Lesson 8-5

Rhombi and Squares

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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 434-436; 14-23, 26-31