Problem:
Two isosceles triangles have equal angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights.

Solution:
Let's assume that the two isosceles triangles have bases of lengths b1 and b2, and heights of lengths h1 and h2 respectively. The given information states that the areas of the triangles are in the ratio 16:25.

We know that the area of a triangle can be calculated using the formula:
Area = (1/2) * base * height

Step 1: Establishing the area ratio:
Since the triangles have equal angles, their bases are proportional. Therefore, we can write:
b2 = kb1
where k is a constant.

The area ratio is given as 16:25, so we can write:
(1/2) * b1 * h1 : (1/2) * b2 * h2 = 16 : 25

Substituting the values of b2 and kb1, we get:
(1/2) * b1 * h1 : (1/2) * kb1 * h2 = 16 : 25

Simplifying the equation, we get:
h1 : h2 = 16 : (25/k)

Step 2: Finding the value of k:
To find the value of k, we can use the fact that the sum of the angles in a triangle is 180 degrees. Since the triangles have equal angles, we can write:
2x + 2x + x = 180 degrees
(where x is the value of each angle)

Simplifying the equation, we get:
5x = 180 degrees
x = 36 degrees

Now, we can use the sine rule to find the value of k:
sin(x) = b1 / h1
sin(36 degrees) = b1 / h1

Similarly, for the second triangle:
sin(x) = b2 / h2
sin(36 degrees) = kb1 / h2

Dividing the two equations, we get:
h1 / h2 = b1 / (kb1)
h1 / h2 = 1 / k

Since h1 / h2 = 16 / (25/k), we can equate the two ratios:
1 / k = 16 / (25/k)

Cross multiplying, we get:
k^2 = 16 * 25
k^2 = 400
k = 20

Step 3: Finding the ratio of heights:
Substituting the value of k in the equation h1 : h2 = 16 : (25/k), we get:
h1 : h2 = 16 : (25/20)
h1 : h2 = 16 : 5

Therefore, the ratio of the corresponding heights of the two isosceles triangles is 16:5, which can be simplified to 4:5. Hence, the correct answer is option D) 4/5.