Final answer:
To determine the values of x that make the pentagons congruent, we need to find the side length of each pentagon and solve an equation involving the areas of the pentagons.
Step-by-step explanation:
To determine the values of x that make the pentagons congruent, we need to find the side length of each pentagon. The area of a regular pentagon can be found using the formula:
A = (5/4) * s^2 * cot(180/5),
where s is the side length. Thus, for the pentagon with an area of 20 mm^2, we have 20 = (5/4) * s^2 * cot(36).
Solving for s, we find s = sqrt((80 * cot(36))/5).
Next, we substitute this value of s into the expression for the area of the second pentagon, x^2 - x, and solve for x.
Setting the expressions for the areas of both pentagons equal to each other, we have
(5/4) * s^2 * cot(36) = x^2 - x.
Solving this equation yields the values of x that make the pentagons congruent.