<Question>
In regular pentagon $FGHIJ$, extending the sides of the pentagon, as shown, forms a star. What is the measure of angle $A$ in the figure?

[asy]
draw((-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)--cycle,linewidth(1));
draw((-10,30.8)--(10,30.8)--(16.2,11.8)--(0,0)--(-16.2,11.8)--cycle,linewidth(1));
label("$A$",(-42.4,30.8),W);
label("$F$",(-10,30.8),NW);
dot((-10,30.8));
label("$G$",(10,30.8),NE);
dot((10,30.8));
label("$H$",(16.2,11.8),E);
dot((16.2,11.8));
label("$I$",(0,0),S);
dot((0,0));
label("$J$",(-16.2,11.8),WSW);
dot((-16.2,11.8));
[/asy]
</Question>

<think>
<Plan_1>
Okay, so I need to find the measure of angle A in this regular pentagon where the sides are extended to form a star. Hmm, let me start by recalling some properties of regular pentagons and the stars formed by their sides.

<prompt_1.1> Recall the properties of regular pentagons and the angles formed by extending their sides. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, a regular pentagon has all sides equal and all internal angles equal. The formula for the internal angle of a regular polygon is ((n-2)*180)/n, where n is the number of sides. For a pentagon, n=5, so each internal angle is ((5-2)*180)/5 = (3*180)/5 = 540/5 = 108 degrees. So each internal angle is 108 degrees.

But angle A in the figure isn't one of the internal angles of the pentagon; it's part of the star formed by extending the sides. So I need to figure out the angle at that intersection point. Let me try to visualize the Asymptote figure. It shows a regular pentagon with sides extended to form a five-pointed star, also known as a pentagram. The points of the star are formed by extending the sides of the pentagon until they meet other extended sides.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
The original pentagon is labeled FGHIJ, and the star is formed by extending these sides. The angle A is labeled at the point (-42.4,30.8), which seems to be one of the outer points of the star.

Wait, actually, in a regular pentagram, each point is an isosceles triangle with a vertex angle. The angle at each point of the star is called the "point angle" of the star. I need to find that point angle.

But let me confirm. The figure is a regular pentagon, so when you extend its sides, the resulting star should also be regular, meaning all the sides and angles of the star are equal. Therefore, angle A is one of these point angles of the pentagram.

So, how do we calculate the point angle of a regular pentagram? I remember that for regular stars, the point angles can be calculated using some trigonometric formulas or by considering the relationships between the sides and angles of the pentagon.

Alternatively, maybe I can use the fact that the pentagram is made up of triangles and other components whose angles we can compute.

First, let me recall that in a regular pentagram, the point angle can be found using the formula: 180 - 2*base angle of the isosceles triangle forming the point.

But perhaps it's better to approach this by considering the central angles or using the properties of the regular pentagon.

Another approach is to use the fact that the regular pentagram is related to the regular pentagon through the concept of golden ratio, but maybe that's complicating things.

Alternatively, perhaps using the exterior angles of the pentagon.

Wait, let's think about the structure. When you extend the sides of the pentagon, each side is extended until it meets another extended side, forming a point of the star. Each of these extensions creates an external angle.

<prompt_2.1> Explore the relationship between the internal angles of the pentagon and the external angles formed by extending its sides. </prompt_2.1>

But I need to be precise here.

Alternatively, perhaps considering the entire figure. The regular pentagram can be thought of as a decagon (10-sided polygon) with its vertices connected in a certain way. Wait, no. A pentagram has 5 points, but it also has intersections inside, creating a 5-pointed star with a pentagon in the center.

<prompt_2.2> Consider the structure of the pentagram and how it relates to the regular pentagon. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
The original internal angle is 108 degrees. When you extend a side, the external angle formed would be 180 - 108 = 72 degrees. But I'm not sure if that's directly helpful.

Wait, but the point of the star is formed by two extended sides. Each extended side forms an angle with the original side. So maybe the angle at the star's point is related to these external angles.
</execution_2.1>

<execution_2.2>
Wait, maybe not. Let me think again.

