A Math Forum Project

Geometry Forum - Problem of the Week

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    Mark Berneburg

    ____________________

    This problem states that there is a quadrilateral with one pair of  
opposite sides congruent and the other pair not.  Also stated is that  
the opposite angles are supplementary.  I started to look at  
different shapes like a parallelogram.  A parallelogram would not  
work because the opposite angles are congruent not supplementary. 
A rectangle would not work even though the angles are 
supplementary the two opposite sides are congruent.  A rhombus and 
a square would not work because their opposite sides are congruent.  
The only shape that is a quadrilateral that would work is a trapezoid. 
A trapezoid has one pair of congruent sides and the opposite angles 
are congruent

    Ernie Leonetti

    ____________________

            The figure can not be a parallelogram because a parallelogram  
has two pairs of opposite sides congruent.  The figure would be a  
trapezoid because the pair of opposite sides that aren't congruent  
would be parallel.   Since a pair of opposite sides are congruent the  
corresponding angles containing these sides are also congruent.   
Also, in a trapezoid the opposite angles are supplementary.  To prove  
the non-congruent pair of sides are parallel I used the theorem that  
says "That if Alternate Interior Angles are congruent then the lines  
are parallel."  So the figure would end up to be an isosceles  
trapezoid.

    Gino Perrotte

    ____________________

    By definition, the answer can not be a parallelogram, square, rectangle,
rhombus.  This is because by definition, these figure's pairs of 
opposite sides are congruent.  Therefore, a trapezoid may work.  But 
it cannot be any ordinary trapezoid, it must be an isosceles trapezoid.
This is because it must have one pair of opposite sides congruent.  If 
the parallel sides are congruent, then the non parallel sides will also 
be congruent.  Therefore, it must be an isosceles trapezoid.  This 
figure is true because the opposite angles are supplementary.  This 
can be found because you use one of the congruent sides as a 
transversal.  The picture explains why the opposite angles are 
supplementary.

A is congruent to B because of PAI.  B is supplementary to C.  
Therefore A is supplementary to C.

               _________
              /       A \
            /             \
          /_C______________ \B

    Ellen Charny

    ____________________

    The quadrilateral would be an isosceles trapezoid.

If both pairs of opposite sides were congruent, then the shape would 
be a parallelogram.  However since one (and only one) pair of 
opposite sides is congruent, it must be a trapezoid.  Since the two 
opposite sides are congruent, the trapezoid is isosceles.  To prove that 
the consecutive angles are supplementary, draw a line perpendicular 
to the two parallel lines of the trapezoid.  The sum of these 
consecutive angles is 180 degrees (90 + 90).  So, since the # of 
degrees in a quadrilateral must equal to 360, and  360-180=180,
the sum of the two remaining consecutive angles must be 180 
degrees.

    Todd Gatnarek

    ____________________

    Assume the figure is a isosceles trapezoid.

        In this figure, it is given by its definition that the two  
nonparallel sides are congruent, and that its two parallel sides are  
not congruent.  (Name each vertex of the isosceles trapezoid 
consecutively from 1 to 4 with numbers 1 & 2 on one transversal 
and 3 & 4 on the other transversal).  Angle 1 and angle 2 are 
supplementary because they are  same side interior angles.  Angle 2 
and angle 3 are congruent by a previously proven theorem on 
isosceles trapezoids.  Angle 1 is supplementary to angle 3 by the 
transitive property.  
        Using this proof, an isosceles trapezoid will fit the requirements.

    Jason Haramic

    ____________________

    Given a quadrilateral with 1 pair of congruent sides, other pair of  
sides not congruent, and opposite angles supplementary, the figure is  
a trapezoid.  This is proven because two sides are congruent, and  
since the other sides aren't, the figure can't be a parallelogram.   
It is also an isosceles trapezoid because with opposite angles  
supplementary, the base angles must be congruent, and their  
supplements are congruent, therefore the non-congruent sides are  
parallel, making the figure an isosceles trapezoid by definition of  
isosceles (base angles congruent;only one pair of opposite sides  
congruent), and by definition of a trapezoid(only one pair of  
parallel sides).

    Chris Stromoski

    ____________________

    The quadrilateral would be an isosceles trapezoid.  
This is because the figure has a pair of opposite parallel sides 
which is proven by the two congruent angles.  These two sides have 
unequal sides, but as is given, the figure has a pair of congruent 
sides, which are not parallel because of only one pair of opposite 
angles being congruent.  Hence forth, a equilateral trapezoid is 
formed.

