To solve this problem, we can use Euler's formula for planar graphs, which states that for a connected planar graph with V vertices, E edges, and F faces:

V - E + F = 2

- We are given that the graph G is connected and has 10 vertices. Therefore, V = 10.
- We are also given that the number of edges on each face is three. Since each edge is incident to two faces, the sum of the degrees of the faces is equal to 2E, where E is the number of edges. Therefore, the average degree of the faces is 2E/F.
- Since each face has degree three, the average degree of the faces is also equal to three. Therefore, 2E/F = 3.
- Since the graph is connected, each face must have at least three edges. Therefore, the number of faces F is at most E/3.

Now, let's substitute these values into Euler's formula:

10 - E + F = 2

Substituting F ≤ E/3:

10 - E + E/3 ≤ 2

Multiplying through by 3:

30 - 3E + E ≤ 6

Combining like terms:

30 - 2E ≤ 6

Subtracting 30 from both sides:

-2E ≤ -24

Dividing through by -2 (and reversing the inequality):

E ≥ 12

Since the number of edges must be an integer, the minimum number of edges is 12.

However, we are looking for the number of edges in G, not just the minimum. Since the graph is planar, it must satisfy the inequality F ≤ E/3. Since each face has degree three, the sum of the degrees of the faces is equal to 2E. Therefore, the average degree of the faces is 2E/F. Since each face has degree three, the average degree of the faces is also equal to three. Therefore, 2E/F = 3.

Substituting this into the inequality F ≤ E/3:

F ≤ 2E/3

Multiplying through by 3:

3F ≤ 2E

Substituting this into Euler's formula:

10 - E + F ≤ 2

Substituting 3F ≤ 2E:

10 - E + 2E/3 ≤ 2

Multiplying through by 3:

30 - 3E + 2E ≤ 6

Combining like terms:

30 - E ≤ 6

Subtracting 30 from both sides:

-E ≤ -24

Dividing through by -1 (and reversing the inequality):

E ≥ 24

Since the number of edges must be an integer, the minimum number of edges is 24.

Therefore, the number of edges in G is at least 24. The only answer choice that satisfies this condition is option A, 24.