Alternatively, perhaps using the concept of supplementary angles. When you extend a side of the pentagon, the angle outside the original pentagon is an external angle. For a regular pentagon, each external angle is 360/5 = 72 degrees. Wait, but when you extend a side, the angle formed is supplementary to the internal angle. Since the internal angle is 108 degrees, the external angle would be 180 - 108 = 72 degrees. So maybe the angles at the star points are related to these external angles.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2:
Let me try to analyze the figure.

Consider one of the points of the star, say point A. This point is formed by the intersection of two extended sides of the pentagon. Each of these sides was originally a side of the pentagon, which had an internal angle of 108 degrees.

<prompt_3.1> Analyze the structure of the pentagram and consider the angles formed by the extended sides of the pentagon. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
When you extend these sides, the angle outside the pentagon is 180 - 108 = 72 degrees. But the angle at the star point is not exactly this 72 degrees. Instead, the star point is formed by two such extended sides, each contributing an angle. Wait, perhaps the angle at the star point is equal to the sum of two such external angles? That is, 72 + 72 = 144 degrees? But that seems too large. Wait, maybe not.

Alternatively, when two extended sides meet at a point, the angle between them is the angle of the star. Let's consider the lines forming these sides.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Each extended side is a side of the pentagon extended beyond a vertex. The original internal angle is 108 degrees, so the external angle is 72 degrees.

<prompt_4.1> Calculate the angle at the star point using the external angles of the pentagon. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
But when you extend two sides, the angle between the extensions is equal to the external angle.

Wait, perhaps the angle at the star point is equal to the external angle of the pentagon. But the external angle is 72 degrees. However, when two sides are extended, the angle between them is actually 180 - 2*72 = 36 degrees? Wait, that might make more sense. Let me draw a mental picture.

Imagine two lines extending from adjacent sides of the pentagon. Each original side is part of the pentagon, and when extended, they form an angle at the star point. The original internal angle is 108 degrees. If we extend one side beyond a vertex, the angle between the extended side and the next side would be supplementary to the internal angle. But since two sides are being extended, maybe the angle at the star is formed by two such extensions.

Wait, perhaps the angle at the star point is the supplement of twice the internal angle? Let me think. If you have a regular pentagon, each internal angle is 108. When you extend a side, the angle outside is 72. But when two sides are extended to meet, the angle between those two extensions would be equal to the internal angle of the pentagram. Wait, I might need to use some diagram.

Alternatively, perhaps using the formula for the point angle of a regular star polygon. The formula for the point angle of a regular star polygon with n points, denoted by {n/m}, where m is an integer such that 2m < n and gcd(n, m) = 1. For a pentagram, it's {5/2}. The formula for the point angle is given by:

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

So for pentagram {5/2}, n=5, m=2:

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

Therefore, angle A is 36 degrees.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Alternatively, another way to compute the point angle is using the formula:

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

<prompt_5.1> Verify the point angle of the pentagram using an alternative formula. </prompt_5.1>

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

Another approach is to consider the fact that the pentagram can be inscribed in a circle, with all its points lying on the circle.

<prompt_5.2> Confirm the point angle of the pentagram by inscribing it in a circle. </prompt_5.2>

Alternatively, let's consider the triangle formed at the tip of the star. If we take one of the points of the star, say point A, and connect it to the two adjacent vertices of the pentagram, forming a triangle.

<prompt_5.3> Analyze the triangle formed at the tip of the star and calculate the angle using trigonometric identities. </prompt_5.3>
</Plan_5>

<Execution_5>
<execution_5.1>
Plugging in n=5, m=2:

Point angle = 180 - (2*180*2)/5 = 180 - 720/5 = 180 - 144 = 36 degrees. So same result.

Therefore, angle A is 36 degrees.

But let me make sure this is correct by considering another method.

Let's consider the regular pentagon and the star it forms. The key idea is that the star is a regular star polygon, which for five points is denoted by {5/2}. The Schläfli symbol {n/m} describes a regular star polygon with n points, each connected to the m-th next point. For a pentagram, m=2, so {5/2}.

The formula for the internal angle (point angle) of a regular star polygon is given by:

Point angle = (180*(n-4*m))/n

Wait, no, that doesn't seem right. Let me check the formula again.

