To determine if the triangles ΔPRQ and ΔFGH are similar, we need to analyze the angles given:
For ΔPRQ:
- ∠P = 30°
- ∠R = 127°
We can find the third angle ∠Q by using the fact that the sum of the angles in a triangle is 180°:
\[ ∠Q = 180° - (∠P + ∠R) = 180° - (30° + 127°) = 180° - 157° = 23° \]
Now we have all the angles for ΔPRQ:
- ∠P = 30°
- ∠R = 127°
- ∠Q = 23°
For ΔFGH:
- ∠H = 30°
- ∠G = 23°
To find the third angle ∠F, we can do the same:
\[ ∠F = 180° - (∠H + ∠G) = 180° - (30° + 23°) = 180° - 53° = 127° \]
Now we have all the angles for ΔFGH:
- ∠F = 127°
- ∠G = 23°
- ∠H = 30°
Now we compare the angles of both triangles:
- ΔPRQ: ∠P = 30°, ∠R = 127°, ∠Q = 23°
- ΔFGH: ∠F = 127°, ∠G = 23°, ∠H = 30°
Both triangles have the same angles:
- 30° in both triangles
- 23° in both triangles
- 127° in both triangles
Since both triangles have two angles that are equal, they fulfill the AA (Angle-Angle) similarity criterion.
Therefore, the criterion that has been met to prove the triangles similar is AA.
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