To determine the criterion for similarity between the triangles PRQ and FGH using the given angle measurements, we can analyze the information as follows:
For triangle PRQ:
- Angle P = 30°
- Angle R = 127°
To find angle Q, we can use the fact that the sum of the angles in a triangle is 180°: \[ \angle Q = 180° - \angle P - \angle R = 180° - 30° - 127° = 23° \]
So the angles in triangle PRQ are:
- Angle P = 30°
- Angle R = 127°
- Angle Q = 23°
For triangle FGH:
- Angle H = 30°
- Angle G = 23°
To find angle F, we again use the sum of the angles in a triangle: \[ \angle F = 180° - \angle H - \angle G = 180° - 30° - 23° = 127° \]
So the angles in triangle FGH are:
- Angle H = 30°
- Angle G = 23°
- Angle F = 127°
Now, we can compare the angles of both triangles:
- Triangle PRQ: 30°, 127°, 23°
- Triangle FGH: 30°, 23°, 127°
We see that both triangles have two pairs of corresponding angles that are equal:
- \(\angle P\) (30°) corresponds to \(\angle H\) (30°)
- \(\angle Q\) (23°) corresponds to \(\angle G\) (23°)
- \(\angle R\) (127°) corresponds to \(\angle F\) (127°)
Since two angles in each triangle are equal, we can conclude that the triangles are similar by the Angle-Angle (AA) criterion.
Therefore, the criterion that has been met to prove the triangles PRQ and FGH similar is AA (Angle-Angle).