Wait, perhaps I confused the formula. Let me look it up in my mind. The formula for the point angle of a regular star polygon {n/m} is:

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

Yes, that's the correct formula. So substituting n=5, m=2:

Point angle = (180*(5 - 4))/5 = 180*1/5 = 36 degrees. Yes, that's correct.
</execution_5.1>

<execution_5.2>
The central angles would be 72 degrees each (since 360/5 = 72). Then, the angle at each point of the star can be calculated using the inscribed angle theorem. Wait, but the point angle is actually the angle between two chords of the circle that intersect at a point outside the circle. The formula for the angle between two chords intersecting outside the circle is half the difference of the measures of the intercepted arcs.

So, if the two arcs intercepted by the angle at point A are the arc between the two points of the pentagon connected to A, and the opposite arc. Wait, in the pentagram, each point is connected to two non-adjacent vertices of the pentagon. The arc between those two vertices would be three sides apart. Since the pentagon is regular, each arc between two connected vertices in the star is 3*72 = 216 degrees? Wait, no. If the central angle between two adjacent points is 72 degrees, then the arc between two points separated by k steps is 72*k degrees. For the pentagram, each point is connected to two points that are two steps away. For example, starting at point A, connected to points C and D in the original pentagon. The arc from C to D would be 72*2 = 144 degrees. Wait, but actually, in the star polygon, each edge skips one vertex. So in terms of steps, moving from one point to the next connected point in the star is two edges away in the original pentagon. Therefore, the arc between the two points connected to A would be 2 steps, which is 2*72 = 144 degrees.

But the angle at point A is formed by two chords: one connecting A to C and another connecting A to D (assuming the original pentagon points are labeled A, B, C, D, E). The angle at A is the angle between these two chords. The formula for the angle between two chords intersecting outside the circle is (arc CD - arc AB)/2, but since the chords are intersecting at A, which is outside the circle, the angle is half the difference of the intercepted arcs. However, in this case, the two arcs intercepted would be the arc from C to D (which is 144 degrees) and the opposite arc, which would be the rest of the circle. Since the circle is 360 degrees, the opposite arc would be 360 - 144 = 216 degrees. Therefore, the angle at A is (216 - 144)/2 = 72/2 = 36 degrees. Therefore, angle A is 36 degrees.
</execution_5.2>

<execution_5.3>
Wait, maybe not. Alternatively, perhaps using the concept of the golden triangle, which is an isosceles triangle with a vertex angle of 36 degrees and base angles of 72 degrees. These triangles are related to the golden ratio and are found in the pentagram.

If the point angle is 36 degrees, then the two base angles would be 72 degrees each. Since the sides of the star are equal, the triangle at the tip would indeed be a golden triangle.
</execution_5.3>
</Execution_5>

<Plan_6>
Based on execution_5.1, execution_5.2, execution_5.3:
Alternatively, let's do a more step-by-step analysis. Let's consider the pentagon FGHIJ. Let's label the points as follows: F, G, H, I, J, and then back to F. When we extend the sides, for example, extending FG and GH beyond G and H respectively, they meet at a point of the star, say A. Wait, but in the Asymptote code provided, the coordinates are given, so maybe point A is at (-42.4,30.8), which is to the left of the pentagon.

<prompt_6.1> Perform a step-by-step analysis of the angles involved in forming the star from the pentagon. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
But perhaps the exact coordinates aren't necessary since we can reason through the angles.

Let me consider two extended sides forming the star point. Each original side of the pentagon is extended, creating an external angle. The angle between the extended sides is the angle of the star point.

Given that the internal angle of the pentagon is 108 degrees, the external angle is 72 degrees. However, when you extend two sides to form the star point, the angle at that point is not the external angle but rather the angle between the two extensions.

To find this angle, consider that each extension forms an angle of 72 degrees with the original side. Therefore, the angle between the two extended sides is 180 - 2*72 = 36 degrees.

Wait, that makes sense. If each original side is extended outward, forming an angle of 72 degrees with the adjacent side, then the angle between the two extensions would be 180 - 2*72 = 36 degrees.
</execution_6.1>
</Execution_6>

<Plan_7>
Based on execution_6.1:
Wait, let me check that again.

