Lesson 6-5
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Transcript of Lesson 6-5
Lesson 6-5
Rhombi and Squares
5-Minute Check on Lesson 6-45-Minute Check on Lesson 6-45-Minute Check on Lesson 6-45-Minute Check on Lesson 6-4
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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 mWXY = 6a² - 6, find a.
RSTU is a rectangle. Find each value.
4. mVRS
5. mRVU
6. What are the coordinates of W if WXYZ is a rectangle and X(2,6), Y(4,3), and Z(1,1)?
Standardized Test Practice:
A CB D
W
Z
X
Y
R
R S
TU
V38°
(-1,4)(-1,-4)(1,4) (1,-4)C
50
3
± 4
52°
104°
Polygon Hierarchy
Polygons
Squares
RhombiRectangles
Parallelograms Kites Trapezoids
IsoscelesTrapezoids
Quadrilaterals
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)
Vocabulary
• Rhombus – quadrilateral with all four sides congruent
• Square – a quadrilateral that is both a rhombus and a rectangle
Rhombi and Squares
Rhombus CharacteristicsAll Parallelogram Properties
All 4 Sides CongruentDiagonals bisect a pair of opposite ’s
Diagonals form right angles with each other
Square Characteristics All Parallelogram Properties
All Rectangle PropertiesAll Rhombus Properties
Diagonals divide into 4 congruent ∆’s
A B
C D
A B
C D
Use rhombus LMNP to find the value of y if m1 = y² - 54.
N
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.
N
Use rhombus LMNP to find mPNL if mMLP = 64
Opposite angles are congruent.
Substitution
The diagonals of a rhombus bisect the angles.
Answer:
Use rhombus ABCD and the given information to find the value of each variable.
Answer: 8 or –8
Answer:
a.
b.
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.
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?
The distance from the center of the pole to a leg is equal to the length of
Answer: The center of the pole is about 1.4 feet from a leg of a table.
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
Quadrilateral Characteristics SummaryConvex Quadrilaterals
Squares
RhombiRectangles
Parallelograms Trapezoids
IsoscelesTrapezoids
Opposite sides parallel and congruentOpposite angles congruentConsecutive angles supplementaryDiagonals bisect each other
Bases ParallelLegs are not ParallelLeg angles are supplementary Median is parallel to basesMedian = ½ (base + base)
Angles all 90°Diagonals congruent
Diagonals divide into 4 congruent triangles
All sides congruentDiagonals perpendicularDiagonals bisect opposite angles
Legs are congruent Base angle pairs congruent Diagonals are congruent
4 sided polygon4 interior angles sum to 3604 exterior angles sum to 360
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 431-33; 1, 2, 7-11, 27-30