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 RhombiWhat makes a quadrilateral a rhombus?

Rhombuses Or RhombiA rhombus is an

equilateral parallelogram.– All sides are

congruent

Rhombus CorollaryA quadrilateral is a

rhombus if and only if it has four congruent sides.

RectanglesWhat makes a quadrilateral a rectangle?

RectanglesA rectangle is an

equiangular parallelogram.

• All angles are congruent

Example 1What must each angle of a rectangle

measure?

Rectangle CorollaryA quadrilateral is a

rectangle if and only if it has four right angles.

SquaresWhat makes a quadrilateral a square?

SquaresA square is a regular

parallelogram.• All angles are

congruent• All sides are

congruent

Square CorollaryA 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 squares2. To find the area of rhombuses, rectangles,

and squares

Example 2Below 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 2Below 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 3For 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 4For 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 5Classify the special quadrilateral. Explain

your reasoning.

Investigation 1We 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 1A parallelogram is a rectangle if and only if

its diagonals are congruent.

Example 6The previous theorem is a biconditional.

Write the two conditional statements that must be proved separately to prove the entire theorem.

Example 7You’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 2A parallelogram is a rhombus if and only if its

diagonals are perpendicular.

Diagonal Theorem 3A parallelogram is a rhombus if and only if

each diagonal bisects a pair of opposite angles.

Example 8Prove 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: SATIn the figure, a small

square is inside a larger square. What is the area, in terms of x, of the shaded region?

Example 10In the diagram below, MRVU SPTV. Let the area of MRVU equal A. Show that A = bh.

Rhombus AreaSince a rhombus is a

parallelogram, we could find its area by multiplying the base and the height.

A b h

Rhombus AreaHowever, 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.

12

A b h

Rhombus AreaSo find the area of

one triangle, and then double the result.

12

A b h

122

A b h

1 21 122 2

A d d

1 2124

A d d

1 212d d

1 212

A d d

Ah, there’s a couple of triangles in there.

Polygon Area Formulas

Exercise 11Find the area of the shaded region.1. 2. 3.

Exercise 12If 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