InterMath
Title: Mystery Quadrilaterals: Classify a quadrilateral by making measurements in a dynamic
construction
Problem Statement
Some quadrilaterals have special names because they have some special properties. For example, a
rectangle is any quadrilateral with four right angles. Alternately, a rectangle is a parallelogram with 1 right
angle (Can you explain why?) A square is "more special" than a rectangle because it has four right angles
and four equal sides
(so a square is a
special rectangle.)
Each of the Geometer's Sketchpad files below contains a different quadrilateral. Your goal is to use your
knowledge about quadrilateral properties and the Measure menu in Geometer's Sketchpad to determine
the MOST specific name of each quadrilateral.
Be careful! All of the quadrilaterals will look like squares, but only one of them will actually be a square.
Justify each of your responses by including properties of the quadrilateral that make it unique.

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


Mystery
Mystery
Mystery
Mystery
Mystery
Mystery
Quadrilateral
Quadrilateral
Quadrilateral
Quadrilateral
Quadrilateral
Quadrilateral
1
2
3
4
5
6
Problem setup
I am trying to determine which of the mystery quadrilaterals is a square. I predict that mystery quadrilateral 1
will be a square. Since all of the quadrilaterals look like squares, using the distinctive properties of specific
quadrilaterals will be vital.
Plans to Solve/Investigate the Problem
Using GSP, I will measure and compare all four angles of each quadrilateral. I will also measure the length of
each side and compare. In addition, I will research the specific properties of quadrilaterals in order to
distinguish each one.
Investigation/Exploration of the Problem
I first measured the angles and sides of each quadrilateral and made comparisons. I then investigated the
properties of quadrilaterals and began to classify each. I also needed to investigate how to show that sides are
parallel mathematically.
Quadrilateral
Sides
Angles
Parallel
Parallelogram
Opposite sides
equal
Opposite angles
equal
Opposite sides
parallel
Rhombus
4 equal sides
Opposite angles
equal
Opposite sides
parallel
Rectangle
Opposite sides
equal
All angles are
right angles/90
degrees
Opposite sides
parallel
Square
4 equal sides
All angles are
right angles
Opposite sides
parallel
Trapezoid
Other
2 sides are
parallel
Kite
Adjacent sides
equal
Diagonals are
perpendicular/
One line of
symmetry
Slope - since slope is a measure of the angle of a line from horizontal and since parallel lines must have the
same angles, parallel lines have the same slope and line that have the same slope are parallel. (Web site)
2
MYSTERY
QUADRILATERAL B1
-10
-5
5
k = 4.24 cm
A
j
AD = 4.17 cm
BC
-2
j = 4.17 cm
BC = 4.17 cm
C
AD
Sl ope k = 0.57
-4
Sl ope BC = -1.69
Sl ope j = 0.57
k
D
Sl ope AD = -1.76
-6
mADC = 89.99
mDCB = 89.01
mCBA = 91.00
mBAD = 90.00
-8
This template is part of the InterMath project, created by the University
of Georgia.
More information about this project can be found at:
http://www.intermath-uga.gatech.edu/
Quadrilateral 1 has 3 equal sides AD, j, and BC. . The measure of the angles is all different. There is one right
angle which is angle BAD. Two opposite sides (j and k ) are parallel because they have the same slope. I
conclude that this is a trapezoid.
MYSTERY QUADRILATERAL 2
6
B4
A
j
q = 5.91 cm
r = 5.86 cm
2
q
s = 5.91 cm
j = 5.86 cm
mADC = 90.00
-5
s
mDCB = 90.00 5
mCBA = 90.00
mBAD = 90.00
-2 C
D
r
Sl ope j = 0.06
Sl ope q = -15.79
Sl ope r = 0.06
Sl ope s = -15.79
Thi s temp late i s part of the InterMath proj ect, created by-4the Uni versi ty of Geo rgi a.
More in formati on abo ut thi s proj ect can be found a t:
http://www.i nte rmath-uga.gatech.edu /
Quadrilateral 2 has 4 right angles. I know this because they all measure 90 degrees. The opposite sides AB
and DC are equal. Opposite sides are parallel because AB and DC have the same slope as well as AD and BC.
I conclude by definition that this is a rectangle. It cannot be a square because all four sides are not equal.
6
MYSTERY QUADRILATERAL 3
m = 4.71 cm
CB = 4.71 cm
o = 4.71 cm
4
B
p = 4.71 cm
mBAC = 90.00
CB
mACD = 90.00
mDBA = 90.00
D
m
2
mBAC = 90.00
Sl ope CB = -0.34
Sl ope m = 2.93
Sl ope p = -0.34
-10
-5
Sl ope o = 2.93
5
A
p
o
-2
C
-4
Thi s temp late i s part of the InterMath proj ect, created by the Uni versi ty of Geo rgi a.
