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Bell Ringer

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Lesson 6-4: Rhombus & Square

Rhombi

Rectangles &

Squares

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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Since a rhombus is a parallelogram the following are true:

Theorem:

The diagonals of a rhombus are perpendicular.

Theorem:

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

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

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.

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

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

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

- 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