Okay, let's try to figure out the measure of angle α in this cyclic hexagon. Hmm, so the hexagon is inscribed in a circle, which means all its vertices lie on the circumference of the circle. The diagram shows some angles labeled: 105° at point F, 110° at point B, and α at point D. The question is asking for the measure of α.

First, I remember that in cyclic polygons, the opposite angles have certain relationships. But wait, this is a hexagon, not a quadrilateral or triangle. For cyclic quadrilaterals, the sum of opposite angles is 180°, but for a hexagon, it's a bit more complicated. Maybe there's a general formula or theorem for cyclic polygons with more sides?

Let me think. For a cyclic polygon, the sum of the measures of all the interior angles is given by (n-2)*180°, where n is the number of sides. For a hexagon, that would be (6-2)*180 = 720°. So the total sum of all interior angles is 720°. But since the hexagon is cyclic, there might be more specific relationships between the angles.

Wait, but cyclic polygons have properties related to their central angles and inscribed angles. Each interior angle of a cyclic polygon can be related to the arcs it subtends. For example, in a cyclic quadrilateral, each angle is half the sum of the arcs intercepted by the angle and its opposite angle.

But for a hexagon, each interior angle is equal to half the sum of the arcs intercepted by the two sides meeting at that vertex. So, if we can figure out the arcs corresponding to each angle, we can set up equations to solve for α.

Let me try to label the arcs. Let's denote the arcs between consecutive points as follows: arc AB = a, arc BC = b, arc CD = c, arc DE = d, arc EF = e, and arc FA = f. Since the hexagon is cyclic, the sum of all these arcs should be 360°, so a + b + c + d + e + f = 360°.

Each interior angle of the hexagon is equal to half the sum of the arcs intercepted by the two sides meeting at that vertex. For example, angle at A is half the sum of arc FB and arc BC? Wait, no, maybe I need to clarify.

Wait, in a cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. For example, angle at vertex A is half the sum of the arcs opposite to the sides AB and AF. Wait, maybe I need to think more carefully.

Let me recall: in a cyclic polygon, the measure of an inscribed angle is half the measure of its intercepted arc. But in a polygon, each interior angle is formed by two adjacent sides. Each side corresponds to an arc. The angle at a vertex is formed by the two arcs that are not adjacent to it. Wait, that might not be right.

Wait, maybe I need to visualize. Suppose we have a cyclic hexagon with vertices A, B, C, D, E, F in order. Then, the angle at vertex A is formed by the sides AB and AF. The arcs intercepted by these sides are the arcs opposite to them. Wait, perhaps the angle at A is half the measure of the arc from B to F, not passing through A. Hmm, I need to verify.

Alternatively, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two sides forming the angle. Wait, that might be the case. For example, in a cyclic quadrilateral, the angle at A is half the sum of the arcs BC and AD. Wait, no, that's not right. In a cyclic quadrilateral, the angle at A is half the difference of the measures of the arcs intercepted by the opposite angles.

Wait, maybe I need to recall the exact formula. For a cyclic polygon, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two sides that form the angle. But in a polygon, each side is a chord of the circle, so each side corresponds to an arc. The angle at a vertex is formed by two sides, each of which corresponds to an arc. The angle itself is equal to half the sum of the measures of the arcs that are opposite to the angle.

Wait, perhaps I need to think in terms of central angles. The central angle corresponding to an arc is equal to the measure of the arc. Then, the inscribed angle subtended by an arc is half the measure of the arc.

But in a polygon, each interior angle is formed by two adjacent sides. Each side is a chord, and each chord subtends an arc. The angle at the vertex is an inscribed angle that subtends the arc opposite to that vertex. Wait, maybe not. Let me think.

Consider a triangle inscribed in a circle: the angle at a vertex is equal to half the measure of the arc opposite to that angle. Similarly, in a quadrilateral, the angle at a vertex is equal to half the measure of the arc opposite to that vertex. So, perhaps in a cyclic polygon, each interior angle is equal to half the measure of the arc opposite to that angle.

