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. always true.

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.

b.

Q R

SOLUTION b.

If rhombus

QRST

is a square, then all four angles are congruent right

R

if

QRST

is a square. Because not all rhombuses are also squares, the statement is sometimes true.

EXAMPLE 2

Classify special quadrilaterals Classify the special quadrilateral. Explain your reasoning.

SOLUTION 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 reasoning.

ANSWER Sometimes; this is only true if

EFGH

is a square.

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

EXAMPLE 3

List properties of special parallelograms Sketch rectangle

ABCD

. List everything that you know about it.

SOLUTION By definition, you need to draw a figure with the following properties: • The figure is a parallelogram.

• The figure has four right angles.

Because

ABCD

is a parallelogram, it also has these properties:

EXAMPLE 3

List properties of special parallelograms • Opposite sides are parallel and congruent.

• Opposite angles are congruent. Consecutive angles are supplementary.

• Diagonals bisect each other.

By Theorem

8.13

, the diagonals of

ABCD

are congruent.

GUIDED PRACTICE

for Example 3 3.

Sketch square

PQRS

. List everything you know about the square.

ANSWER

P

1. PQRS is a parallelogram, rectangle and a rhombus.

2.

Opposite pairs of sides are parallel and all four sides are congruent.

3.

All four angles are right angles.

S

4.

Diagonals are congruent and bisect each other.

Q R

Homework: 537: 3-17, 19-24, 27-51 odd