Name:
Date:
Math 2
U
Special Quadrilaterals
Q
What do we know about convex quadrilaterals so far:
 four sides
 four angles
 the angles sum to 360
 we can always draw diagonals that will be in the interior of
the quadrilateral
D
A
What if we start specifying things? We have seven special quadrilaterals that are defined by the
specific information we know about them. They all have special names and the specific
information means we can prove additional properties about them.
For example, if one (and only one) pair of sides are parallel, do we know more about the sides,
angles, diagonals?
1 Pair of Parallel Sides (Trapezoid)
base
T
T
R
R
leg
leg
A
P
P
A
base
We just proved that consecutive angles between parallel lines are supplementary so the angles on
the same leg (T and P, R and A) are supplementary. We cannot determine anything else
about the sides or the diagonals.
You will complete the exploration given various pieces of information about the special
quadrilaterals. Using the theorems and postulates that we have studied so far, consider if the
sides are equal/parallel/perpendicular, if the angles are equal/complimentary/supplementary, and
if the diagonals are equal/parallel/perpendicular/bisectors of each other. As you go along, fill in
what you have proven in the table on the last page.
Special Quadrilateral Exploration
page 2
1 Pair of Parallel Sides, other sides are equal (Isosceles Trapezoid)
P
I
R
T
Start by drawing the diagonals to form
triangles. Can you prove anything about the
lengths of the diagonals? anything about the
angles of the trapezoid? do the diagonals
bisect each other? do they form any congruent
triangles?
Two Pairs of Consecutive Congruent Sides (Kite)
I
T
K
E
Start by drawing diagonal KT. What can you
prove about the sides and angles of the kite?
Add the other diagonal. What can you prove
about the diagonals (bisect each other, angle
bisectors, perpendicular, congruent triangles
formed, etc.)?
Special Quadrilateral Exploration
page 3
Two Pairs of Parallel Sides (Parallelogram)
P
L
A
R
Start by drawing one diagonal. What can you
prove about the sides and angles of the
parallelogram? Add the other diagonal. What
can you prove about the diagonals (bisect each
other, perpendicular, congruent triangles
formed, etc.)?
For all of the rest of the problems, start with the given information (from the definition) and see
what you can prove about the sides (equal? parallel?) and the angles (equal? complementary?
supplementary? bisected?). Do the same with the diagonals (equal? bisected? perpendicular?).
Equiangular (Rectangle)
T
C
R
E
Special Quadrilateral Exploration
page 4
Equilateral (Rhombus)
R
H
O
M
Regular (Square)
Q
S
R
E
Consider this: If knowing the defined properties gives various other attributes, does the converse
also hold true? Does knowing some attributes give you the properties of the definition? If so,
how many of them do you need to know?
Special Quadrilateral Exploration
Name
Trapezoid
Isosceles
Trapezoid
Kite
Parallelogram
Rectangle
Rhombus
Square
Defined
Properties
1 Pair of Parallel
Sides
1 Pair of Parallel
Sides, other sides
are equal
Two Pairs of
Consecutive
Congruent Sides
Two Pairs of
Parallel Sides
Equiangular
Equilateral
Regular
page 5
Side Attributes
Angle Attributes
Diagonal Attributes
 1 pair parallel
 2 pairs parallel
 supplementary
 right
 perpendicular
 equal
 1 pair equal
 2 pairs equal
 1 pair equal
 2 pairs equal
 one bisects other
 both bisect
 1 pair parallel
 2 pairs parallel
 supplementary
 right
 perpendicular
 equal
 1 pair equal
 2 pairs equal
 1 pair equal
 2 pairs equal
 one bisects other
 both bisect
 1 pair parallel
 2 pairs parallel
 supplementary
 right
 perpendicular
 equal
 1 pair equal
 2 pairs equal
 1 pair equal
 2 pairs equal
 one bisects other
 both bisect
 1 pair parallel
 2 pairs parallel
 supplementary
 right
 perpendicular
 equal
 1 pair equal
 2 pairs equal
 1 pair equal
 2 pairs equal
 one bisects other
 both bisect
 1 pair parallel
 2 pairs parallel
 supplementary
 right
 perpendicular
 equal
 1 pair equal
 2 pairs equal
 1 pair equal
 2 pairs equal
 one bisects other
 both bisect
 1 pair parallel
 2 pairs parallel
 supplementary
 right
 perpendicular
 equal
 1 pair equal
 2 pairs equal
 1 pair equal
 2 pairs equal
 one bisects other
 both bisect
 supplementary
 right
 perpendicular
 equal
 1 pair equal
 2 pairs equal
 one bisects other
 both bisect
 1 pair parallel
 2 pairs parallel
 1 pair equal
 2 pairs equal
Special Quadrilateral Exploration
page 6