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```									    InterMath

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.)

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.

Problem setup:
I am going to look at the properties of each of the mystery quadrilaterals and
determine what type of quadrilateral each one is. Because all the quadrilaterals
look like square, it is hard to make a predication. I predict that there is at least
one rectangle, and one parallelogram.

Plans to Solve/Investigate the Problem:
I am going to use the GSP to measure the segments, angles, and slope of each
mystery quadrilateral to determine the properties of the mystery figure. Then, I
will use the properties to classify the figure appropriately.
MYSTERY

j = 4.17 cm
B m ABC = 91.00         Three line segments are congruent.
m DAB = 90.00
A                                                            One pair of parallel s ides
j    Slope j = 0.02                                Only one right angle
BC = 4.17 cm

BC   Slope BC = -24.71

C    m BCD = 89.01
m CDA = 89.99   D          k     Slope k = 0.02

k = 4.25 cm

The first quadrilateral is not a square because the there are only three congruent
line segments. It’s not a rectangle because it does not have four right angles, or
opposite sides are not parallel. This figure only has one pair of parallel sides.
According to the slope segment AB and DC are parallel. Since these sides are
parallel, this figure is a trapezoid.

j = 5.86 cm                                m ABC = 90.00
B
m DAB = 90.00
A
j
Slope j = 0.06

Slope q = -15.79                                                   s = 5.91 cm
q
q = 5.91 cm
Slope s = -15.79
s

C
m CDA = 90.00 D                 r Slope r = 0.06                         m BCD = 90.00
r = 5.86 cm

Oppos ite sides are congruent.
All 4 angles are right angles.
There are two pair of parallel lines.

The second quadrilateral has four right angles, but the sides are not congruent.
So, this figure is not a square. According to the slopes, the opposite sides are
parallel. Although this figure has four right angles, the segments are not
congruent. This quadrilateral must be a rectangle.

m ABD = 90.00            Slope CB = -0.34
B
CB = 4.71 cm
CB
m BDC = 90.00
m = 4.71 cm      m                                 D

Slope m = 2.93

o = 4.71 cm
A
m CAB = 90.00
Slope o = 2.93
p                         o
p = 4.71 cm
C
Slope p = -0.34                  m DCA = 90.00

There are 4 congruent s ides and 4 right angles .
The s lopes indicates that opposite sides are
parallel.

The third quadrilateral has four right angles, and all four sides are congruent. The
slopes indicate that the opposites are parallel. This quadrilateral must be a square.
m ABC = 91.00

B
BA = 3.92 cm

BA                         BC = 3.92 cm
Slope BA = 0.82
BC
Slope BC = -1.17
m DAB = 89.00   A
C      m BCD = 89.00
DA = 3.92 cm
CD = 3.92 cm
DA
CD
Slope DA = -1.17
Slope CD = 0.82
D
m CDA = 91.00

All four s ides are congruent.
This quadrilateral has two pairs of congruent angles .
The s lope indicates that there are two pairs of parallel
lines.
This quadrilateral is a rhombus because the s ides
are congruent, but the angles are not congruent.

The fourth quadrilateral has four congruent sides, and two pair of congruent
angles. The slopes indicate that opposite sides are parallel. The opposite angles
are also congruent. This quadrilateral is a rhombus.
AB = 2.30 in.              B    m ABC = 90.94
m DAB = 89.06     A                       AB

Slope AB = 0.11
Oppos ite segments are congruent.
There are 2 congruent angles .
l = 2.25 in.       There are 2 pairs of parallel sides.

Slope l = -7.92
m = 2.25 in.

l

Slope m = -7.92         m
Slope n = 0.11
C   m BCD = 89.06
n
m CDA = 90.94      D                    n = 2.30 in.

diagonals form 2congruent triangles
diagonals als o bis ect each other
The measurements are s hown below.

Perimeter   CDA = 7.79 in.       Perimeter      ABC = 7.79 in.
Area   CDA = 2.59 in2            Area ABC = 2.59 in2
m DAC = 45.15                   m ACB = 45.15
m ACD = 43.91                  m BAC = 43.91

The fifth quadrilateral has 2 sets of congruence sides and opposites angles.
According to the slopes this figure has 2 sets of parallel sides. The diagonals
forms 2 congruent triangles and the diagonals bisect each other. This

mDAB = 88       A            Slope AD = 0                             Perimeter   ADC = 15 cm
Perimeter   ABC = 15 cm                                              Slope CD = -6

Area   ABC = 9 cm   2           AB                                      CD
Slope AB = -5
A                CD = 4.27 cm
AB = 4.23 cm

Slope BC = 0
C    mBCD = 88
BC
mBAC = 44     B
BC = 4.23 cm

mDAB+mBAC+mBCD+mCDA = 312
Diagonals are perpendicular
Diagonals create 2 congruence triangles
According to the s lope, there are a pair of parallel lines segment AB is parallal
to BC.

m DAA = 90
m DAC = 90
m AAB = 90
m BAC = 90

The sixth quadrilateral has adjacent sides that are equal in length. The diagonals
form 2 congruent triangles and the diagonals perpendicular. I measured the
diagonals to see if they formed angles that measured 90°. So I concluded that
I predicted that there was at least one rectangle (2) and one parallelogram (5).
Extensions of the Problem:
Discuss possible extensions for the problem and explore/investigate at least one
of the extensions you discussed.
Since the properties of quadrilaterals are difficult and confusing, I thought an
extension would be to have the students create a chart to keep in their notebook
to use as a guide until their proficient with the characteristics. I inserted a simple
chart with the basic quadrilaterals and some of their properties.

Quadrilateral     Opposite Opposite Opposite Diagonal Diagonals Diagonals   Diagonals   A           All    All
sides || sides    angles   forms    bisect each are       are         diagonal    angles sides
other                             bisects 2   are
angles      right

Parallelogram     x        x        x        x         x

Rectangle         x        x        x        x         x                    x                       x

Rhombus           x        x        x        x         x         x                      x                   x

Square            x        x        x        x         x         x          x           x           x       x

Trapezoid

Isosceles Trap.                                                             x

Kite                                                             x          x

M4M2    Students will understand the concept of angle and how to measure angles.
a.    Use tools, such as a protractor or angle ruler, and other methods, such as paper folding
or drawing a diagonal in a square, to measure angles.

M4G1    Students will define and identify the characteristics of geometric figure through
examination and construction.
b. Examine and classify quadrilaterals (including parallelograms, squares, rectangles,
trapezoids, and rhombi).
d. Compare and contrast the relationship among quadrilaterals.

M5G1    Students will understand congruence of geometric figures and the correspondence of their
vertices, sides, and angles.

M4-     Students will solve problems (using appropriate technology)
8P1        a.    Build new mathematical knowledge through problem solving
b. Make and investigate mathematics and in other contexts.
c.    Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.

M4-     Students will reason and evaluate mathematical arguments.
8P2         a.   Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c.   Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.

Author & Contact
Norma Smith
norma_smith19@yahoo.com

Link(s) to resources, references, lesson plans, and/or other materials