Okay, so I need to find the measure of angle A in this regular pentagon FGHIJ where extending the sides forms a star. Let me start by recalling some properties of a regular pentagon and the star formed by extending its sides. 

First, a regular pentagon has all sides equal and all interior angles equal. The measure of each interior angle in a regular pentagon is 108 degrees. That's calculated using the formula for interior angles of a regular polygon: (n-2)*180/n, where n=5. So, (5-2)*180/5 = 3*36 = 108 degrees. 

Now, when you extend the sides of a regular pentagon, they form a five-pointed star, also known as a pentagram. The points of the star are where the extended sides meet. In the Asymptote code provided, the figure is drawn with coordinates, but since it's a regular pentagon, all the sides and angles should be consistent.

Looking at the Asymptote code, the figure is drawn with coordinates for points F, G, H, I, J, and some other points. The label A is at (-42.4,30.8), which seems to be one of the outer vertices of the star. So angle A is at that vertex. 

In a regular pentagram, each point of the star has an angle. I remember that the points of a regular pentagram are 36 degrees each. But wait, let me verify that. 

Alternatively, maybe I can calculate it using the properties of the pentagon and the star. Let me think. 

When you extend the sides of the regular pentagon, the angles at the points of the star are formed by the intersection of two extended sides. In a regular pentagram, the angle at each tip is 36 degrees. But let me confirm this. 

Another approach is to consider the triangle formed at each tip. Since the pentagram is formed by connecting every other vertex of a regular pentagon, each tip of the star is essentially the vertex of an isosceles triangle whose base is a side of the pentagon and whose other sides are the extensions of the pentagon's sides. 

Wait, maybe I need to use some trigonometry here. Let me recall that in a regular pentagon, the central angles are 72 degrees because the full circle is 360 degrees divided by 5, which is 72. The central angles correspond to the angles at the center of the pentagon between two adjacent vertices. 

But how does that relate to the angles in the star? Let me consider the relationship between the internal angles of the pentagon and the angles of the star. 

When you extend the sides of the pentagon, each side is extended until it meets another extended side. The angle formed at the tip of the star is the angle between two such extended sides. 

Perhaps I can use the fact that the extended sides form supplementary angles with the original sides of the pentagon. Since each interior angle of the pentagon is 108 degrees, the exterior angle is 72 degrees (since exterior angles of a polygon sum up to 360, so 360/5=72). 

But when you extend a side of the pentagon, you're essentially creating a line that is an extension beyond the original vertex. The angle between the side and its extension is 180 degrees minus the interior angle. Wait, no. The interior angle is 108 degrees, so the exterior angle is 72 degrees. But when you extend a side, the angle outside the pentagon is 180 - 108 = 72 degrees. 

But how does that help with angle A? 

Wait, maybe I need to consider the triangle formed at the tip of the star. Since the pentagon is regular, all the triangles formed at the tips of the star are congruent. Each tip is formed by two extended sides of the pentagon. 

If I can figure out the angles of this triangle, maybe I can find angle A. Let's consider one such triangle. The two sides of the triangle are extensions of the pentagon's sides, and the base is the side of the pentagon. 

Wait, but actually, the base of each triangle would not be the side of the pentagon but a longer segment. Hmm, maybe this approach is complicating things. 

Alternatively, I remember that in a regular pentagram, the angles at the points are 36 degrees. Let me check this again. 

The pentagram can be thought of as a collection of triangles. Each of these triangles is an isosceles triangle with a vertex angle of 36 degrees. Wait, is that right? 

Alternatively, let's use the fact that the sum of the angles in a pentagram is related to the sum of the angles in a pentagon. But maybe that's not directly helpful. 

Another method is to use the fact that the pentagram is a regular star polygon, denoted by {5/2}, meaning it connects every second vertex of a regular pentagon. The internal angles of a regular star polygon can be calculated using the formula: 

Internal angle = 180 - (180*(n-2*k))/n 

But I might be misremembering this. Wait, another formula for the point angle of a regular star polygon {n/m} is:

Point angle = (180*(n-2*m))/n 

But for a pentagram, which is {5/2}, so substituting n=5 and m=2:

Point angle = (180*(5 - 2*2))/5 = (180*(5 -4))/5 = (180*1)/5 = 36 degrees. 

Yes, that's correct. So each tip of the pentagram has a point angle of 36 degrees. Therefore, angle A, which is one of these tips, should be 36 degrees. 

But let me verify this with another method to be sure. 

Consider the regular pentagon with all sides equal and internal angles 108 degrees. When you extend the sides, the angle formed outside the pentagon at each vertex is equal to the external angle of the pentagon, which is 72 degrees. But angle A is not exactly the external angle. 

Wait, perhaps I can consider the triangle formed by two adjacent extended sides. Let me imagine extending two adjacent sides of the pentagon. These two extended sides meet at a point outside the pentagon, forming angle A. 

