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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≅∠C Statements Reasons 1. Isosceles △ABC with AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ 1. Given 2. D, the midpoint of BC¯¯¯¯¯¯¯¯ AD¯¯¯¯¯¯¯¯ through points A and D 2. Construction 3. BD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ 3. Definition of midpoint 4. AD¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯ 4. Reflexive property of congruence 5. △ABD≅△ACD 5. SAS Congruence Theorem 6. ∠B≅∠C 6. CPCTC Theorem (1 point) Responses 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 ." Reason 3 has a mistake. It should say “Definition of angle bisector.” Reason 3 has a mistake. It should say “Definition of angle bisector.” Reason 5 has a mistake. It should say “SSS Congruence Theorem.” Reason 5 has a mistake. It should say “SSS Congruence Theorem.”

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Let's analyze Fox's reasoning in his proof of the Base Angles Theorem step by step:

  1. Statement 1 and Reason 1: These are correct as they state the given information.

  2. Statement 2 and Reason 2: The construction of point D as the midpoint of segment BC is appropriate for the proof.

  3. Statement 3 and Reason 3: Fox asserts that \( BD \cong DC \) based on the definition of a midpoint. This statement is correct.

  4. Statement 4 and Reason 4: Fox uses the Reflexive Property of Congruence, claiming \( AD \cong AD \). This is also correct.

  5. Statement 5 and Reason 5: Here, Fox claims \( \triangle ABD \cong \triangle ACD \) by using the SAS Congruence Theorem. Let’s examine this claim further:

    • For SAS, he has two sides and the included angle:
      • \( AB \cong AC \) (given, as \( AB \cong AC \))
      • \( BD \cong DC \) (from the midpoint definition)
      • The included angle \( \angle ADB \) is common to both triangles.
    • Thus, he correctly applied the SAS Congruence Theorem.

  6. Statement 6 and Reason 6: This is where the mistake occurs. Fox concludes \( \angle B \cong \angle C \) based on the corresponding parts of the congruent triangles (CPCTC). However, his congruence specifically concerns triangles \( \triangle ABD \) and \( \triangle ACD \), but it is essential to note that he should be referring to \( \angle B \) and \( \angle C \).

Critique and Correction:

  • Critique: The conclusion \( \angle B \cong \angle C \) does hold true under the assumption that triangles \( \triangle ABD \) and \( \triangle ACD \) are congruent. Since \( \angle CAD \) and \( \angle BAD \) are the angles opposite the congruent sides in each respective triangle, Fox should have framed his notation correctly in the conclusion, specifically referencing the correct angles.

  • Correction: Instead of concluding \( \angle B \cong \angle C \) directly, Fox should ensure that he has clearly defined all angles involved, especially that \( \angle ADB \) and angles \( \angle B \) and \( \angle C \) are the angles being compared in the context of congruent triangles. The correct conclusion should state: "Since \( \triangle ABD \cong \triangle ACD \) by SAS, it follows that \( \angle B \cong \angle C \) (CPCTC).”

In conclusion:

The reasoning in Statement 6 is acceptable but can be misleading, so the language should emphasize clearly that the angles being referenced are in the context of the triangles in question. No fundamental mistake exists, but clarity needs to be improved.