    Nick High

    ____________________

            If given a quadrilateral with one pair of congruent sides,  
the other pair or sides not congruent, and opposite angles  
supplementary, the figure described would be an isosceles trapezoid.   
This can be stated because if non parallel sides are congruent, then  
the opposite sides are congruent.  Because in a trapezoid the non  
parallel sides serve as transversals of the parallel sides, then the  
consecutive angles of the trapezoid are supplementary.  Since the  
base angles opposite the congruent sides are congruent, they will be  
supplementary to their opposite angles because the base angles have  
the same measure, and so do the supplements.  Finally, with the one  
set of sides not congruent, that would not make it a parallelogram  
according to the definition of a parallelogram.  Therefore it is an  
isosceles trapezoid.

    Thomas Niebel

    ____________________

            Given a quadrilateral with two opposite sides congruent, the  
other pair not congruent, and a pair of opposite angles  
supplementary, I believe the figure is an isosceles trapezoid.  This  
can be said because if the non-parallel sides are congruent, then the  
angles opposite them are congruent.  Then by relying on the fact that  
the other two remaining sides are parallel, then it can be said the  
congruent sides are transversals, meaning that the same-side interior  
angles are supplementary.  Meaning the angles contained in the  
figure, along the congruent sides are supplementary.  Then since the  
two base angles are congruent, they are both supplementary to the  
same angles, making the opposite angles supplementary.  Finally, in  
closing, since on set of sides is not congruent, that rules out the  
chance of a parallelogram by definition making the figure a  
trapezoid.

    Bipin Mujumdar

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            The resulting figure had to be an isosceles trapezoid.  First of all,
if the bases were the congruent side and they were parallel the 
resulting figure would be a parallelogram. This cannot be true 
because only one pair may be congruent. I the base angles were 
congruent and not parallel, the alternate angles would not be 
supplementary. The only time you may have only one pair of
congruent sides and have opposite angles supplementary in an 
isosceles triangle. To prove that opposite angles are supplementary is 
simple.  Since the nonparallel sides are congruent angle 2 and 4 are 
congruent. Since the other sides are parallel  1 and 2 are 
supplementary by Pai. Since 1 supplementary 2 and 2 congruent to 
4, 1 is supplementary 4 and the opposite angles are congruent and 
the figure is an isosceles triangle.

                     -----------------------------------------
                   /   1                                    3  \
                 /                                               \
               /                                                   \
             /    2                                              4   \
             ---------------------------------------------------------

    Paul Curcio

    ____________________

    By its definition, an isosceles trapezoid has a pair of congruent sides 
and supplementary opposite angles.  The opposite angles are 
supplementary since the non-congruent sides are parallel, and 
parallel lines form supplementary same-side interior angles.

    Keith Monteleone

    ____________________

            The only way in which you could have one pair of opposite  
sides congruent, the other pair not congruent, and a pair of opposite  
angles that are supplementary. is in a trapezoid.  Being that side  
FH=EG, then it must be an isosceles trapezoid, because if the two  
non-parallel sides of a trapezoid are congruent, then it is an  
isosceles trapezoid.

    Rachel Laughlin

    ____________________

    This figure must be an isosceles trapezoid. Other types of  
quadrilaterals do not fit the description.
1. It can't be a parallelogram (including rectangle, square, and  
rhombus) because both sets of opposite sides would be congruent.
2. It can't be a right trapezoid because neither set of opposite  
sides would be congruent.
3. It can't be a kite because the opposite angles wouldn't be  
supplementary.
4. It can't be a nonconvex figure because the opposite angles  
wouldn't be supplementary.

    Barb Lutka

    ____________________

    The figure is an isosceles trapezoid. The noncongruent sides are 
parallel, the two congruent sides are the transversals.  These 
segments form two pairs of consecutive, supplementary angles. 
Opposite angles are also supplementary.  

In my example shown below,  angles A and B are 110 degrees each 
and angles C and D are 70 degrees each.  

                         A              B
                          [------------]     





                   [-----------------------------]
                  C                                 D

    Ryan Ferchak and Rob Gallagher

    ____________________

    The figure is an isosceles trapezoid.  Two sides are congruent and the two bases are 
parallel.  When there are two parallel lines cut by a transversal you can have same side 
interior angles, which means they're supplementary.  There are 360 degrees in a 
trapezoid, and in an isosceles trapezoid base angles are congruent.  180 degrees  for each 
leg.  Two numbers we used were 120 and 60.

