ABCDEF is a regular hexagon. Square ABXY is constructed on the outside of the hexagon. Let M be the midpoint of DX. What is the measure (in degrees) of ∠CMD?

Geometrically, I have no idea. Algebraically, if we let

A = (0,0)
B = (1,0)
then
D = (1,√3)
X = (1,-1)
so
M = (1,(√3-1)/2)
C = (3/2,√3/2)

slope of MC is 1
slope of MD is √3/2

∠CMD = 60°-45° = 15°

Maybe by working with all those 60° and 30° angles in the figure this will become clear geometrically.

Well, I sort of get a different answer: ∠CMD = 45

Here's how I got it:
Let CD = x. It is trivial by law of cosine that: BD = x√3
Whence DM = (√3 + 1)/2 * x. Let us denote the midpoint of BD as K. Then it is easy to see that:
MK = DM - DK = x/2. Now it is trivial to end of to get:
<CMD = tan^(-1) 1 = 45.

Well, since the live period is long over I suppose we can discuss. In the future, please refrain from posting live brilliant questions :)

To find the measure of ∠CMD, we need to observe the given diagram and make use of the properties of regular hexagons and squares.

Let's start by noting some important properties:

1. In a regular hexagon, all interior angles are equal. Each interior angle of a regular hexagon measures 120 degrees (360 degrees ÷ 6).

2. In a square, all four interior angles are right angles, each measuring 90 degrees.

Now, let's break down the problem step by step:

1. Since ABXY is a square constructed on the outside of the hexagon, we can conclude that ∠YAX and ∠YBX are right angles (90 degrees).

2. Considering vertical angles, we can see that ∠YAX is also equal to ∠ABX since they share the same vertex, A. Therefore, ∠ABX is also 90 degrees.

3. From the properties of a regular hexagon, we know that ∠BAC (interior angle of regular hexagon ABCDEF) measures 120 degrees.

4. Now, consider the triangle ABX. Since ∠ABX and ∠YBX are both right angles (90 degrees each), the remaining angle ∠ABY can be found by subtracting the sum of the other two angles from 180 degrees:
∠ABY = 180 degrees - ∠ABX - ∠YBX = 180 degrees - 90 degrees - 90 degrees = 0 degrees.

This means that ABY is a degenerate triangle, where the three vertices lie on a straight line, and ∠ABY is essentially nonexistent.

5. Since ABY is a degenerate triangle, we can conclude that AB and AY are collinear, meaning that line BY is an extension of line AB.

6. From step 5, we can conclude that line BX is parallel to line AY. Since ∠ABY is 0 degrees, the alternate interior angles ∠YBM and ∠ABX are also 0 degrees.

7. Since ∠YBX and ∠YBM are both 90 degrees, we can conclude that BYMD is a rectangle (opposite sides are parallel and congruent), and the opposite angles are equal. Therefore, ∠BMD is also 90 degrees.

8. Finally, let's consider triangle CMD. From step 7, we know that ∠BMD is 90 degrees. Also, since M is the midpoint of line DX, BM and MD are congruent.

In an isosceles triangle, the angle opposite the congruent sides is equal. Therefore, ∠CMD = ∠BMD = 90 degrees.

So, the measure of ∠CMD is 90 degrees.

Note: It's important to carefully observe the given diagram and apply the properties of regular hexagons and squares to solve geometry problems like this one.