ox 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 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 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 ." Statement 6 has a mistake. It should say “∠A≅∠C ."

Answer 1

The key mistake in the proof is in Reason 5. The proof states that \( \triangle ABD \cong \triangle ACD \) using SAS (Side-Angle-Side) congruence theorem, but it does not correctly support this statement with appropriate reasoning.

Reason 5 should indeed reference the SAS Congruence Theorem, which requires two sides and the included angle to be congruent. In this scenario, angle \( \angle A \) is not mentioned or included to prove that the triangles are congruent.

Here's how to fix the mistake:

1. To correctly apply SAS, you should confirm that:
   • \( AB \cong AC \) (given)
   • \( AD \) is common to both triangles (\( AD \cong AD \) by the reflexive property)
   • \( BD \cong DC \) (from Reason 3: Definition of midpoint)

The triangles would then be considered congruent as follows for the correct reasoning:

Corrected Reason 5: \( \triangle ABD \cong \triangle ACD \) (by SAS Congruence Theorem, where \( AB \cong AC, AD \cong AD, \) and \( BD \cong DC \)).

As a result, the conclusion \( \angle B \cong \angle C \) (Reason 6) would then correctly follow from the congruence of the triangles (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

To sum up, the main issue was the application and justification of the triangle congruence theorem in Reason 5. The proof should clearly establish the relationships and the included angle to apply SAS appropriately.

Answer 2

Choose an option

Answer 3

Based on the critique provided, the correct option for the mistake in the proof is:

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

However, upon further analysis, the accurate reasoning in the context of the proof is that it actually correctly applies the SAS (Side-Angle-Side) property if all criteria are stated correctly, but there seems to be a misunderstanding in the choices provided.

If based purely on your given options, I would select:

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

This option highlights that the reasoning for the triangle congruence theorem is misused, but it's worth noting that its application could be valid if done correctly, specifically by validating the included angle. Still, since the options focus on a mistake, this answer aligns best with the intent of the critique.