Wait, that might make sense. For a triangle, the angle is half the measure of the opposite arc. For a quadrilateral, the angle is half the measure of the opposite arc. So, by extension, in a cyclic hexagon, each interior angle is equal to half the measure of the arc opposite to that angle.

But in a hexagon, each angle has an opposite arc. For example, angle at A would be opposite to the arc that is not adjacent to A. Wait, but in a hexagon, each angle is adjacent to two arcs, and the opposite arc would be the one that's three arcs away? Wait, maybe not. Let's try to visualize.

Suppose the hexagon is labeled A, B, C, D, E, F in order around the circle. Then, the arc opposite to angle A would be the arc that is not adjacent to A, which would be the arc from D to A, but that's not correct because arcs are between two points. Wait, maybe the arc opposite to angle A is the arc that is not adjacent to A, which would be the arc from C to E. Wait, maybe.

Alternatively, perhaps the opposite arc for angle A is the arc that is opposite to the angle, meaning the arc that is not adjacent to A. But in a hexagon, each angle is between two adjacent sides, each of which is a chord. The angle at A is formed by chords AB and AF. Each chord AB and AF subtends two arcs: arc AB and arc AF. Wait, but the angle at A is an inscribed angle that subtends the arc opposite to angle A, which is the arc from B to F. Because the angle at A is formed by two chords, AB and AF, which intercept arcs BF and... Wait, maybe not.

Wait, let me think of a triangle. In a triangle inscribed in a circle, the angle at A is equal to half the measure of the arc opposite to A. So, in a triangle, angle A subtends arc BC. Similarly, in a quadrilateral, angle A subtends arc C. Wait, but in a quadrilateral, angle A and angle C are opposite angles, and each subtends the same arc? Wait, maybe not.

Wait, perhaps in a quadrilateral, the opposite angles each subtend the same arc. For example, in a cyclic quadrilateral ABCD, angle A and angle C both subtend arc BD. But since ABCD is cyclic, angle A + angle C = 180°, which is because they are supplementary. So, angle A is half the measure of arc BD, and angle C is also half the measure of arc BD. But wait, that would mean angle A and angle C are equal? That can't be right unless the quadrilateral is a rectangle.

Wait, maybe I need to correct this. In a cyclic quadrilateral, the sum of angle A and angle C is 180°, because they are opposite angles. Each angle is equal to half the measure of the arc that is opposite to it. So, angle A is half the measure of arc BCD, and angle C is half the measure of arc BAD. Wait, but arc BCD and arc BAD together make the whole circle, 360°. So, if angle A is half of arc BCD, and angle C is half of arc BAD, then angle A + angle C = (arc BCD + arc BAD)/2 = 360°/2 = 180°, which matches the cyclic quadrilateral property.

So, generalizing this to a cyclic hexagon, each interior angle is equal to half the measure of the arc opposite to it. Therefore, angle A is half the measure of the arc opposite to A, which is arc CDE. Similarly, angle B is half the measure of arc DEF, angle C is half the measure of arc EFA, and so on.

Wait, let me check with the hexagon. Let's label the hexagon as A, B, C, D, E, F in order around the circle. Then, the arc opposite to angle A would be the arc that is not adjacent to A. Since A is connected to B and F, the opposite arc would be the one that's not near A, which would be arc CD, but that's only two arcs. Wait, maybe the arc opposite to angle A is the arc that is three arcs away? Because in a hexagon, the opposite vertex would be three apart. So, opposite to A is D. Then, the arc opposite to angle A is the arc from C to E? Wait, no. Wait, maybe the arc opposite to angle A is the arc that is intercepted by the angle's sides.

Wait, perhaps I need to think in terms of the intercepted arcs by the sides. The angle at A is formed by sides AB and AF. Each side is a chord, and each chord intercepts an arc. The angle at A is equal to half the measure of the arc opposite to angle A. Wait, but which arc is opposite? The arc that is not adjacent to A.

Alternatively, maybe the angle at A is equal to half the measure of the arc that is intercepted by the angle's sides. Wait, that might not make sense.

