ABCDEF is a regular hexagon (all sides and interior angles are equal). What is the measure, in degrees, of angle ADB, where D is the vertex opposite B and A, B are adjacent vertices of the hexagon?

Introduction / Context:
This question involves a regular hexagon, a polygon that has six equal sides and six equal interior angles. Regular polygons have many symmetrical properties, and understanding them allows us to calculate angles formed by drawing diagonals between vertices. Here, we must find the angle formed at vertex D by the segments DA and DB inside the regular hexagon ABCDEF.

Given Data / Assumptions:

• ABCDEF is a regular hexagon.
• All side lengths are equal and all interior angles are equal.
• Vertices are labelled consecutively around the hexagon as A, B, C, D, E, F.
• We are interested in angle ADB, formed at vertex D by segments DA and DB.
• The hexagon is assumed to be convex and regular in the standard Euclidean plane.

Concept / Approach:
For a regular hexagon, the interior angle at each vertex is: Interior angle = ((n − 2) * 180°) / n where n = 6, giving 120°. Another important fact is that a regular hexagon can be inscribed in a circle with equal central angles of 60° between adjacent vertices. We can place the hexagon on a circle and use coordinates or symmetry to compute angle ADB. In a regular hexagon, opposite vertices are separated by three edges, and geometry reveals that angle ADB equals 30°.

Step-by-Step Solution:
Step 1: Consider a regular hexagon ABCDEF inscribed in a circle of some radius R, with centre O. Step 2: The six vertices are evenly spaced on the circle. The central angle between consecutive vertices (for example ∠AOB) is 360° / 6 = 60°. Step 3: Arrange labels so that A, B, C, D, E, F go around the circle in order. Then B and D are separated by two intermediate vertices (C between them on one side, or E on the other), while D is opposite A and B is next to A. Step 4: Draw diagonals AD and BD. These form triangle ADB inside the hexagon. Because the points lie on the circle, we can relate the inscribed angles to the central angles. Step 5: The arc AB corresponds to 60° at the centre. The arc AD corresponds to 180°, since A and D are opposite vertices. By careful geometric or coordinate analysis, or by symmetry, it can be shown that angle ADB is half of the difference between two central angles, leading to a value of 30°. Step 6: Hence, ∠ADB = 30°.

Verification / Alternative check:
A quick verification can be done with a coordinate approach. Place the regular hexagon on the unit circle with A at (1, 0). Then B, C, D, E and F lie at angles 60°, 120°, 180°, 240° and 300° respectively. Computing the coordinates of A, D and B and then using vector methods or the dot product to determine angle ADB confirms that the angle is π/6 radians, which equals 30°. This more algebraic method reinforces the geometric reasoning.

Why Other Options Are Wrong:
Option 1: 15° is too small and does not correspond to any natural subdivision of the hexagon angles. Option 3: 45° is a common angle in square and right triangle problems but does not arise from the hexagon symmetries here. Option 4: 60° is the central angle between adjacent vertices, not the internal angle ADB that we are asked to find. Option 5: 75° has no standard connection to regular hexagon geometry in this configuration.

Common Pitfalls:
One common mistake is to confuse interior angles at the vertices of the hexagon (which are 120°) with various angles formed by diagonals. Another pitfall is to assume that angle ADB equals one of the central angles directly, such as 60° or 120°, without analyzing the geometry of the triangle formed by the diagonals. Students may also incorrectly assume that all diagonals create the same angles, which is not true; the angle depends on which vertices are joined. Careful visualization or a small sketch is very helpful in avoiding these errors.

Final Answer:
The measure of angle ADB is 30°.