Use the image to answer the question. Fox is working to prove the Base Angles Theorem. His proof is shown below. Critique his reasoning. Which statement or reason in his proof has a mistake? How can he fix his mistake? Given: Isosceles △ABC with AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯Prove: ∠B≅∠CStatements Reasons1. Isosceles △ABC with AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ 1. Given2. D, the midpoint of BC¯¯¯¯¯¯¯¯AD¯¯¯¯¯¯¯¯ through points A and D2. Construction3. BD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ 3. Definition of midpoint4. AD¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯ 4. Reflexive property of congruence5. △ABD≅△ACD 5. SAS Congruence Theorem6. ∠B≅∠C 6. CPCTC Theorem(1 point)ResponsesReason 5 has a mistake. It should say “SSS Congruence Theorem.” Reason 5 has a mistake. It should say “SSS Congruence Theorem.” Reason 3 has a mistake. It should say “Definition of angle bisector.” Reason 3 has a mistake. It should say “Definition of angle bisector.” Statement 6 has a mistake. It should say “∠A≅∠C."Statement 6 has a mistake. It should say “ angle upper A congruent to angle upper C ."Statement 3 has a mistake. It should say “∠BAD≅∠CAD."Statement 3 has a mistake. It should say “ angle upper B upper A upper D congruent to angle upper C upper A upper D ."

Answer 1

Let's analyze Fox's proof step-by-step to identify the mistake.

1. Statements and Reasons:

   • 1. Isosceles △ABC with \( AB \cong AC \) — (Given)
   • 2. D, the midpoint of \( BC \) — (Construction)
   • 3. \( BD \cong DC \) — (Definition of midpoint)
   • 4. \( AD \cong AD \) — (Reflexive property of congruence)
   • 5. \( \triangle ABD \cong \triangle ACD \) — (SAS Congruence Theorem)
   • 6. \( \angle B \cong \angle C \) — (CPCTC Theorem)

2. Critique of Reasoning:

   • All statements up to 3 are correct.
   • Statement 3 correctly states that since D is the midpoint, \( BD \cong DC \).
   • Statement 4 uses the reflexive property accurately, noting that segment \( AD \) is congruent to itself.
   • In Statement 5, the assertion that \( \triangle ABD \cong \triangle ACD \) is based on the lengths of two sides and the included angle, which is directly valid here; hence there is no mistake in the congruence criteria used here. This is valid for SAS since we have \( AB \cong AC \), \( AD \cong AD \) (reflexive), and \( BD \cong DC \).
   • In Statement 6, the conclusion correctly follows from the earlier step. Thus, \( \angle B \cong \angle C \) is valid.

3. Conclusion: There appears to be no error in Fox's proof as it follows logically from each of the statements and reasons provided. The confusion might arise in the critique responses, suggesting that reason 5 should say "SSS" instead of "SAS." However, since two sides and the included angle are taken into account, SAS is indeed appropriate here.

Hence, the proof is sound. If a critique were to be given, it could emphasize confirming the rational use of triangle congruence criteria rather than suggesting incorrect criteria.

The best fix would be to clarify the justification of statement 5 with the context of why SAS is used (with all sides considered) but not change from SAS to SSS.

In conclusion, reason 5 is accurate, and Fox's proof is valid without any mistakes.