Wait, let's take a specific example. Suppose we have a regular hexagon inscribed in a circle. Each interior angle in a regular hexagon is 120°, and each arc between vertices is 60°. Then, according to the formula, each angle should be half the measure of the opposite arc. In a regular hexagon, the opposite arc to angle A would be the arc that is three arcs away, which is 180°, so half of that is 90°, but that's not 120°. Wait, that's a problem. So my previous assumption must be wrong.

Alternatively, in a regular hexagon, each interior angle is 120°, which is equal to half the measure of the arc intercepted by the two sides. Wait, but in a regular hexagon, each side subtends a 60° arc. The angle at A is formed by sides AB and AF, which subtend arcs AB and AF. The angle at A is equal to half the measure of the arc opposite to angle A, which is the arc that is not adjacent to A. In a regular hexagon, the arc opposite to angle A would be the arc from C to D to E to F? Wait, that's too long. Wait, maybe the arc opposite is the one that's directly across, which would be the arc from B to C to D to E to F to A? No, that's the whole circle.

Wait, maybe in a regular hexagon, the angle at A is formed by sides AB and AF, each of which subtends 60° arcs. The angle at A is the angle between these two sides, which in a regular hexagon is 120°. But according to the formula, the angle should be half the measure of the opposite arc. If the opposite arc is the arc from B to F, which is 240°, then half of that is 120°, which matches the angle. So, yes, in this case, the angle at A is half the measure of the arc opposite to angle A, which is the arc from B to F.

Therefore, generalizing, in a cyclic hexagon, each interior angle is equal to half the measure of the arc opposite to that angle. So, angle α at vertex D is equal to half the measure of the arc opposite to D. The arc opposite to D would be the arc that is not adjacent to D, which in the hexagon would be the arc from B to C to E to F to A? Wait, no. Wait, the arc opposite to D would be the arc that is intercepted by the sides opposite to D. Wait, in a hexagon, each vertex has two adjacent vertices, so the arc opposite to D would be the arc that's not adjacent to D. Let's list the hexagon vertices in order: A, B, C, D, E, F.

So, vertex D is connected to C and E. The arc opposite to D would be the arc that is not adjacent to D, which would be the arc from A to B to C. Wait, no, that's adjacent. Wait, maybe the arc opposite to D is the arc that is three arcs away. Since a hexagon has six sides, the opposite arc would be the arc that is three steps away from D. So starting at D, moving three steps: D to E to F to A. So the opposite arc would be from A to F? Wait, no. Wait, if we start at D, moving three steps clockwise would bring us to A. So the arc from D to A is three arcs. But the angle at D is formed by sides DC and DF. Each side DC and DF subtends arcs. The angle at D is equal to half the measure of the arc opposite to D, which is the arc that is not adjacent to D. The adjacent arcs to D are DC and DF. So the opposite arc would be the one that's not adjacent, which is the arc from A to B to C. Wait, but that's three arcs. Wait, maybe the arc opposite to angle D is the arc from A to B to C to E? Wait, that's not right.

Alternatively, maybe the opposite arc is the one that's intercepted by the non-adjacent sides. The sides at D are DC and DF. The angle at D is formed by these two sides. The arcs opposite to these sides would be the arcs that are not adjacent to D. So, side DC subtends arc DA, and side DF subtends arc DB. Wait, no. Wait, side DC connects D to C, so it subtends arc DC. Similarly, side DF connects D to F, subtending arc DF. The angle at D is equal to half the measure of the arc that is opposite to D, which would be the arc that is not adjacent to D. Wait, but in a hexagon, each side subtends an arc equal to the number of sides between them. For example, side DC subtends arc DC, which is one side, so 60°, but in a regular hexagon. Wait, maybe this is getting too convoluted.

Alternatively, let's use the given information. The problem provides some angles: 105° at point F, 110° at point B, and α at point D. The hexagon is inscribed in a circle, so we can use the fact that each interior angle is half the measure of the opposite arc.

Let me denote the arcs between the vertices as follows:

arc AB = a

arc BC = b

arc CD = c

arc DE = d

arc EF = e

arc FA = f

Since the hexagon is cyclic, the sum of all arcs is 360°, so a + b + c + d + e + f = 360°.

