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 S2. 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 CD2. 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 BDProve: 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