ABCD is a square and CDE is an equilateral triangle constructed outside the square on side CD. What is the measure, in degrees, of angle BEC?

Introduction / Context:
This problem combines properties of a square and an equilateral triangle. The equilateral triangle is constructed on one side of the square, and we are asked to find a specific angle formed between a vertex of the square and a vertex of the equilateral triangle. Such questions are common in geometry sections because they test the ability to use basic angle facts, symmetry and sometimes coordinate geometry.

Given Data / Assumptions:

• ABCD is a square.
• Side CD of the square is used to construct an equilateral triangle CDE outside the square.
• CDE is equilateral, so each angle in triangle CDE is 60°.
• We want the measure of angle BEC, where B is a vertex of the square, E is the apex of the equilateral triangle and C is the common vertex between the square and the triangle.
• All constructions lie in a plane and lengths are consistent.

Concept / Approach:
In a square, each interior angle is 90°. In an equilateral triangle, each interior angle is 60°. The figure can be analyzed using vector or coordinate geometry for precision, or by cleverly decomposing angles in a well drawn diagram. A helpful method is to place the square on the coordinate plane, compute coordinates of all points, and then use vector dot product to find angle BEC. This more algebraic method confirms the geometric intuition that the angle is 15°.

Step-by-Step Solution:
Step 1: Let the side length of the square be 1 unit for convenience (any positive number works because the answer is angle based). Step 2: Place the square so that A = (0, 0), B = (1, 0), C = (1, 1) and D = (0, 1). Step 3: Construct equilateral triangle CDE outside the square on side CD. Side CD goes from (1, 1) to (0, 1). The midpoint of CD is at (0.5, 1). The height of an equilateral triangle with side 1 is √3 / 2. Step 4: Since the triangle is constructed outside the square, point E lies above side CD at coordinates (0.5, 1 + √3 / 2). Step 5: Now consider angle BEC. It is the angle at E between lines EB and EC. Step 6: Compute or reason geometrically using symmetry and standard angle results; a detailed vector computation shows that angle BEC equals 15°.

Verification / Alternative check:
Using vector methods, we can form vectors EB and EC from E and compute the angle between them with the dot product formula: cos θ = (EB · EC) / (|EB| * |EC|) Substituting the exact coordinates leads to cos θ = cos(15°), which confirms that θ = 15°. This matches the geometric expectation that combining a 90° angle from the square and 60° from the equilateral triangle can create smaller special angles such as 30° and 15° in the composite figure.

Why Other Options Are Wrong:
Option 2: 30° is a common angle in problems involving equilateral triangles and is half of 60°, but the detailed geometry of this configuration yields 15°, not 30°. Option 3: 25° is not associated with any standard geometric combination of 60° and 90° in this type of construction. Option 4: 10° is too small and does not arise naturally from the given right and equilateral triangle angle structure. Option 5: 20° is also not consistent with the exact calculations based on coordinates or detailed angle chasing.

Common Pitfalls:
Students often guess that the resulting angle must be 30° or 45° because these angles are common in right triangle and equilateral triangle problems. Without a careful diagram or analytic method, it is easy to overlook the creation of a 15° angle. Another pitfall is not carefully distinguishing between internal angles of the square and the triangle and the angle formed at E by connecting B and C. Using coordinate geometry or systematic angle chasing helps avoid these errors.

Final Answer:
The measure of angle BEC is 15°.