Each interior angle is half the measure of the opposite arc. Let's write equations for each angle:

Angle at A: opposite arc is arc BC + arc CD + arc DE + arc EF + arc FA? Wait, no. Wait, according to the earlier logic, angle at A is half the measure of the arc opposite to A, which would be the arc that is not adjacent to A. Since A is connected to B and F, the opposite arc would be the arc from C to D to E. So, angle A = 1/2 (c + d + e). Similarly:

Angle at B: opposite arc is arc CD + arc DE + arc EF + arc FA. Wait, no. Wait, angle at B is formed by sides BA and BC. The opposite arc would be the arc not adjacent to B, which is from D to E to F. So angle B = 1/2 (d + e + f).

Angle at C: opposite arc is arc EF + arc FA + arc AB. Angle C = 1/2 (e + f + a).

Angle at D: opposite arc is arc FA + arc AB + arc BC. Angle D = 1/2 (f + a + b).

Angle at E: opposite arc is arc AB + arc BC + arc CD. Angle E = 1/2 (a + b + c).

Angle at F: opposite arc is arc BC + arc CD + arc DE. Angle F = 1/2 (b + c + d).

But we are given angle F = 105°, angle B = 110°, and angle D = α. So let's plug these into the equations.

First, angle F = 1/2 (b + c + d) = 105°, so b + c + d = 210°.

Angle B = 1/2 (d + e + f) = 110°, so d + e + f = 220°.

Angle D = 1/2 (f + a + b) = α, so f + a + b = 2α.

We also have the total sum of arcs: a + b + c + d + e + f = 360°.

Let me write down the equations:

1. b + c + d = 210° (from angle F)
2. d + e + f = 220° (from angle B)
3. f + a + b = 2α (from angle D)
4. a + b + c + d + e + f = 360° (total arcs)

We need to solve for α. Let's see how we can manipulate these equations.

First, let's add equations 1 and 2:

(b + c + d) + (d + e + f) = 210° + 220° = 430°

Which simplifies to:

b + c + d + d + e + f = 430°

So, b + c + 2d + e + f = 430°

But from equation 4, a + b + c + d + e + f = 360°, so we can express a as:

a = 360° - (b + c + d + e + f) = 360° - (360° - a) ? Wait, no. Wait, a = 360° - (b + c + d + e + f). Wait, no, equation 4 is a + b + c + d + e + f = 360°, so a = 360° - (b + c + d + e + f).

But from the sum of equations 1 and 2, we have b + c + 2d + e + f = 430°. Let's subtract equation 4 from this:

(b + c + 2d + e + f) - (a + b + c + d + e + f) = 430° - 360°

Simplifies to:

d - a = 70°

So, d = a + 70°

That's one equation.

Now, let's see if we can express other variables in terms of a.

From equation 4: a + b + c + d + e + f = 360°

We can substitute d = a + 70° into this:

a + b + c + (a + 70°) + e + f = 360°

Which simplifies to:

2a + b + c + e + f + 70° = 360°

So, 2a + b + c + e + f = 290°

But from equation 1: b + c + d = 210°, and d = a + 70°, so:

b + c + (a + 70°) = 210°

Which gives:

a + b + c = 140°

So, a + b + c = 140°

Substituting this into the previous equation:

2a + (140°) + e + f = 290°

So,

2a + e + f = 290° - 140° = 150°

Therefore,

2a + e + f = 150° (Equation 5)

From equation 2: d + e + f = 220°, and d = a + 70°, so:

(a + 70°) + e + f = 220°

Which simplifies to:

a + e + f = 220° - 70° = 150° (Equation 6)

Now, equation 5 and equation 6 are both 2a + e + f = 150° and a + e + f = 150°. Subtract equation 6 from equation 5:

(2a + e + f) - (a + e + f) = 150° - 150°

Which gives:

a = 0°

Wait, that can't be. A length of arc can't be zero. So there must be a mistake in the calculations.

Let me check the steps again.

