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 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 ." 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 5 has a mistake. It should say “SSS Congruence Theorem.” Reason 5 has a mistake. It should say “SSS Congruence Theorem.” Reason 3 has a mistake. It should say “Definition of angle bisector.” Reason 3 has a mistake. It should say “Definition of angle bisector.”

Question

GPT-4o mini · Answer

Fox's proof involves showing that the base angles of an isosceles triangle are congruent. The proof itself is mostly correct, but it contains an error in the interpretation of angles and the congruence relationship.

Critique of the Reasoning:

Statement 6: The conclusion is flawed. The angle B measured in triangle ABD is not equal to angle C measured in triangle ACD since the angle notation and congruence are mistyped. In fact, in triangle ABC, you want to prove that angles B and C are congruent, not A and C.

Incorrect Statements:

Statement 6 has a mistake.
It should say “∠B ≅ ∠C.” Instead of stating a different relationship involving angle A.

How to Fix the Mistake:

To fix the mistake, Fox should ensure that once he proves triangles ABD and ACD are congruent, he clearly states that from the congruence of those triangles, the corresponding angles are also congruent.

Corrected Conclusion:

Add a step after statement 5:

After establishing that triangles ABD and ACD are congruent, you would explicitly state:
- 6. ∠B ≅ ∠C (because corresponding parts of congruent triangles are congruent).

Thus, the reasoning of Fox's proof should clearly indicate that he is showing angles B and C are congruent as the conclusion of the proof, confirming the Base Angles Theorem for triangle ABC.