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 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 ." Statement 6 has a mistake. It should say “ angle upper A congruent to angle upper C ."

Answer 1

Let's go through Fox's proof step-by-step and identify the errors.

1. Statement 1: Given is correct.
2. Statement 2: The construction of point D as the midpoint of line segment BC is correct.
3. Statement 3: This states that \( BD \cong DC \). This is correct as it follows from the definition of midpoint.
4. Statement 4: This states \( AD \cong AD \) is correct due to the reflexive property of congruence.
5. Statement 5: The statement uses the SAS (Side-Angle-Side) Congruence Theorem, claiming that \(\triangle ABD \cong \triangle ACD\). This is correct because it is established that \(AB \cong AC\) (given), \(\angle BAD\) is common (which we have not explicitly mentioned but can be implied), and \(BD \cong DC\) (from Statement 3).
6. Statement 6: This concludes that \(\angle B \cong \angle C\) using CPCTC (Corresponding Parts of Congruent Triangles are Congruent), which is correct.

The primary error in the proof lies with the language of the statements rather than the logic itself. The critique options suggest possible mistakes at Statement 3, where it should clarify that it is referring to angles rather than the segments related to the triangle. However, the proof ultimately maintains its validity.

Critique Summary:

• Respondent Concerns: Reason 3 does not relate to the angles but rather to the segments. The identification of angles, which are relevant for proving the base angles of the isosceles triangle, should indeed be part of the proof's clear statement.

How to Fix the Mistake: To clarify Statement 3, instead of saying it relates to the segments \(BD\) and \(DC\):

• It should correctly emphasize the angles, as the angles at A are more relevant for the proof of base angles. Thus, it could say something like: "Angles \(\angle BAD \cong \angle CAD\) can be assumed for proper application of the triangle congruence."

In summary, while Fox's reasoning does appear logically sound, the articulation of some statements could be improved for clarity regarding the angles, specifically in Statement 3 and its implications for further statements.

Answer 2

what's the answer from the options given

Answer 3

Based on the options you provided, the most appropriate critique of Fox's proof is:

Reason 3 has a mistake. It should say “Definition of angle bisector.”

This option correctly highlights that Statement 3 should be focused on angles and refers to the angles created by the segments rather than just discussing the midpoints or segments directly. Although the proof uses the midpoint correctly to conclude the triangle congruence, making clear connections to the angles involved is essential for proving the Base Angles Theorem correctly.