Aim: Properties of Square & Rhombus Course: Applied Geo.

Do Now:

Aim: What are the properties of a rhombus and a square?

Find the length of AD in rectangle ABCD, if AB = 2 and diagonal BD = 4.

A rectangle has 4 right angles.A rectangle has 4 right angles.

42

x

B C

DA

a2 + b2 = c2

Pythagorean Theorem

a2 + b2 = c2

ABD is a right triangle

22 + x2 = 42

4 + x2 = 16x2 = 12

12 3.46x

Aim: Properties of Square & Rhombus Course: Applied Geo.

A B

D C

Properties of a Rhombus

A rhombus has all the properties of a parallelogram,A rhombus has all the properties of a parallelogram,

PLUS,A rhombus is equilateral.A rhombus is equilateral.

The diagonals of a rhombus are perpendicular to each other.The diagonals of a rhombus are perpendicular to each other.

A rhombus is a parallelogram that has two congruent consecutive sides.

A rhombus is a parallelogram that has two congruent consecutive sides.

The diagonals of a rhombus bisect its anglesThe diagonals of a rhombus bisect its angles

Aim: Properties of Square & Rhombus Course: Applied Geo.

Properties of a QuadrilateralsProperty Parallel. Rectangle Rhombus

All sides are

Opposite sides are

Opposite sides are | |

Opposite angles are

All angles are right

Diagonals bisect each other

Diagonals are

Diagonals are

Each Diagonal bisects opposite

Aim: Properties of Square & Rhombus Course: Applied Geo.

Model Problem

Find the measures of the numbered angles in the rhombus

500

1

324

1 = 900 Diagonals of a rhombus are Perpendicular

2 = 500 Alternate Interior s

3 = 500 Diagonals of rhombus bisect the s

4 = 400 The sum of the s of a equal 1800

The diagonals of a rhombus are perpendicular to each other.

The diagonals of a rhombus are perpendicular to each other.

900

500

500400

900

Aim: Properties of Square & Rhombus Course: Applied Geo.

Regents Question

In the diagram below of rhombus ABCD, m∠C = 100.

What is m∠DBC? 40o

40o

Aim: Properties of Square & Rhombus Course: Applied Geo.

Model Problem

Find the value of the variables (2x)º

(x + y)º(3z)º

Since all sides are congruent this quadrilateral is a rhombus.

Property Parallel. Rectangle Rhombus

2x = 90 x = 453z = 90 z = 30

x + y = 90 45 + y = 90 y = 45

Diagonals are

Aim: Properties of Square & Rhombus Course: Applied Geo.

Model Problem

Explain how and why ABCD is a rhombus

Given: ABCD is a parallelogramAB = 2x + 1, DC = 3x – 11, AD = x + 13

Plan: Show that 2 consecutive sides are congruent. (AB AD)

A B

D C

1) Since ABCD is a parallelogram, opposite sides are equal in length.DC = AB 3x – 11 = 2x + 1

3x – 2x = 11 + 1 x = 12

2) Substitute x = 12 to find the length of AB and AD: AB = 2x + 1 = 2(12) + 1 = 25

AD = x + 13 = 12 + 13 = 25

AB AD. Since parallelogram ABCD hastwo consecutive congruent sides, it’s a rhombus.

Aim: Properties of Square & Rhombus Course: Applied Geo.

Do Now:

Aim: What are the properties of a rhombus and a square?

In rhombus KLMN, KL = 3x, LM = 2(x + 3). Find the length of each side of the rhombus.

K N

L M

Aim: Properties of Square & Rhombus Course: Applied Geo.

Properties of a Square

A square is a rectangle that has two congruent sides.

A square is a rectangle that has two congruent sides.

C

B

D

A

A square has all the properties of a rectangle,A square has all the properties of a rectangle,

PLUS,

A square has all the properties of a rhombus.A square has all the properties of a rhombus.

Aim: Properties of Square & Rhombus Course: Applied Geo.

Properties of a SquareProperty Parallel. Rhombus Rectangle Square

All sides are

Opposite sides are

Opposite sides are | |

Opposite angles are

All angles are right

Diagonals bisect each other

Diagonals are

Diagonals are

Each Diagonal bisects opposite

Aim: Properties of Square & Rhombus Course: Applied Geo.

Model Problem

ABCD is a square with diagonal BD. Determine if True or False.

A. AB BC

B. AB CD

C. AB AC

D. 1 2

E. 1 3

F. B 4

G. ABC is isosceles

H. ABC is right triangle

I. ABC ACD

A B

CD

12

34

Aim: Properties of Square & Rhombus Course: Applied Geo.

Model Problem

Find the value of the variables.

In a square: diagonals are to each otherdiagonals bisect opposite angles

In a square: diagonals are to each otherdiagonals bisect opposite angles

x = 5

19x 6z

1 = 3y – 6

y = 32x = 7.5

Aim: Properties of Square & Rhombus Course: Applied Geo.

Model Problem

ABCD is a square. If AB = 8x – 6 and BC = 5x + 12, find the length of each side of the square.

Which statement is false?

a) a square is a rectangle

b) a square is a rhombus

c) a rhombus is a square

d) a square is a parallelogram

Aim: Properties of Square & Rhombus Course: Applied Geo.

Model Problem

In square ABCD diagonal AC is drawn. How many degrees are there in the measure of ACB?

If the side of a square is 4, find the length of the diagonal.