SOLUTION: The lengths of the sides of convex quadrilateral ABCD are 5, 6, 7, and x. if sinA = sinB = sinC = sinD, what are all possible values of x?

All internal angles of a convex polygon are less than 180° 

Angles whose measures are less than 180° have the same sine if and only if 
they are either equal in measure or are supplementary. (Remember the sines
are positive in quadrants I and II, and the angles in the 2nd quadrant are
obtuse and their reference angles are in the 1st quadrant and have the same
sines as their supplements). 

All 4 angles cannot be equal in measure for then they would be 90° each, and
the figure would be a rectangle.  But we can rule out a rectangle because a
rectangle cannot have sides with 3 different measures 5, 6, and 7.  We can
rule out any 90° angles.

So at least one pair of internal angles are supplementary.  Since rhe sum of the
measures of all 4 angles of a quadrilateral is 360°, then if two of the angles
have measures totaling 180°, then the other two angles must have measures
totaling 180°, and so they are supplementary.

There cannot be three angles with equal measures and the 4th angle
supplementary to them all, because

Let the 3 angles with the same measure have measure y and the fourth
be suplementary to them. Then the fourth angle would have measure 180-y.
Then using the fact that the four angles must have sum 360°, we have: 

      y + y + y + (180 - y) = 360 
               3y + 180 - y = 360
                   2y + 180 = 360
                         2y = 180
                          y = 90°

But we have ruled out right angles.  Therefore one pair of angles are
acute and have equal measures, and the other pair are obtuse and have
equal measures, and each of the acute angles is supplementary to each
of the obtuse angles.

Only parallelograms and isosceles trapezoids have that property.
Parallelograms are ruled out because a parallelogram cannot have sides
with 3 different measures 5, 6, and 7. 

So you can have only the three isosceles trapezoids drawn below. Neither of
the parallel bases can have measure x because the two legs have
equal measures and none of the three measures 5, 6, or 7 are equal. 
So x must be the measure of one of the legs.



So x is the measure of one of the congruent legs, and can only be

5, 6, or 7.

Edwin