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### Lesson 8-5

Rhombi and Squares

50

3

± 4

52°

104°

C

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 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)?

X

W

R

Y

Z

S

R

V

38°

U

T

Standardized Test Practice:

(1,4)

(1,-4)

(-1,-4)

(-1,4)

A

B

C

D

Click the mouse button or press the Space Bar to display the answers.

Polygon Hierarchy

Polygons

Quadrilaterals

Parallelograms

Kites

Trapezoids

IsoscelesTrapezoids

Rectangles

Rhombi

Squares

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

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 CharacteristicsAll Parallelogram PropertiesAll Rectangle Properties

All Rhombus Properties

Diagonals divide into 4 congruent ∆’s

D

C

Example 5-2a

Use rhombus LMNP to find the value of y if m1 = 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.

Answer:

Example 5-2cUse rhombus LMNP to find mPNL if mMLP = 64

Opposite angles are congruent.

Substitution

The diagonals of a rhombus bisect the angles.

b.

Answer:

Example 5-2eUse rhombus ABCD and the given information to find the value of each variable.

Answer: 8 or –8

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.

The distance from the center of the pole to a leg is equal to the length of

Example 5-4bAnswer: The center of the pole is about 1.4 feet from a leg of a table.

Example 5-4d to the length of

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 Summary to the length of

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

Rectangles

Rhombi

IsoscelesTrapezoids

All sides congruent

Diagonals perpendicular

Diagonals 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

Summary & Homework to the length of

- 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

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