<prompt_7.1> Re-evaluate the angle calculation using the external angles of the pentagon. </prompt_7.1>
</Plan_7>

<Execution_7>
<execution_7.1>
Suppose at vertex G of the pentagon, the internal angle is 108 degrees. When we extend the side FG beyond G, the angle between the extended FG and the next side GH is 180 - 108 = 72 degrees. Similarly, if we extend the side HG beyond H, the angle between the extended HG and the next side HI is also 72 degrees. Wait, but how does this relate to the angle at point A?

Wait, point A is formed by the extension of two sides: one extended from FG and another extended from another side. Wait, maybe I need to clarify which sides are being extended.

Looking back at the Asymptote code, the star is drawn with the command:

draw((-10,30.8)--(10,30.8)--(16.2,11.8)--(0,0)--(-16.2,11.8)--cycle,linewidth(1));

Which connects points F, G, H, I, J, and back. Then another draw command for the inner pentagon. But perhaps the coordinates can help us. Let me parse the coordinates:

The outer points are:

F: (-10,30.8)

G: (10,30.8)

H: (16.2,11.8)

I: (0,0)

J: (-16.2,11.8)

Then back to F: (-10,30.8)

So this seems like a star with points at F, G, H, I, J. But the label A is at (-42.4,30.8), which is outside the original pentagon.

Looking at the coordinates, the point A is to the left of the original pentagon. The original pentagon has its leftmost point at J: (-16.2,11.8). So point A is further left at (-42.4,30.8). So this is a separate point outside the original pentagon, part of another star or a larger star.

Wait, perhaps the figure is a combination of two stars? Or maybe the Asymptote code is drawing a star with multiple layers.

Alternatively, perhaps the figure is a five-pointed star (pentagram) with the original pentagon inscribed in it. The coordinates given might be part of such a star. Let me try to visualize the coordinates.

Point F is at (-10,30.8), G at (10,30.8), H at (16.2,11.8), I at (0,0), J at (-16.2,11.8). So connecting F to G to H to I to J and back to F. Then another draw command connects (-10,30.8)--(10,30.8)--(16.2,11.8)--(0,0)--(-16.2,11.8)--cycle. Wait, this seems to be the same as the first draw command. Maybe it's a different layer.

But perhaps the figure is a combination of a pentagon and a pentagram. However, the label A is at (-42.4,30.8), which is far to the left. The original pentagon's leftmost point is J at (-16.2,11.8). So point A is further left, so maybe it's part of another, larger star or a different component.

Alternatively, maybe the Asymptote code is drawing a more complex figure with multiple overlapping stars.

But perhaps instead of relying on the coordinates, which might be specific to the drawing, I can reason purely geometrically.

Given that the original figure is a regular pentagon with sides extended to form a star, the measure of angle A is the point angle of the star. As per the earlier calculations, this should be 36 degrees. But let me confirm once again.

Another approach is to consider that in a regular pentagram, the ratio of the length of the sides and the diagonals is the golden ratio. However, maybe that's not directly helpful here.

Alternatively, let's consider the triangle formed at the tip of the star. If we take point A and connect it to two adjacent points of the original pentagon, say F and G, then triangle AFG would be an isosceles triangle with apex at A. The base FG is a side of the pentagon, and the two sides AF and AG are the extended sides.
</execution_7.1>
</Execution_7>

<Plan_8>
Based on execution_7.1:
But since F and G are part of the original pentagon, the length of FG is known. However, without specific measurements, maybe we can use angles.

<prompt_8.1> Analyze the triangle formed by extending the sides of the pentagon to find angle A. </prompt_8.1>
</Plan_8>

<Execution_8>
<execution_8.1>
Alternatively, perhaps using the fact that the central angles and inscribed angles can help. Since the pentagon is regular, all sides and angles are equal, and the central angles are 72 degrees.

If we consider the star point A, and the two adjacent points F and G, the lines AF and AG are extensions of the sides of the pentagon. The angle at A is the angle between these two extended sides.

