To determine if triangles \( \triangle FHG \) and \( \triangle PRQ \) are similar, we can use the Angle-Angle (AA) Similarity Theorem, which states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
Given:
- \( \angle H = 38^\circ \)
- \( \angle P = 38^\circ \)
- \( \angle G = 41^\circ \)
- \( \angle Q = 41^\circ \)
So we have:
- \( \angle H \) corresponds to \( \angle P \)
- \( \angle G \) corresponds to \( \angle Q \)
We need to determine \( \angle F \) and \( \angle R \) next to see if the third angles are equal. The sum of the angles in a triangle is \( 180^\circ \).
For \( \triangle FHG \): \[ \angle F + \angle H + \angle G = 180^\circ \] \[ \angle F + 38^\circ + 41^\circ = 180^\circ \] \[ \angle F + 79^\circ = 180^\circ \] \[ \angle F = 180^\circ - 79^\circ = 101^\circ \]
For \( \triangle PRQ \): \[ \angle R + \angle P + \angle Q = 180^\circ \] \[ \angle R + 38^\circ + 41^\circ = 180^\circ \] \[ \angle R + 79^\circ = 180^\circ \] \[ \angle R = 180^\circ - 79^\circ = 101^\circ \]
Now we have:
- \( \angle F = 101^\circ \)
- \( \angle R = 101^\circ \)
Since:
- \( \angle H = \angle P \)
- \( \angle G = \angle Q \)
- \( \angle F = \angle R \)
By the AA Similarity Theorem, the triangles \( \triangle FHG \) and \( \triangle PRQ \) are similar.
Thus, the correct response is: yes because of the AA Similarity Theorem.
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