15. How many of the following polygons could exist?
A triangle with all three sides the same length, but three different interior angles. A quadrilateral with all four sides the same length, but four different interior angles. A pentagon with all five sides the same length, but five different interior angles.
A only the pentagon
C the quadrilateral and the pentagon
E none of them
B only the quadrilateral
D all three
15. A A triangle with all sides the same length is an equilateral triangle, which has three equal interior angles. So the triangle described does not exist.
A quadrilateral with all sides the same length is a rhombus, which has two pairs of equal interior angles, or a square, which has four equal interior angles. So the quadrilateral described does not exist.
However, it is possible for a pentagon to have all its sides the same length and yet have five different interior angles.

To demonstrate this, the diagram shows four circles of equal radius. The centre of each circle lies on the circumference of one or more of the other circles. Four of the vertices of the pentagon shown are the centres of the circles and the fifth vertex is the point of intersection of the two circles on the left of the diagram. So the five sides of the pentagon are equal in length. By varying the positions of the circles relative to each other, the interior angles of the pentagon may be changed and the diagram shows one configuration in which these interior angles are different. It is left to the reader to draw such a configuration and to check that the interior angles of the pentagon are indeed all different.

An alternative solution.
The left-hand diagram shows isosceles triangle $A C E$ in which $A E$ has length 1 , while $A C$ and $E C$ both have length 2 . The midpoints of $A C$ and $E C$ are $B$ and $D$ respectively. In the right-hand diagram, $B, C$ and $D$ have been moved very slightly to positions denoted by $B^{\prime}, C^{\prime}$ and $D^{\prime}$ such that $A B^{\prime}, B^{\prime} C^{\prime}, C^{\prime} D^{\prime}$ and $D^{\prime} E$ are all of length 1 . Hence $A B^{\prime} C^{\prime} D^{\prime} E$ is a pentagon in which all of the sides have the same length. However, the interior angles at $A$ and $E$ are now different from each other as the former has increased and the latter decreased. Also, those at $B^{\prime}$ and $D^{\prime}$ are different from each other as the former is slightly smaller than $180^{\circ}$ and the latter slightly greater. Any change in the interior angle at $C$ is very slight, so, as can be seen, the interior angles of the pentagon are all different.