Sebastian, Aisha, and Conner are analyzing the statement △BGH≅△PCF


𝐵
𝐺
𝐻


𝑃
𝐶
𝐹
.

Sebastian says ∠B≅∠F

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𝐹
because they are both on endpoints of their respective triangles.

Aisha says ∠G≅∠C

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𝐶
because both letters are in the middle of their respective triangles.

Conner says ∠H≅∠P

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𝑃
because both letters are closest to the congruency sign in their respective triangles.

Which student is correct in their reasoning?

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
2 hours ago

In the congruence statement \( \triangle BGH \cong \triangle PCF \), the letters represent the corresponding vertices of the two triangles. The order of the letters in the congruence statement indicates how the corresponding angles and sides match up.

  1. Sebastian's reasoning: He stated that \( \angle B \cong \angle F \). This is incorrect because in the congruence statement, \( \angle B \) corresponds to \( \angle P \), not \( \angle F \).

  2. Aisha's reasoning: She claimed that \( \angle G \cong \angle C \). This is also incorrect because \( \angle G \) corresponds to \( \angle C \) in the congruence, but the reasoning does not support the correspondence based solely on the location within the triangle. In this case, although her conclusion is correct—the angles are indeed congruent—the basis of her reasoning is weak.

  3. Conner's reasoning: He mentioned that \( \angle H \cong \angle P \). This is incorrect reasoning as well, since \( \angle H \) corresponds to \( \angle F \) in the congruence statement.

The correct correspondences given the congruence \( \triangle BGH \cong \triangle PCF \) are:

  • \( \angle B \cong \angle P \)
  • \( \angle G \cong \angle C \)
  • \( \angle H \cong \angle F \)

From this analysis, Aisha was the only one who correctly identified the congruence between the angles, though her reasoning was technically flawed. The others misidentified the corresponding angles entirely.