Question

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 Reasons1. Isosceles  △ABC  with  AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ 1. Given2.  D , the midpoint of  BC¯¯¯¯¯¯¯¯ AD¯¯¯¯¯¯¯¯  through points  A  and D 2. Construction3.  BD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ 3. Definition of midpoint4.  AD¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯ 4. Reflexive property of congruence5.  △ABD≅△ACD 5. SAS Congruence Theorem6.  ∠B≅∠C 6. CPCTC Theorem(1 point)ResponsesStatement 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 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 ."Reason 3 has a mistake. It should say “Definition of angle bisector.” 

Answer 1

Let's examine Fox's proof carefully to identify any mistakes in his reasoning and what corrections can be made.

1. Review of Statements and Reasons:
   • Statement 1: The fact that triangle ABC is isosceles with AB ≅ AC is given.
   • Statement 2: D is defined as the midpoint of BC, with AD being drawn from point A to point D.
   • Statement 3: BD ≅ DC due to the definition of midpoint.
   • Statement 4: AD ≅ AD is true by the reflexive property of congruence.
   • Statement 5: This states that triangles ABD and ACD are congruent using the SAS congruence theorem.
   • Statement 6: The conclusion drawn is that ∠B ≅ ∠C by CPCTC (corresponding parts of congruent triangles are congruent).

Critique of Reasoning:

• Checking the Lines of Reasoning:
  • In Statement 5, Fox claims congruence of triangles ABD and ACD using the SAS Theorem. The criteria for SAS (Side-Angle-Side) are satisfied because:
    • AD is shared (and therefore congruent by the reflexive property).
    • BD ≅ DC (by definition of the midpoint).
    • The angles ∠BAD and ∠CAD are not directly addressed in the proof provided and should have been established for SAS justification.
  • While the construction and midpoint definition are correct, the validity of the congruence from Statement 5 relies on angle congruences that aren't stated.

In particular, Statement 6 is accurate in stating the conclusion, but it relies on a potentially flawed Statement 5 that assumes the necessary angle congruences without explicitly mentioning or proving them.

2. Assessing Options:

   • The critiques suggested in your query focus on possible changes to Statements and Reasons:
     • Suggesting that Statement 6 should specify angles A or that Reason 5 should be sourced to SSS are misleading because the triangles in consideration must establish angle congruence or ensure that the included angle is accounted.
   • The remark on Statement 3 discussing angles could be relevant but misses the core reasoning about triangle congruence.

3. Improvement Suggestions:

   • Statement and Reason Focus: To fix the reasoning, Fox should explicitly state that ∠BAD ≅ ∠CAD due to the properties of isosceles triangles (angles opposite equal sides are congruent).
   • Upon establishing that these angles are congruent, Statement 5 would function correctly under the SAS theorem as long as the angles are identified.

Conclusion

The mistake lies in the reliance on angle congruences not explicitly stated, particularly evident in Statement 5. Fox should supply additional justification about ∠BAD being congruent to ∠CAD stemming from the properties of isosceles triangles to finalize his proof correctly. The appropriate revisions should clarify these angles to validate the triangle congruence and conclusions that follow.