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?”
Theorem 4.6.1
If both pairs of opposite sides of a
quadrilateral are congruent, then
the quadrilateral is a
parallelogram.
3) GL  GL
3) Reflexive Prop.
4)  GFL   LOG
4) SSS
5)  GLF   LGO
5) CPCTC
 GLO   LGF
6)  GLF &  LGO are AIAs
6) Def. of AIAs
 GLO &  LGF are AIAs
) GO // FL & GF // OL
) Quad. GFLO is a //gram
7) AIAT Converse
) Def. of //gram
Theorem 4.6.2
If one pair of opposite sides of a
quadrilateral are both parallel and
congruent, then the quadrilateral is
a parallelogram.
Theorem 4.6.3
If the diagonals of a quadrilateral
bisect each other, then the
quadrilateral is a parallelogram.
Note: we also have our definition
of a parallelogram: If two pairs of
opposite sides of a quadrilateral
are parallel, 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!
Theorem 4.6.4
If one angle of a parallelogram is a
right angle, then the parallelogram
is a rectangle.
Theorem 4.6.5
If the diagonals of a parallelogram
are congruent, then the
parallelogram is a rectangle.
The previous 2 statements 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.
Theorem 4.6.6
If one pair of adjacent sides of a
parallelogram are congruent, then
the parallelogram is a rhombus.
Theorem 4.6.7
If the diagonals of a parallelogram
bisect the angles of the
parallelogram, then the
parallelogram is a rhombus.
Theorem 4.6.8
If the diagonals of a parallelogram
are perpendicular then the
parallelogram is a rhombus.
The previous 3 statements will 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.
Paralle log ram
Paralleogr am
Rectangle
Paralle log ram
Re c tan gle
R hom bus
Square