Starting from equation 1: b + c + d = 210°

Equation 2: d + e + f = 220°

Equation 4: a + b + c + d + e + f = 360°

Adding equations 1 and 2:

b + c + d + d + e + f = 430° => b + c + 2d + e + f = 430°

Equation 4: a + b + c + d + e + f = 360°

Subtract equation 4 from the sum of 1 and 2:

(b + c + 2d + e + f) - (a + b + c + d + e + f) = 430° - 360°

Simplifies to:

d - a = 70° => d = a + 70°

That's correct.

Then, substituting d = a + 70° into equation 4:

a + b + c + (a + 70°) + e + f = 360° => 2a + b + c + e + f + 70° = 360° => 2a + b + c + e + f = 290°

From equation 1: b + c + d = 210°, substituting d = a + 70°:

b + c + a + 70° = 210° => a + b + c = 140°

Substituting into the previous equation:

2a + (140°) + e + f = 290° => 2a + e + f = 150° (Equation 5)

From equation 2: d + e + f = 220°, substituting d = a + 70°:

a + 70° + e + f = 220° => a + e + f = 150° (Equation 6)

So, equation 5: 2a + e + f = 150°

Equation 6: a + e + f = 150°

Subtracting equation 6 from equation 5:

(2a + e + f) - (a + e + f) = 150° - 150°

This gives a = 0°, which is impossible. Therefore, there must be an error in the initial assumption.

Wait, maybe the angle opposite to D is not f + a + b? Let's re-examine the angle at D.

Earlier, I thought angle D is half the measure of the arc opposite to D, which is the arc from A to B to C. But if the hexagon is labeled in order A, B, C, D, E, F around the circle, then the arc opposite to D would actually be the arc from A to B to C to D? Wait, no. Wait, the opposite arc should be the one that's not adjacent to D. Since D is connected to C and E, the opposite arc would be the one that's not near D. The arcs adjacent to D are DC and DE. The opposite arc would be the one that's opposite to D, which would be the arc from B to C to E? Wait, but that's three arcs. Wait, maybe the opposite arc is the one that's three arcs away? In a hexagon, each vertex has an opposite vertex three apart. So, opposite to D is A. Then, the opposite arc would be from A to B to C to D? Wait, no. Wait, if D is opposite to A, then the arc opposite to D would be the arc from A to B to C to D? Wait, that can't be. Wait, in a circle, the opposite arc to D would be the arc that doesn't include D. If D is opposite to A, then the arc opposite to D would be the arc from A to B to C. Wait, but that's three arcs, which is 180°, but in a regular hexagon, the opposite arc is 180°, so angle D would be half of that, which is 90°, but in a regular hexagon, angle D is 120°. So that can't be.

Wait, I think there's confusion here. Maybe the angle at D is half the measure of the arc that is intercepted by the sides forming the angle. So, angle D is formed by sides DC and DF. Each side DC and DF subtends arcs DC and DF. The angle at D is equal to half the measure of the arc that is opposite to D. Wait, but the opposite arc to D would be the arc that is intercepted by the sides opposite to the sides at D. Wait, the sides at D are DC and DF. The sides opposite to DC would be BA, and the sides opposite to DF would be CB? No, this is getting too convoluted.

Alternatively, maybe the measure of an angle formed by two chords in a circle is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical opposite. Wait, that might be the case. For example, in a cyclic quadrilateral, the angle is half the sum of the intercepted arcs. Wait, but in the case of a cyclic quadrilateral, the angle is half the sum of the opposite arcs. Wait, maybe in a cyclic hexagon, the angle is half the sum of the intercepted arcs.

Wait, let me check a reference in my mind. In a cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two sides forming the angle. So, in a triangle inscribed in a circle, the angle is half the measure of the arc opposite. In a quadrilateral, the angle is half the sum of the intercepted arcs. Wait, but in a quadrilateral, each angle is equal to half the sum of the arcs intercepted by the two opposite sides. Wait, no, in a cyclic quadrilateral, opposite angles are supplementary, and each angle is half the measure of the arc opposite to it.

Wait, perhaps in a cyclic polygon with an even number of sides, each angle is half the sum of the opposite arcs, but for odd number of sides, it's different. Wait, maybe this is not a standard formula. Let me think differently.

