To determine whether the triangles \( \triangle PRQ \) and \( \triangle FGH \) are similar, we can use the Angle-Angle (AA) similarity criterion, which states that if two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar.
In triangle \( \triangle PRQ \):
- \( \angle P = 30^\circ \)
- \( \angle R = 127^\circ \)
We can find the third angle \( \angle Q \) using the fact that the sum of the angles in a triangle is \( 180^\circ \): \[ \angle Q = 180^\circ - \angle P - \angle R = 180^\circ - 30^\circ - 127^\circ = 23^\circ \]
So the angles in triangle \( \triangle PRQ \) are:
- \( \angle P = 30^\circ \)
- \( \angle R = 127^\circ \)
- \( \angle Q = 23^\circ \)
In triangle \( \triangle FGH \):
- \( \angle H = 30^\circ \)
- \( \angle G = 23^\circ \)
We can find the third angle \( \angle F \): \[ \angle F = 180^\circ - \angle H - \angle G = 180^\circ - 30^\circ - 23^\circ = 127^\circ \]
So the angles in triangle \( \triangle FGH \) are:
- \( \angle H = 30^\circ \)
- \( \angle G = 23^\circ \)
- \( \angle F = 127^\circ \)
Now we compare the angles of both triangles:
- \( \angle P = 30^\circ \) is congruent to \( \angle H = 30^\circ \)
- \( \angle Q = 23^\circ \) is congruent to \( \angle G = 23^\circ \)
- \( \angle R = 127^\circ \) is congruent to \( \angle F = 127^\circ \)
Since all three pairs of corresponding angles are equal, both triangles \( \triangle PRQ \) and \( \triangle FGH \) satisfy the AA criterion for similarity. Therefore, we can conclude that the triangles are similar by the AA similarity criterion.