More in formati on abo ut thi s proj ect can be found a t:
http://www.i nte rmath-uga.gatech.edu /
Quadrilateral 3 has 4 right angles. They all measure 90 degrees. The sides all measure the same length of 4.71
c.m. Both pairs of opposite sides( CB and p/ m and o) are parallel because they have the same slope. By
definition, this is a square.
MYSTERY
QUADRILATERAL 4
6
B
CD = 3.92 cm
4
k = 3.92 cm
BA
BA = 3.92 cm
BC
BC = 3.92 cm
A
2
mABC = 91.00
C
mBCD = 89.00
mCDA = 91.00
mDAB = 89.00
CD
-10
-5
Sl ope BC = -1.17
5
k
D
Sl ope BA = 0.82
Sl ope k = -1.17
Sl ope CD = 0.82
-2
-4
This template is part of the InterMath project, created by the University
of Georgia.
More information about this project can be found at:
Quadrilateral 4 has 4 equal sides. They all measure 3.92 c.m. There are no right angles. However, opposite
angles are equal. Angle BCD and angle DAB both measure 89 degrees. Angle ABC and Angle CDA both
measure 91 degrees. Opposite sides are parallel because the opposite sides have the same slope. This is a
rhombus.
2.5
MYSTERY
QUADRILATERAL 5
2
B
A
AB
AB = 2.30 in.
m = 2.25 in.
1.5
n = 2.30 in.
l = 2.25 in.
1
mABD = 90.94
mBDC = 89.06
mDCA = 90.94
0.5
mCAB = 89.06
Sl ope AB = 0.11
-1
-2
-3
1Sl ope m = -7.92 2
l
3
Sl ope n = 0.11
m
-0.5
Sl ope l = -7.92
D
C
n
-1
-1.5
Thi s temp late i s part of the InterMath proj ect, created by the Uni versi ty of Geo rgi a.
More in formati on abo ut thi s proj ect can be found a t:
-2
http://www.i nte rmath-uga.gatech.edu /
Quadrilateral 5 has no right angles because none of them measures 90 degrees. Sides m and l are opposite sides
and are equal because they both measure 2.25 inches. Sides AB and n are opposites and equal because they
measure 2.30 in. Opposite angles are equal. Opposite sides are parallel because they have the same slope. This
is a parallelogram.
MYSTERY
QUADRILATERAL 6
BC = 4.23 cm
D4
AD
A
AB = 4.23 cm
AD = 4.27 cm
2
CD
AB
CD = 4.27 cm
mABC = 92.15
mBCD = 88.42
mCDA = 91.00
mDAB = 88.42
5
-5
B
C
BC
Sl ope AD = 0.15
Sl ope AB = -5.47
-2
Sl ope BC = 0.14
Sl ope CD = -5.79
This template is part of the InterMath project, created by the University
-4
of Georgia.
More information about this project can be found at:
http://www.intermath-uga.gatech.edu/
-6
Quadrilateral 6 has two pairs of sides that are equal. The sides AD and CD are adjacent and equal and
measure 4.27 c.m. BC and AB are also equal sides that are adjacent. They both measure 4.23 c.m.. There
are no right angles; however, there is one pair of opposite angles( Angle BCD and Angle DAB) that both
measure 88.42 degrees. There are no parallel sides because the slopes are all different. This is a kite.
Extensions of the Problem
Explore some overlapping relationships by answering the following questions: Why is the relationship between
a square and a rectangle? Why is the relationship between a rectangle and a parallelogram? Why is the
relationship between a rhombus and a parallelogram?
Rectangle
Opposite sides
equal
All angles are
right angles/90
degrees
Opposite sides
parallel
Square
4 equal sides
All angles are
right angles
Opposite sides
parallel
*A square is a special type of rectangle.
Parallelogram
Opposite sides
equal
Opposite angles
equal
Opposite sides
parallel
Rectangle
Opposite sides
equal
All angles are
right angles/90
degrees
Opposite sides
parallel
*A rectangle is a special case of a parallelogram
Parallelogram
Opposite sides
equal
Opposite angles
equal
Opposite sides
parallel
Rhombus
4 equal sides
Opposite angles
equal
Opposite sides
parallel
*A rhombus is a special case of a parallelogram with four equal sides
Author & Contact
Chantel Lewis
chantel_lewis@putnam.k12.ga.us
GPS Connection: Students must understand quadrilaterals and the properties that make them unique.
GPS:
M5G1. Students will understand congruence of geometric figures and the correspondence of their vertices,
sides, and angles.
M5P2. Students will investigate, develop, and evaluate mathematical arguments.
M5P3. Students will use the language of mathematics to express ideas precisely.
Link(s) to resources, references, lesson plans, and/or other materials
http://www.onlinemathlearning.com/quadrilaterals.html
http://www.purplemath.com/modules/slope2.html
Important Note: You should compose your write-up targeting an audience in mind rather than just the
instructor for the course. You are creating a page to publish it on the web.