If triangle PRG has ∠P = 30 and ∠R = 127 and triangle FGH has ∠H = 30 and ∠G = 23, what criterion has been met to prove the triangles similar?

To determine if the triangles PRG and FGH are similar, we can look at their angles.

In triangle PRG:

• ∠P = 30°
• ∠R = 127°

To find ∠G, we can calculate it as follows: ∠G = 180° - ∠P - ∠R = 180° - 30° - 127° = 23°.

In triangle FGH:

• ∠H = 30°
• ∠G = 23°

Now we can check the angles:

• ∠P (30°) is equal to ∠H (30°).
• ∠R (127°) is equal to the third angle in triangle FGH which we just calculated as 127°.

So the third angle in triangle FGH can also be derived as follows: ∠F = 180° - ∠H - ∠G = 180° - 30° - 23° = 127°.

Both triangles share two angles:

• ∠P = ∠H (30°)
• ∠R = ∠F (127°)

Since both triangles have two corresponding angles that are equal, we can use the Angle-Angle (AA) similarity criterion.

Thus, the answer is:

d. AA