All Rhombus
Properties Diagonals divide into 4 congruent ?s
D
C
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Example 5-2a
Use rhombus LMNP to find the value of y if m?1
y² - 54.
Diagonals of a rhombus are perpendicular.
Substitution
Add 54 to each side.
Take the square root of each side.
Answer The value of y can be 12 or 12.
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Example 5-2c
Use rhombus LMNP to find m?PNL if m?MLP 64
Opposite angles are congruent.
Substitution
The diagonals of a rhombus bisect the angles.
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Example 5-2e
Use rhombus ABCD and the given information to
find the value of each variable.
Answer 8 or 8
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Example 5-4a
A square table has four legs that are 2 feet
apart. The table is placed over an umbrella stand
so that the hole in the center of the table lines
up with the hole in the stand. How far away from
a leg is the center of the hole?
Let ABCD be the square formed by the legs of the
table. Since a square is a parallelogram, the
diagonals bisect each other. Since the umbrella
stand is placed so that its hole lines up with
the hole in the table, the center of the umbrella
pole is at point E, the point where the diagonals
intersect. Use the Pythagorean Theorem to find
the length of a diagonal.
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Example 5-4b
Answer The center of the pole is about 1.4 feet
from a leg of a table.
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Example 5-4d
Kayla has a garden whose length and width are
each 25 feet. If she places a fountain exactly in
the center of the garden, how far is the center
of the fountain from one of the corners of the
garden?
Answer about 17.7 feet
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Quadrilateral Characteristics Summary
Convex Quadrilaterals
4 sided polygon 4 interior angles sum to 360 4
exterior angles sum to 360
Parallelograms
Trapezoids
Bases Parallel Legs are not Parallel Leg angles
are supplementary Median is parallel to
basesMedian ½ (base base)
Opposite sides parallel and congruent Opposite
angles congruent Consecutive angles
supplementary Diagonals bisect each other
Rhombi
Rectangles
IsoscelesTrapezoids
All sides congruent Diagonals perpendicular Diagon
als bisect opposite angles
Angles all 90 Diagonals congruent
Legs are congruent Base angle pairs congruent
Diagonals are congruent
Squares
Diagonals divide into 4 congruent triangles
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Summary Homework

• Summary
• A rhombus is a quadrilateral with each side
  congruent, diagonals that are perpendicular, and
  each diagonal bisecting a pair of opposite
  angles.
• A quadrilateral that is both a rhombus and a
  rectangle is a square.
• Homework
• pg 434-436 14-23, 26-31