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Bell Ringer. Rhombi Rectangles & Squares. Rhombus. Definition:. A rhombus is a parallelogram with four congruent sides. Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. ≡. ≡.

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Bell Ringer

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### Bell Ringer

Lesson 6-4: Rhombus & Square

Rhombi

Rectangles &

Squares

### Rhombus

Definition:

A rhombus is a parallelogram with four congruent sides.

• Opposite sides are parallel.

• Opposite sides are congruent.

• Opposite angles are congruent.

• Consecutive angles are supplementary.

• Diagonals bisect each other

Since a rhombus is a parallelogram the following are true:

### Properties of a Rhombus

Theorem:

The diagonals of a rhombus are perpendicular.

Theorem:

Each diagonal of a rhombus bisects a pair of opposite angles.

### Rhombus Examples .....

Given: ABCD is a rhombus. Complete the following.

• If AB = 9, then AD = ______.

• If m<1 = 65, the m<2 = _____.

• m<3 = ______.

• If m<ADC = 80, the m<DAB = ______.

• If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.

9 units

65°

90°

100°

10

### Square

Definition:

A square is a parallelogram with four congruent angles and four congruent sides.

• Opposite sides are parallel.

• Four right angles.

• Four congruent sides.

• Consecutive angles are supplementary.

• Diagonals are congruent.

• Diagonals bisect each other.

• Diagonals are perpendicular.

• Each diagonal bisects a pair of opposite angles.

Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals.

### Squares – Examples…...

Given: ABCD is a square. Complete the following.

• If AB = 10, then AD = _____ and DC = _____.

• If CE = 5, then DE = _____.

• m<ABC = _____.

• m<ACD = _____.

• m<AED = _____.

10 units

10 units

5 units

90°

45°

90°

Rectangles

### Rectangles

Definition:

A rectangle is a parallelogram with four right angles.

• Opposite sides are parallel.

• Opposite sides are congruent.

• Opposite angles are congruent.

• Consecutive angles are supplementary.

• Diagonals bisect each other.

A rectangle is a special type of parallelogram.

Thus a rectangle has all the properties of a parallelogram.

A

B

E

D

C

### Properties of Rectangles

Theorem:

If a parallelogram is a rectangle, then its diagonals are congruent.

Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles.

Converse:

If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.

A

B

2

3

1

E

4

5

6

D

C

### Examples…….

• If AE = 3x +2 and BE = 29, find the value of x.

• If AC = 21, then BE = _______.

• If m<1 = 4x and m<4 = 2x, find the value of x.

• If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.

x = 9 units

10.5 units

x = 18 units

m<1=50,

m<3=40,

m<4=80,

m<5=100,

m<6=40