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?

Question

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

Reason 5 has a mistake. It should say “SSS Congruence Theorem.”
Reason 5 has a mistake. It should say “SSS Congruence Theorem.”

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 ."

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 3 has a mistake. It should say “∠BAD≅∠CAD."

Answer 1

The critique of Fox's proof highlights a mistake in reasoning related to the application of the SAS Congruence Theorem in step 5.

1. Mistake: The problem arises because in an isosceles triangle (like ∆ABC), the congruence of sides AB and AC does not directly show that triangles ABD and ACD are congruent through a Side-Angle-Side (SAS) argument without knowing that angle A is common to both triangles.

2. Correction: The correct reasoning should involve showing that the triangles ABD and ACD are congruent by using the fact that the two sides (AB and AC) are equal (as given), the segments BD and DC are equal (by construction), and pointing out that AD is common in both triangles (reflexive property). Thus, we can indeed use the SAS Congruence Theorem correctly.

So Statement 5 should remain SAS, not change to SSS. The proof itself should look something like this:

• Revised 5: "ΔABD ≅ ΔACD by SAS Congruence Theorem (AB ≅ AC, AD ≅ AD, BD ≅ DC)."

The critiques relating to statement 3 and statement 6 (about angles A and C) are not directly relevant to the error in the proof. The desired conclusion of congruence of angles B and C (statement 6) is indeed proper if triangles ABD and ACD are proven congruent.

To summarize, the main issue is with an incorrect label on step 5, which should correctly invoke the SAS Congruence Theorem rather than requiring a mention of side lengths leading to angles B and C.

Therefore, the critique that "Reason 5 has a mistake. It should say 'SSS Congruence Theorem.'" is incorrect. The SAS Congruence Theorem is the appropriate reference in this context when triangulating those respective segments of the triangles.

Thus, there is no mistake in the application of the theorem so the proof as structured is correct with the necessary reasoning adjustments.