Honors Geometry Section 4.6 (1)

Conditions for Special Quadrilaterals

In section 4.5, we answered questions such as “If a quadrilateral is a parallelogram,

what are its properties?” or “If a quadrilateral is a rhombus, what are its properties?” In this section we look to

reverse the process, and answer the question “What must we know about a

quadrilateral in order to say it is a parallelogram or a rectangle or a

whatever?”

What does it take to make a parallelogram?

State whether the following conjectures are true or false. If it is

false, draw a counterexample.

If one pair of opposite sides of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.

If one pair of opposite sides of a quadrilateral are parallel,then the quadrilateral is a parallelogram.

If one pair of opposite sides of a quadrilateral are both parallel and

congruent, then the quadrilateral is a parallelogram.

If both pairs of opposite sides of a quadrilateral are parallel,then the quadrilateral is a parallelogram.

If both pairs of opposite sides of a quadrilateral are congruent, then

the quadrilateral is a parallelogram.

If the diagonals of a quadrilateral bisect each other, then the

quadrilateral is a parallelogram.

The last 4 statements will be our tests for determining if a

quadrilateral is a parallelogram.

If a quadrilateral does not satisfy one of these 4 tests, then we

cannot say that it is a parallelogram!

What does it take to make a rectangle?

State whether the following conjectures are true or false. If it is

false, draw a counterexample.

If one angle of a quadrilateral is a right angle, then the quadrilateral

is a rectangle.

If one angle of a parallelogram is a right angle, then the parallelogram

is a rectangle.

If the diagonals of a quadrilateral are congruent, then the

quadrilateral is a rectangle.

If the diagonals of a parallelogram are congruent, then the

parallelogram is a rectangle.

If the diagonals of a parallelogram are perpendicular , then the parallelogram is a rectangle.

Statements 2 and 4 will be our tests for determining if a

quadrilateral is a rectangle.

Notice that in both of those statements you must know that

the quadrilateral is a parallelogram before you can say that it is a

rectangle.

What does it take to make a rhombus?

State whether the following conjectures are true or false. If it is

false, draw a counterexample.

If one pair of adjacent sides of a quadrilateral are congruent, then

the quadrilateral is a rhombus.

If one pair of adjacent sides of a parallelogram are congruent, then

the parallelogram is a rhombus.

If the diagonals of a parallelogram are congruent, then the

parallelogram is a rhombus.

If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.

If the diagonals of a parallelogram bisect the angles of the parallelogram, then the

parallelogram is a rhombus.

Statements 2, 4 and 5will be our tests for determining if a

quadrilateral is a rhombus.

Notice that in each of these statements you must know that

the quadrilateral is a parallelogram before you can say that it is a

rhombus.

What does it take to make a square?

It must be a parallelogram, rectangle and rhombus.

Examples: Consider quad. OHMY with diagonals that intersect at point S. Determine if the given

information allows you to conclude that quad. OHMY is a

parallelogram, rectangle, rhombus or square. List all that apply.