Geometry Section 6.3 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?”

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

Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 6.10 If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.

Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

What does it take to make a rectangle? In order for a quadrilateral to be a rectangle, it must first be a parallelogram. We then look for one of the following.

1. one angle is a right angle 2. the diagonals are congruent

What does it take to make a rhombus? In order for a quadrilateral to be a rhombus, it must first be a parallelogram. We then look for one of the following.

1. one pair of adjacent sides of are congruent 2. the diagonals are perpendicular

What does it take to make a square? In order for a quadrilateral to be a square, it must be a parallelogram, rectangle and rhombus.

If two lines are parallel, their slopes are ______ If two lines are perpendicular, their slopes are _________________ equal opposite reciprocals

You are given the coordinates of the vertices of quad ABCD in the following examples. Determine if quad ABCD is a parallelogram, rectangle, rhombus or square. If more than one applies, list each of them.

When given the coordinates of 4 points in the coordinate plane, how can we determine if the quadrilateral with these vertices is a ………….. parallelogram:

When given the coordinates of 4 points in the coordinate plane, how can we determine if the quadrilateral with these vertices is a ………….. rectangle: (you must first make sure the figure is a parallelogram)

When given the coordinates of 4 points in the coordinate plane, how can we determine if the quadrilateral with these vertices is a ………….. rhombus: (you must first make sure the figure is a parallelogram)