8.4 Properties of Rhombuses, Rectangles, and Squares€¦ · Squares A square is a regular parallelogram. •All angles are congruent •All sides are congruent

Assignment

• P. 537-540: 1, 2, 3-

48 M3, 49, 52, 55,

Pick one (56, 60, 61,

63)

• P. 723: 5, 18, 25, 27,

40

• P. 732: 8, 11, 15, 20,

28, 36

• Challenge Problems

Rhombuses Or Rhombi

What makes a quadrilateral a rhombus?

Rhombuses Or Rhombi

A rhombus is an

equilateral

parallelogram.

– All sides are

congruent

Rhombus Corollary

A quadrilateral is a

rhombus if and only

if it has four

congruent sides.

Rectangles

What makes a quadrilateral a rectangle?

Rectangles

A rectangle is an

equiangular

parallelogram.

• All angles are

congruent

Example 1

What must each angle of a rectangle

measure?

Rectangle Corollary

A quadrilateral is a

rectangle if and only

if it has four right

angles.

Squares

What makes a quadrilateral a square?

Squares

A square is a regular

parallelogram.

• All angles are

congruent

• All sides are

congruent

Square Corollary

A quadrilateral is a

square if and only if

it is a rhombus and

a rectangle.

8.4 Properties of Rhombuses, Rectangles, and

Squares

Objectives:

1. To discover and use properties of

rhombuses, rectangles, and squares

2. To find the area of rhombuses,

rectangles, and squares

Example 2

Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram. Fill in the missing names.

Example 2

Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram.

Example 3

For any rhombus QRST, decide whether the

statement is always or sometimes true.

Draw a sketch and explain your reasoning.

1. Q S

2. Q R

Example 4

For any rectangle ABCD, decide whether the

statement is always or sometimes true.

Draw a sketch and explain your reasoning.

1. AB CD

2. AB BC

Example 5

Classify the special quadrilateral. Explain

your reasoning.

Investigation 1

We know that the

diagonals of

parallelograms bisect

each other. The

diagonal of rectangles

and rhombuses have

a few other properties

we will discover using

GSP.

Diagonal Theorem 1

A parallelogram is a rectangle if and only if

its diagonals are congruent.

Example 6

The previous theorem is a biconditional.

Write the two conditional statements that

must be proved separately to prove the

entire theorem.

Example 7

You’ve just had a new door installed, but it

doesn’t seem to fit into the door jamb

properly. What could you do to determine

if your new door is rectangular?

Diagonal Theorem 2

A parallelogram is a rhombus if and only if its

diagonals are perpendicular.

Diagonal Theorem 3

A parallelogram is a rhombus if and only if

each diagonal bisects a pair of opposite

angles.

Example 8

Prove that if a parallelogram has

perpendicular diagonals, then it is a

rhombus.

Given: ABCD is a

parallelogram;

AC BD

Prove: ABCD is a

rhombus

Example 9: SAT

In the figure, a small

square is inside a

larger square.

What is the area, in

terms of x, of the

shaded region?

Example 10

In the diagram below,

MRVU SPTV. Let

the area of MRVU

equal A. Show that

A = bh.

Rhombus Area

Since a rhombus is a

parallelogram, we

could find its area

by multiplying the

base and the

height.A b h

Rhombus Area

However, you’re not

always given the

base and height, so

let’s look at the two

diagonals. Notice

that d1 divides the

rhombus into 2

congruent triangles.

Ah, there’s a couple of

triangles in there.

1

2A b h

Rhombus Area

So find the area of

one triangle, and

then double the

result.

1

2A b h

12

2A b h

1 2

1 12

2 2A d d

1 2

12

4A d d

1 2

1

2d d

1 2

1

2A d d

Ah, there’s a couple of

triangles in there.

Polygon Area Formulas

Exercise 11

Find the area of the shaded region.1. 2. 3.

Exercise 12

If the length of each diagonal of a rhombus is

doubled, how is the area of the rhombus

affected?

Assignment

• P. 537-540: 1, 2, 3-

48 M3, 49, 52, 55,

Pick one (56, 60, 61,

63)

• P. 723: 5, 18, 25, 27,

40

• P. 732: 8, 11, 15, 20,

28, 36

• Challenge Problems