Line segment AB intersects the straight line CD at point B. For which value of angle ABD (in degrees) will the length of AB be the smallest, that is, the distance from A to line CD is minimised?

Introduction / Context:
This question is about the shortest distance from a point to a straight line in geometry. When a line segment from a point A meets a line CD at B, the distance AB is smallest when AB is perpendicular to CD. This is a classic property often used in geometric proofs and also appears in optimisation problems.

Given Data / Assumptions:
- CD is a straight line in the plane.
- Point A lies off the line CD somewhere in the plane.
- Line segment AB intersects line CD at point B.
- Angle ABD is the angle at B between segment BA and the part of line CD that extends toward D.
- We want the length AB to be as small as possible, that is, the shortest distance from A to line CD.

Concept / Approach:
In Euclidean geometry, the shortest distance from a point to a straight line is along the perpendicular from the point to the line. This means that the segment from the point to the line must form a 90 degree angle with the line. If the angle is anything other than 90 degrees, the segment from the point will be longer because it will have both a perpendicular component and a component along the line.

Step-by-Step Solution:
Step 1: Visualise or sketch a line CD and a point A not on that line. Step 2: Draw several segments from A to the line CD, meeting the line at different points such as B. Step 3: Observe that the length of AB is smallest when the segment AB is perpendicular to CD. Step 4: A perpendicular intersection corresponds to an angle of 90 degrees between AB and CD at B. Step 5: Therefore, AB is minimised when angle ABD is exactly 90 degrees.

Verification / Alternative check:
If you imagine rotating the point of intersection B along the line CD, any tilt away from the perpendicular makes AB the hypotenuse of a right triangle whose one leg is the shortest distance to the line. Since the hypotenuse of a right triangle is always longer than either leg, any non perpendicular segment will be longer than the perpendicular one. This reasoning confirms that the smallest possible length occurs at a 90 degree angle.

Why Other Options Are Wrong:
Angles 30, 45, 60 and 75 degrees correspond to oblique intersections between AB and CD. In each case, AB is the hypotenuse of a right triangle formed by the perpendicular distance and a segment along the line CD. The hypotenuse is always longer than the perpendicular distance, so these angles cannot give the minimal length.

Common Pitfalls:
A common misunderstanding is to think that any acute angle may minimise the distance, but only the right angle gives the true shortest distance. Some learners also confuse this with projection along a line. Always remember that the perpendicular from a point to a line represents the minimum distance between them.

Final Answer:
AB is smallest when angle ABD is 90 degrees.