Question
Use the image to answer the question.
Triangles upper F upper G upper H and upper P upper Q upper R are plotted on a grid. Triangle upper P upper Q upper R is larger.
If ∠H
and ∠P
each equal 38 degrees and ∠G
and ∠Q
each equal 41 degrees, is △FHG∼△PRQ
?
(1 point)
Responses
yes because of the AA Similarity Theorem
yes because of the AA Similarity Theorem
no because the ratio of corresponding side lengths is unknown
no because the ratio of corresponding side lengths is unknown
no because m∠F
and m∠R
are unknown
no because m angle cap f and m angle cap r are unknown
yes because of the SSS Congruence Theorem
Triangles upper F upper G upper H and upper P upper Q upper R are plotted on a grid. Triangle upper P upper Q upper R is larger.
If ∠H
and ∠P
each equal 38 degrees and ∠G
and ∠Q
each equal 41 degrees, is △FHG∼△PRQ
?
(1 point)
Responses
yes because of the AA Similarity Theorem
yes because of the AA Similarity Theorem
no because the ratio of corresponding side lengths is unknown
no because the ratio of corresponding side lengths is unknown
no because m∠F
and m∠R
are unknown
no because m angle cap f and m angle cap r are unknown
yes because of the SSS Congruence Theorem
Answers
GPT-4o mini
To determine whether triangles \( \triangle FHG \) and \( \triangle PRQ \) are similar, we can use the AA (Angle-Angle) Similarity Theorem, which states that if two triangles have two pairs of corresponding angles that are equal, then the triangles are similar.
From the information provided:
- \( \angle H = \angle P = 38^\circ \)
- \( \angle G = \angle Q = 41^\circ \)
Since we have two pairs of angles that are equal:
1. \( \angle H \cong \angle P \)
2. \( \angle G \cong \angle Q \)
The third angles of both triangles can be found using the fact that the sum of the angles in a triangle is \( 180^\circ \):
- For \( \triangle FHG \):
\[
\angle F = 180^\circ - (38^\circ + 41^\circ) = 180^\circ - 79^\circ = 101^\circ
\]
- For \( \triangle PRQ \):
\[
\angle R = 180^\circ - (38^\circ + 41^\circ) = 180^\circ - 79^\circ = 101^\circ
\]
Thus, we also have \( \angle F \cong \angle R \).
With two pairs of angles being equal, we can conclude that:
\[
\triangle FHG \sim \triangle PRQ \text{ by the AA Similarity Theorem.}
\]
The correct response is:
**yes because of the AA Similarity Theorem.**
From the information provided:
- \( \angle H = \angle P = 38^\circ \)
- \( \angle G = \angle Q = 41^\circ \)
Since we have two pairs of angles that are equal:
1. \( \angle H \cong \angle P \)
2. \( \angle G \cong \angle Q \)
The third angles of both triangles can be found using the fact that the sum of the angles in a triangle is \( 180^\circ \):
- For \( \triangle FHG \):
\[
\angle F = 180^\circ - (38^\circ + 41^\circ) = 180^\circ - 79^\circ = 101^\circ
\]
- For \( \triangle PRQ \):
\[
\angle R = 180^\circ - (38^\circ + 41^\circ) = 180^\circ - 79^\circ = 101^\circ
\]
Thus, we also have \( \angle F \cong \angle R \).
With two pairs of angles being equal, we can conclude that:
\[
\triangle FHG \sim \triangle PRQ \text{ by the AA Similarity Theorem.}
\]
The correct response is:
**yes because of the AA Similarity Theorem.**