The two sides being extended are sides of the pentagon. Let's denote the original pentagon as ABCDE, with each internal angle 108 degrees. If we extend side AB beyond B and side BC beyond C, they meet at a point, say F, forming angle F. 

Wait, but in the given figure, angle A is at a vertex that's not part of the original pentagon. Wait, in the Asymptote code, the label A is at (-42.4,30.8), which seems to be a point outside the original pentagon. 

Wait, perhaps the figure is a pentagram where each tip is a point of the star, and angle A is one of those tips. 

Alternatively, maybe angle A is an intersection point of two extended sides, but not necessarily a tip of the star. Wait, the Asymptote code draws lines between various points, so maybe angle A is at one of the outer intersections. 

Wait, the Asymptote code draws the main pentagon and then another smaller pentagon inside, connected to form the star. The label A is at (-42.4,30.8), which is one of the outer vertices. 

Looking at the coordinates, the main pentagon is drawn with coordinates like (-42.4,30.8), (-10,30.8), (0,63.2), (10,30.8), (42.4,30.8), (16.2,11.8), (24.9,-18.1), (0,0), (-24.9,-18.1), (-16.2,11.8), and back to (-42.4,30.8). 

So the figure is a star polygon with 10 points? Wait, no. Wait, the Asymptote code seems to draw a decagon? Wait, the coordinates are 10 points? Let me count: 

1. (-42.4,30.8)
2. (-10,30.8)
3. (0,63.2)
4. (10,30.8)
5. (42.4,30.8)
6. (16.2,11.8)
7. (24.9,-18.1)
8. (0,0)
9. (-24.9,-18.1)
10. (-16.2,11.8)
11. Back to (-42.4,30.8)

So that's 10 points, forming a 10-sided polygon? Wait, but the figure is supposed to be a pentagon with extensions forming a star. Maybe the Asymptote code is drawing a more complex figure. 

Alternatively, perhaps the figure is a five-pointed star (pentagram) with additional lines connecting other points. 

Alternatively, maybe I should focus on the properties of a regular pentagram. 

In a regular pentagram, each internal angle at the tips is 36 degrees. Since angle A is labeled at one of these tips, it should be 36 degrees. 

But let me confirm this with another approach. 

Consider the regular pentagon. Each internal angle is 108 degrees. When we extend the sides, they intersect at points outside the pentagon, forming the pentagram. Each of these intersection points forms an angle. 

To find the measure of angle A, we can consider the triangle formed by the intersection of two extended sides. Let's say we have a regular pentagon ABCDE, and we extend sides AB and AE. These extended sides meet at a point F, forming angle A at F. 

In triangle AFB (assuming F is the intersection point), we can find angle A. 

But to find the measure of angle A, we need to find the angles at point F. 

Alternatively, since the pentagram is a regular star polygon, we can use the formula for its internal angles. As mentioned earlier, the formula for the internal angle of a regular star polygon {n/m} is 180 - (180*(n-2*m))/n. For a pentagram, n=5 and m=2. 

So plugging in the values: 

Internal angle = 180 - (180*(5 - 4))/5 = 180 - (180*1)/5 = 180 - 36 = 144 degrees. 

Wait, that's the internal angle of the star polygon. But angle A in the figure is the tip of the star, which is the point angle. 

Wait, maybe I confused the formulas. The formula I used gives the internal angle of the polygon, but in the case of a star polygon, the point angle is different. 

Alternatively, the point angle (the angle at each tip) of a regular star polygon {n/m} is given by:

Point angle = (180*(2*m))/n 

Wait, for a pentagram {5/2}, point angle would be (180*4)/5 = 144 degrees. But that contradicts my previous knowledge. 

Wait, I think I need to verify this. 

Another approach: in a regular pentagram, each tip is formed by two edges at an angle. The angle between these two edges is 36 degrees. 

Alternatively, considering the regular pentagon and extending its sides, the angle between two extended sides at the tip is 36 degrees. 

Alternatively, let's consider the coordinates given in the Asymptote code. 

Looking at the coordinates:

- Point F is at (-10,30.8)
- Point G is at (10,30.8)
- Point H is at (16.2,11.8)
- Point I is at (0,0)
- Point J is at (-16.2,11.8)

So the main pentagon is F, G, H, I, J, and back to F. Then, there are other points (-24.9,-18.1), (0,0), etc. 

Wait, the main figure is drawn with these coordinates, and the label A is at (-42.4,30.8), which is another point. 

So perhaps the figure is a combination of a pentagon and a star, with multiple intersections. 

Alternatively, maybe the figure is constructed by extending the sides of the pentagon until they meet, forming a star with five points. 

In that case, each tip of the star is formed by the intersection of two extended sides. 

To find angle A, which is at one of these tips, we need to figure out the angle between two extended sides. 

Since each side of the pentagon is extended, forming an angle with the adjacent side. 

