Understanding Constructible Angles
In classical geometry, an angle is said to be constructible if it can be created using only a straightedge (ruler) and a compass. The constructibility of angles is determined by the properties of the numbers involved.

Constructible Angles and Their Properties
To determine whether a given angle can be constructed, we use the following criteria:
- An angle θ is constructible if and only if the cosine of θ can be expressed in terms of integers and square roots.
- This is linked to the degree of the field extension over the rationals, specifically, the angle must be a submultiple of 180° that can be achieved through a series of angle bisections.

Analysis of Given Angles
Let's analyze the angles provided in the options:
- **20°**: Constructible (can be obtained through angle bisection).
- **30°**: Constructible (as it is one of the standard angles).
- **40°**: Constructible (can be obtained through angle bisection).
- **50°**: Not constructible.

Why is 50° Not Constructible?
The underlying reason that 50° is not constructible relates to the following:
- **Cosine of 50°**: The cosine of this angle cannot be expressed in terms of rational numbers and square roots, as it does not correspond to a constructible angle.
- **Field Extension**: The angle 50° leads to a polynomial equation that is not solvable by radicals, indicating it cannot be constructed with just a ruler and a compass.

Conclusion
Thus, among the angles provided, **50° is the only angle that cannot be constructed using a ruler and compass**, making option 'D' the correct answer.