Alternatively, perhaps the measure of the angle at each vertex is equal to half the difference of the measures of the arcs intercepted by the two sides forming the angle. Wait, that might be the case.

For example, in a triangle inscribed in a circle, the angle is half the measure of the arc opposite. In a quadrilateral, each angle is half the measure of the arc opposite to it. Wait, but in a quadrilateral, the sum of the angles is 360°, and each angle is half the measure of the opposite arc. So, in a quadrilateral, angle A = 1/2 arc C, angle B = 1/2 arc D, etc., but the sum of all angles is 1/2(arc C + arc D + arc A + arc B) = 1/2(360°) = 180°. Wait, that can't be. Wait, no, in a quadrilateral, the sum of angles is 360°, but if each angle is half the measure of the opposite arc, then the sum of angles would be 1/2*(sum of opposite arcs). But in a quadrilateral, there are two pairs of opposite arcs, so sum of angles would be 1/2*(arc A + arc C + arc B + arc D) = 1/2*(360°) = 180°, which contradicts the fact that the sum of angles in a quadrilateral is 360°. Therefore, my initial assumption must be wrong.

Therefore, the formula for cyclic polygons must be different. Let me recall that in a cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two sides forming the angle. Wait, that might make sense.

For example, in a cyclic quadrilateral, each angle is equal to half the sum of the measures of the arcs intercepted by the two sides. Wait, but in a quadrilateral, each angle is formed by two sides, each of which intercepts an arc. The angle at the vertex is equal to half the sum of the intercepted arcs. Wait, but in a quadrilateral, the opposite angles would intercept arcs that are not adjacent. Maybe that's the key.

Wait, let's take a specific example. Suppose we have a cyclic quadrilateral ABCD. The angle at A is formed by sides AB and AD. These sides intercept arcs BC and CD. So, angle A is equal to half the sum of arcs BC and CD. Similarly, angle C is formed by sides CB and CD, intercepting arcs BA and AD. So angle C is half the sum of arcs BA and AD. Therefore, angle A + angle C = 1/2(arc BC + CD + arc BA + arc AD) = 1/2(arc BC + CD + arc BA + arc AD). But since arc BA + arc AD is the same as arc BAD, but arc BC + CD is arc BCD. Wait, but arc BA is the same as arc AB, and arc AD is the same as arc DA. Wait, maybe not. Wait, in a quadrilateral, the sum of angle A and angle C would be 1/2(arc BC + CD + arc BA + arc DA). But arc BC + arc BA = arc BAC, and arc CD + arc DA = arc CDA. But in a circle, arc BAC + arc CDA = 360° - arc AB - arc CD. Wait, this is getting too confusing.

Alternatively, perhaps in a cyclic polygon, each interior angle is equal to half the measure of the arc that is opposite to the angle. For example, in a triangle, angle A is half the measure of arc BC. In a quadrilateral, angle A is half the measure of arc C. In a pentagon, angle A is half the measure of arc CDE. In a hexagon, angle A is half the measure of arc DEF. Wait, that seems possible. Let's test this with a square. In a square, each angle is 90°, and the opposite arc would be 180°, so half of that is 90°, which matches. In a regular pentagon, each interior angle is 108°, and the opposite arc would be 216°, half of that is 108°, which matches. In a regular hexagon, each interior angle is 120°, and the opposite arc would be 240°, half of that is 120°, which matches. So this seems to hold.

Therefore, generalizing, in a cyclic n-gon, each interior angle is equal to half the measure of the arc opposite to that angle. The opposite arc is the one that is not adjacent to the angle. For a triangle, the opposite arc is the one opposite the angle. For a quadrilateral, the opposite arc is the one opposite the angle. For a pentagon, the opposite arc is three arcs away, and so on.

Therefore, applying this to a cyclic hexagon:

- Angle at A is half the measure of arc opposite to A, which is arc CDE.
- Angle at B is half the measure of arc opposite to B, which is arc DEF.
- Angle at C is half the measure of arc opposite to C, which is arc EFA.
- Angle at D is half the measure of arc opposite to D, which is arc FAB.
- Angle at E is half the measure of arc opposite to E, which is arc ABC.
- Angle at F is half the measure of arc opposite to F, which is arc BCD.

