Warm-up

• Pg 520 #39, 40

• Pg 529 #43-45

Properties of Rhombuses, Rectangles, and Squares

8.4

Rhombus

• A rhombus is a parallelogram with four congruent sides.

• A quadrilateral is a rhombus if and only if it has four congruent sides

Rectangle

• A rectangle is a parallelogram with four right angles.

• A quadrilateral is a rectangle if and only if it has four right angles

Square

• A square is a parallelogram with four congruent sides and four right angles.

• A quadrilateral is a square if and only if it is a rhombus and a rectangle

EXAMPLE 1 Use properties of special quadrilaterals

For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.

a. Q S

SOLUTION

a. By definition, a rhombus is a parallelogram with four

congruent sides. By Theorem 8.4, opposite angles of a parallelogram are congruent.

So, .The statement is always true.

Q S

EXAMPLE 1 Use properties of special quadrilaterals

b. If rhombus QRST is a square, then all four angles are congruent right angles. So, if QRST is a square. Because not all rhombuses are also squares, the statement is sometimes true.

Q R

For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.

Q Rb.

SOLUTION

EXAMPLE 2 Classify special quadrilaterals

SOLUTION

Classify the special quadrilateral. Explain your reasoning.

The quadrilateral has four congruent sides. One of the angles is not a right angle, so the rhombus is not also a square. By the Rhombus Corollary, the quadrilateral is a rhombus.

GUIDED PRACTICE for Examples 1 and 2

1. For any rectangle EFGH, is it always or sometimes true that Explain your reasoning.

FG GH ?

Sometimes; this is only true if EFGH is a square.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

2. A quadrilateral has four congruent sides and four congruent angles. Sketch the quadrilateral and classify it.

ANSWER square

Properties of Diagonals

• A parallelogram is a rhombus if and only if its diagonals are perpendicular.

• A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

• A parallelogram is a rectangle if and only if its diagonals are congruent.

EXAMPLE 3 List properties of special parallelograms

Sketch rectangle ABCD. List everything that you know about it.

SOLUTION

• The figure is a parallelogram.

• The figure has four right angles.

Because ABCD is a parallelogram, it also has these properties:

• Opposite sides are parallel and congruent.

• Opposite angles are congruent. Consecutive angles are supplementary.

• Diagonals bisect each other, and are congruent

GUIDED PRACTICE for Example 3

3. Sketch square PQRS. List everything you know about the square.

ANSWERP Q

S R

1. PQRS is a parallelogram, rectangle and a rhombus.2. Opposite pairs of sides areparallel and all four sides arecongruent.3. All four angles are right angles.4. Diagonals are congruent and bisecteach other.

EXAMPLE 4 Solve a real-world problem

You are building a frame for a window. The window will be installed in the opening shown in the diagram.

Carpentry

a. The opening must be a rectangle. Given the

measurements in the diagram, can you assume

that it is? Explain.

b. You measure the diagonals of the opening. The diagonals are 54.8 inches and 55.3 inches. What can you conclude about the shape of the opening?

Class/Group Work

• Pg. 538• #26,28• #32,34,36,38,40,42,44,46,48• #53• #55

Homework

• Pg 537• 3-15 mult of 3• 19-23 odd• 27,29• 33-48 mult of 3