6.4 Rhombuses, Rectangles, and Squares

Review

Find the value of the variables.

52°

68°

h

p

(2p-14)° 50°

52° + 68° + h = 180°

120° + h = 180 °

h = 60°

p + 50° + (2p – 14)° = 180°p + 2p + 50° - 14° = 180° 3p + 36° = 180° 3p = 144 °

p = 48 °

Special Parallelograms

Rhombus A rhombus is a parallelogram with four

congruent sides.

Special Parallelograms

Rectangle A rectangle is a parallelogram with four right

angles.

Special Parallelogram

Square A square is a parallelogram with four

congruent sides and four right angles.

Corollaries

Rhombus corollary A quadrilateral is a rhombus if and only if it

has four congruent sides.

Rectangle corollary A quadrilateral is a rectangle if and only if it

has four right angles.

Square corollary A quadrilateral is a square if and only if it is a

rhombus and a rectangle.

Example

PQRS is a rhombus. What is the value of b?

P Q

RS

2b + 3

5b – 6

2b + 3 = 5b – 6 9 = 3b 3 = b

Review

In rectangle ABCD, if AB = 7f – 3 and CD = 4f + 9, then f = ___

A) 1

B) 2

C) 3

D) 4

E) 5

7f – 3 = 4f + 9

3f – 3 = 9

3f = 12

f = 4

Example

PQRS is a rhombus. What is the value of b?

P Q

RS

3b + 12

5b – 6

3b + 12 = 5b – 6 18 = 2b 9 = b

Theorems for rhombus

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

L

Theorem of rectangle

A parallelogram is a rectangle if and only if its diagonals are congruent.

A B

CD

Match the properties of a quadrilateral

1. The diagonals are congruent

2. Both pairs of opposite sides are congruent

3. Both pairs of opposite sides are parallel

4. All angles are congruent

5. All sides are congruent

6. Diagonals bisect the angles

A. Parallelogram

B. Rectangle

C. Rhombus

D. Square

B,D

A,B,C,D

A,B,C,D

B,D

C,D

C

Decide if the statement is sometimes, always, or never true.1. A rhombus is equilateral.

2. The diagonals of a rectangle are _|_.

3. The opposite angles of a rhombus are supplementary.

4. A square is a rectangle.

5. The diagonals of a rectangle bisect each other.

6. The consecutive angles of a square are supplementary.

Always

Quadrilateral ABCD is Rhombus.7. If m <BAE = 32o, find m<ECD.8. If m<EDC = 43o, find m<CBA.9. If m<EAB = 57o, find m<ADC.10. If m<BEC = (3x -15)o, solve for x.11. If m<ADE = ((5x – 8)o and m<CBE = (3x +24)o, solve for x12. If m<BAD = (4x + 14)o and m<ABC = (2x + 10)o, solve for x.

A B

E

D C

32o

Sometimes

Sometimes

Always

Always

Always

86o

66o

35o

16 26

Coordinate Proofs Using the Properties of Rhombuses, Rectangles and Squares

Using the distance formula and slope, how can we determine the specific shape of a parallelogram?

Rhombus –

Rectangle –

Square -

Based on the following Coordinate values, determine if each parallelogramis a rhombus, a rectangle, or square.

P (-2, 3) P(-4, 0)Q(-2, -4) Q(3, 7)R(2, -4) R(6, 4)S(2, 3) S(-1, -3)

1. Show all sides are equal distance

2. Show all diagonals are perpendicular.

1. Show diagonals are equal distance

2. Show opposite sides are perpendicular

Show one of the above four ways.

RECTANGLE RECTANGLE