In a regular pentagon, each internal angle is 108 degrees. When you extend a side, the external angle is 72 degrees. But how does that relate to the angle at the tip of the star? 

Alternatively, consider that when two sides are extended, the angle outside the pentagon is formed. Since each side is extended, the angle between the extended sides is supplementary to the internal angle. 

But the internal angle is 108 degrees, so the external angle would be 72 degrees. But angle A is not the external angle; it's the angle at the tip of the star. 

Wait, maybe the angle at the tip of the star is the angle between two extended sides. Let's consider that. 

Each extended side forms an angle with the adjacent extended side. Since the sides of the pentagon are being extended, the angle between two adjacent extended sides would be 180 - 108 = 72 degrees. But that's the angle between the extended side and the original side. 

Wait, perhaps the angle at the tip is formed by two extended sides, each of which is a continuation of the original sides. 

Wait, maybe I should use the concept of intersecting lines. When two lines intersect, the sum of the angles around the point is 360 degrees. 

In the case of the star formed by the pentagon, each tip of the star is formed by the intersection of two extended sides. The angle at that tip is the angle between the two extended sides. 

To find this angle, we can consider the geometry of the regular pentagon. Each internal angle is 108 degrees. The sides are extended until they meet. 

Alternatively, consider the triangle formed by the tip of the star and two adjacent vertices of the pentagon. 

Wait, maybe if I can compute the distance between the tip and the vertices, but that might be complicated. 

Alternatively, perhaps using the fact that in a regular pentagon, the length of the diagonal is related to the golden ratio. 

In a regular pentagon, the ratio of the diagonal to the side is the golden ratio φ = (1 + sqrt(5))/2 ≈ 1.618. 

But how does that help with the angles? 

Alternatively, considering the regular pentagram, which is composed of triangles. Each of these triangles is an isosceles triangle with a vertex angle of 36 degrees. 

Wait, if that's the case, then the angle at the tip of the star is 36 degrees. 

Alternatively, perhaps I can calculate the central angles. 

In a regular pentagram, the central angles between the tips are 72 degrees. Wait, but that's the same as the central angles of the original pentagon. Hmm. 

Alternatively, thinking about the fact that the pentagram can be inscribed in a circle, with each tip touching the circle. The central angles between the tips would be 72 degrees. 

But the angle at each tip is formed by two chords of the circle. The angle between two chords can be calculated using the formula: 

Angle = (arc between the two points)/2 

But if the central angle between two tips is 72 degrees, then the angle at the tip of the star would be half of that, which is 36 degrees. 

Yes, that makes sense. The angle at the tip is an inscribed angle subtended by the arc between two adjacent tips. Since the central angle is 72 degrees, the inscribed angle is half of that, which is 36 degrees. 

Therefore, angle A, being one of these tips, is 36 degrees. 

But let me check this again. 

If two chords intersect at a point on the circumference of a circle, the angle between the chords is equal to half the measure of the arc intercepted by the angle. 

In this case, the tip of the star is a point on the circumference, and the two chords are the lines connecting to the adjacent tips. Since the central angle between the tips is 72 degrees, the inscribed angle at the tip is 36 degrees. 

Therefore, angle A is 36 degrees. 

Alternatively, if I construct a regular pentagram, each tip angle is indeed 36 degrees. 

Therefore, the measure of angle A is 36 degrees. 

But to ensure this is correct, let me consider another approach. 

Consider the regular pentagon inscribed in a unit circle. The coordinates of the vertices can be calculated using trigonometric functions. 

Each vertex of a regular pentagon inscribed in a unit circle has coordinates (cos θ, sin θ), where θ = 0°, 72°, 144°, 216°, 288°, etc. 

If we extend the sides, the points where the extended sides intersect will form the tips of the pentagram. The angle at each tip can be calculated by finding the angle between the two extended sides. 

Alternatively, perhaps using complex numbers or vectors. 

But maybe this is getting too complicated. 

Alternatively, let's use the fact that in a regular pentagon, the distance between non-adjacent vertices (the diagonals) are in the golden ratio to the sides. 

The golden ratio φ is (1 + sqrt(5))/2 ≈ 1.618. 

If the side length of the pentagon is s, then the diagonal length is φ*s. 

But how does this help with the angle? 

Alternatively, in the regular pentagram, the triangles formed are golden triangles, which are isosceles triangles with a base angle of 72 degrees and a vertex angle of 36 degrees. 

Yes, that's correct. A golden triangle has base angles of 72 degrees and a vertex angle of 36 degrees. 

Therefore, the tip angle is 36 degrees. 

Therefore, angle A is 36 degrees. 

Therefore, after considering multiple approaches and confirming through different properties of the regular pentagon and pentagram, I conclude that the measure of angle A is 36 degrees.
Therefore, the measure of angle $A$ in the figure is $\boxed{36^\circ}$.
</think>