    Tim Thiel

    ____________________

    In the figure segment dc is da and cb are congruent opposite sides.  
Angle 3 and angle 2 are opposite angles and are supplementary.  
(given)  I will assume that the figure is an isosceles trapezoid.  By  
definition angle 1 is congruent to angle 2.  Since this is true then  
angle one is supplementary to angle two by the transitive property.   
since one and three are supplementary and same side interior angles  
this proves that segment  dc is parallel to ab.  Since it was given  
that the two others sides  are not congruent  but they are proven to  
be parallel then the sides ad and bc can not be parallel to  
each other. Since only a pair of sides are parallel the figure is not   
a parallelogram.  The figure can not be a kite because opposite  
angles in a kite are not congruent.  The most descriptive name for  
the figure is a isosceles trapezoid because 2 sides are congruent and  
the other two are shown to be parallel.  the other pair of sides are  
not congruent because if they were then the congruent ones would 
be parallel and this is already shown that they are not.

    Annie Chan & Daniel Chan

    ____________________

                A                 B
             /------------------\
            /            .       \
           /        .             \
          /    .                   \
         /                          \
      D----------------------------- C
               Join BC
               Opposite angles are supplementary 
               Therefore   ABCD is cyclic.
               /_ ABD  =  /_BDC   (equal angles from equal chords -   
               AD=BC)
               Therefore  AB//DC   (alternate interior angles equal)
               Therefore ABCD is an isosceles trapezoid.

    Carrie Tappe

    ____________________

    ABCD is a quadrilateral with one pair of opposite sides that are  
congruent.  It also has one set of opposite angles that are  
supplementary.  The congruent sides are AD and BC.  Angle D and 
angle B are supplementary.  Assume that the figure is an isosceles 
 trapezoid, therefore angle D and angle C are congruent.  Angle A  
would be congruent to angle C because of the transitive property.  AB  
and CD are parallel because the same side interior angles are  
supplementary.  AD and BC are not parallel.  If they were parallel,  
their alternate interior angles would be congruent:  angles 1 and 3  
would be congruent, but they are not.  Since AD and BC are not  
parallel, they are not congruent, therefore, the figure is not a  
parallelogram.  By definition, the quadrilateral would be an  
isosceles trapezoid.

    Kate Rusbasan

    ____________________

            My name is Kate Rusbasan and I'm in tenth grade at Shaler  
Area High school.  This is my solution of the problem of the week.  
From the given, the two non-congruent lines would have to be 
parallel for the other sides to be congruent.  That would make the  
quadrilateral an isosceles trapezoid.

    Mandy Balcer & Nicole Dunlap

    ____________________

    The figure that has one pair of opposite sides congruent, the other 
sides not  congruent, and a pair of opposite angles that are 
supplementary is an:

Isosceles Trapezoid

     1.  angle1 is supplementary to angle2  1. given
     2. angle2 is congruent to angle3   2. given
     3. angle1 is supplementary to angle3   3. transitive
     4.  AB is parallel to CB            4. Theorem 9-8
     5.  ABCD is a trapezoid             5. definition of trapezoid
     6.  ABCD is an isosceles trapezoid  6. definition of isosceles trapezoid

Theorem 9-8
   Given tow lines cut by a transversal, if one pair of interior  
angles on the same side as the transversal  are supplementary  the  
lines are parallel.

    Erin O'Connor

    ____________________

    My name is Erin O'Connor. I attend 8th grade at Saint Michael School 
in Olympia, Washington. My answer to the problem of the week is 
the following:  the figure is an isosceles trapezoid. By definition of an 
isosceles trapezoid, one pair of opposite sides is congruent and the 
other pair is not.  Opposite angles are supplementary, also by 
definition of a isosceles trapezoid.

    Kathy Medlin

    ____________________

        Quadrilateral ABCD is inscribed in circle P.  Call the top left angle 
A, the top right angle B, the bottom right angle C, and the bottom left 
angle D.  Segment AD is congruent to segment BC.  Segments AB and 
DC are not congruent.  Angle A and angle C are supplementary.  
    Since segment AD is congruent to BC, arc AD is congruent to arc BC. 
If you use the common arc theorem, arc DB will be congruent to arc 
AC.  Angle ADC intercepts arc AC and angle BCD intercepts arc BD; 
therefore, since arc AC is congruent to arc BD, angle BCD is congruent 
to angle ADC.  Since angle A and angle C are supplementary, and 
angle C and angle D are congruent, then angle A must be 
supplementary to angle D.  If you use the theorem, if two interior
angles on the same side of the transversal are supplementary, then 
the lines are parallel.  That means the segment AB is parallel to 
segment DC, and that shows that quadrilateral ABCD is a trapezoid. 
Since segment AD is congruent to segment BC, the trapezoid must be 
isosceles.
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30 June 1995