Wait, let me verify with a regular hexagon. In a regular hexagon, each interior angle is 120°, and the opposite arc would be 240°, so half of that is 120°, which matches. So this seems correct.

Therefore, for our problem:

Angle F = 1/2(arc CDE) = 105°, so arc CDE = 210°.

Angle B = 1/2(arc DEF) = 110°, so arc DEF = 220°.

Angle D = 1/2(arc FAB) = α, so arc FAB = 2α.

But we need to relate these arcs. Let's note the arcs:

arc AB = a

arc BC = b

arc CD = c

arc DE = d

arc EF = e

arc FA = f

Total arcs: a + b + c + d + e + f = 360°

Given that:

arc CDE = c + d + e = 210° (from angle F = 105°)

arc DEF = d + e + f = 220° (from angle B = 110°)

arc FAB = f + a + b = 2α (from angle D = α)

So, we have three equations:

1. c + d + e = 210°

2. d + e + f = 220°

3. f + a + b = 2α

Also, the total arcs:

4. a + b + c + d + e + f = 360°

Our goal is to find α. Let's try to express all variables in terms of a single variable.

From equation 1: c + d + e = 210°

From equation 2: d + e + f = 220°

Subtract equation 1 from equation 2:

(d + e + f) - (c + d + e) = 220° - 210°

Which simplifies to:

f - c = 10° => f = c + 10° (Equation 5)

From equation 1: c + d + e = 210°, so d + e = 210° - c (Equation 6)

From equation 2: d + e + f = 220°, substituting from equation 6 and equation 5:

(210° - c) + (c + 10°) = 220°

Simplify:

210° - c + c + 10° = 220°

220° = 220°, which is an identity. So, this doesn't give us new information.

Now, from equation 4: a + b + c + d + e + f = 360°

We can substitute c + d + e = 210° and f = c + 10°:

a + b + 210° + c + 10° = 360°

So,

a + b + c = 360° - 210° -10° = 140° (Equation 7)

From equation 3: f + a + b = 2α

But f = c + 10°, so substituting:

(c + 10°) + a + b = 2α

But from equation 7: a + b + c = 140°, so:

140° + 10° = 2α => 150° = 2α => α = 75°

Wait, that seems straightforward. Let me check again.

From equations 1 and 2, we found that f = c + 10°, and c + d + e = 210°, d + e + f = 220°, which gives f = c + 10°. Then, total arcs equation: a + b + c + d + e + f = 360°, substituting c + d + e = 210°, and f = c + 10°, gives a + b + 210° + c + 10° = 360°, so a + b + c = 140°. Then, equation 3: f + a + b = 2α => (c + 10°) + a + b = 2α. But since a + b + c = 140°, then (a + b + c) + 10° = 140° + 10° = 150° = 2α. Therefore, α = 75°.

Therefore, the measure of angle α is 75 degrees. 

But let me verify this with another approach to ensure there's no mistake.

Suppose α = 75°, then arc FAB = 2α = 150°. From equation 4: a + b + c + d + e + f = 360°. We have arc CDE = c + d + e = 210°, arc DEF = d + e + f = 220°, and arc FAB = f + a + b = 150°.

Let me solve for each arc:

From arc DEF = d + e + f = 220°

From arc CDE = c + d + e = 210°

Subtract: (d + e + f) - (c + d + e) = 220° - 210° => f - c = 10° => f = c + 10° (same as before)

From arc FAB = f + a + b = 150°, and f = c + 10°, so (c + 10°) + a + b = 150° => a + b + c = 140° (same as equation 7)

From total arcs: a + b + c + d + e + f = 360°. Substitute c + d + e = 210°, and f = c + 10°, so:

a + b + 210° + c + 10° = 360° => a + b + c = 140°, which matches equation 7.

Therefore, the solution is consistent. So, α = 75°.

Therefore, the measure of angle α is 75 degrees.
</think>