To find angle A, we can analyze the angles around point A. The sum of angles around point A is 360 degrees. However, angle A is just one of those angles. Wait, no. Actually, in the star figure, each point of the star is a triangle, and the angle at A is the vertex angle of that triangle.

Alternatively, perhaps considering the entire star as a combination of triangles and pentagons. The key is that the point angle of a regular pentagram is 36 degrees, as calculated earlier.
</execution_8.1>
</Execution_8>

<Plan_9>
Based on execution_8.1:
Alternatively, maybe using the concept of the internal angles of the star polygon. The internal angle of a regular star polygon can be calculated by the formula:

Internal angle = 180 - Point angle

<prompt_9.1> Verify the angle calculation using the relationship between internal and external angles of the star polygon. </prompt_9.1>
</Plan_9>

<Execution_9>
<execution_9.1>
But not sure. Wait, in a regular star polygon, the internal angle and the point angle are the same. Wait, no. The internal angle is the angle you turn when walking along the edges of the star. Wait, perhaps confusing.

Alternatively, perhaps considering the fact that the regular pentagram can be inscribed in a circle, and each point of the star subtends a central angle. The central angle between two adjacent points of the star is 144 degrees (since 360/2.5 = 144). Then, using the inscribed angle theorem, the angle at point A would be half the central angle. Wait, but the inscribed angle theorem states that the angle subtended at the center is twice any inscribed angle subtended by the same arc. However, in this case, the angle at point A is not an inscribed angle but rather an angle formed by two chords.

Wait, if the central angle between two points connected to A is 144 degrees, then the inscribed angle at A would be half of that, which is 72 degrees. But that contradicts our previous result. Therefore, there must be a mistake here.

Wait, perhaps the central angle is different. Let me think again. If the star is a pentagram, each point is connected to two non-adjacent vertices of the pentagon. The original pentagon has vertices every 72 degrees. So connecting every other vertex would create the star. The central angle between two connected vertices is 2*72 = 144 degrees. Therefore, the arc between two connected vertices is 144 degrees. Then, the angle at point A, which is outside the circle, formed by two chords, would be half the difference of the intercepted arcs. The formula is (arc far - arc near)/2. The far arc is 360 - 144 = 216 degrees, and the near arc is 144 degrees. Therefore, the angle at A is (216 - 144)/2 = 36 degrees.
</execution_9.1>
</Execution_9>

<Plan_10>
Based on execution_9.1:
Therefore, angle A is 36 degrees.

Alternatively, another way to think about it is using the fact that in a regular pentagon, the diagonals intersect at angles of 36 degrees. The ratio of the diagonal to the side in a regular pentagon is the golden ratio φ = (1 + sqrt(5))/2 ≈ 1.618.

<prompt_10.1> Confirm the angle calculation using the properties of the golden ratio and trigonometric identities. </prompt_10.1>
</Plan_10>

<Execution_10>
<execution_10.1>
However, perhaps this is complicating things.

Alternatively, let's use coordinates to verify. Suppose we place the pentagon in a coordinate system. Let's assume a regular pentagon centered at the origin with a certain radius. Then, the coordinates of the vertices can be calculated using polar coordinates. However, this might be time-consuming, but let's attempt it.

Let’s assume the original pentagon is centered at the origin with a vertex at (1,0). The coordinates of the vertices can be given by (cos(72*k), sin(72*k)) for k=0,1,2,3,4. Then, the star is formed by connecting every other vertex. The angle at each star point can be calculated using vector analysis or trigonometry.

However, this might be too involved. Alternatively, since we already have multiple methods leading to 36 degrees, and the calculations are consistent, we can be confident that angle A is 36 degrees.
</execution_10.1>
</Execution_10>

<Plan_11>
Based on execution_10.1:
Therefore, the measure of angle A is 36 degrees. Converting this to the required format, the final answer is:

\boxed{36^\circ}

**Final Answer**
\boxed{36^\circ}
</Plan_11>

<Execution_11>
No prompt to execute.
</Execution_11>

<Final_answer>
Therefore, the measure of angle $A$ in the figure is $\boxed{36^\circ}$.
</Final_answer>
</think>