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 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.” 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 ." Skip to navigation

Answer 1

Fox's proof contains a mistake primarily in Statement 3 and the reasoning associated with it. Let's break down the issues:

1. Statement 3: The statement "BD≅DC" is correct as it correctly explains that since D is the midpoint of segment BC, then segments BD and DC are congruent. However, there's confusion in your critique regarding the actual issue.

2. Correction for Statement 3: This statement itself is not the problem. The critique about “∠BAD≅∠CAD" or angles being incorrectly referenced does not apply here; they don't directly pertain to the proof regarding base angles. The midpoint statement is essential and correct.

3. Reason 3: The reasoning associated with Statement 3 ("Definition of midpoint") is accurate and does not need to be changed.

4. Statement 5: This is where a significant misconception arises. The congruence of triangles \(\triangle ABD\) and \(\triangle ACD\) is justified correctly via the Side-Angle-Side (SAS) theorem because you have two sides (AD is common to both triangles, BD≅DC from statement 3), and the included angle (∠BAD is equal to ∠CAD).

5. Correction for Statement 5: Since the triangles are proven congruent by SAS, stating it as SSS (Side-Side-Side) changes the reasoning and is incorrect. The mistake is suggesting SSS when SAS is appropriate given the common side AD is included in both triangles.

6. Statement 6: This statement is not inherently wrong, as it correctly follows from the congruent triangles \(\triangle ABD\) and \(\triangle ACD\) that \(\angle B\) is congruent to \(\angle C\) using the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) principle.

In conclusion: The proof has no wrong statements per se, but the rationale presented for congruency by Statement 5 is incorrect.

To fix the mistakes:

• Maintain Statement 3 as BD≅DC with the correct definition of midpoint.
• In Statement 5, retain the reason as "SAS Congruence Theorem" because that's labeled correctly as two sides and the included angle is used.

So statement 5 should remain as it is, and there isn’t a mistake that needs changing. This proof is indeed valid.