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

Answer 1

The critique of Fox's proof can be broken down as follows:

1. Statement 1: Correctly states the given information.
2. Statement 2: Correctly identifies the construction of point D as the midpoint of segment BC.
3. Statement 3: This statement asserts that $BD\cong DC$ based on the definition of a midpoint. This part is correct.
4. Statement 4: The statement that $AD\cong AD$ uses the reflexive property correctly.
5. Statement 5: States that triangles $\mathrm{△}ABD$ and $\mathrm{△}ACD$ are congruent. This is correct based on the SAS (Side-Angle-Side) Congruence Theorem since they have two sides and the included angle equal.
6. Statement 6: This states that $\mathrm{\angle }B\cong \mathrm{\angle }C$, which follows from the congruence of the triangles established in the previous step (CPCTC - Corresponding Parts of Congruent Triangles are Congruent). This is also correct.

Mistakes in Reasoning:

From the options provided, Reason 3 is the one that has a mistake. It should not address angles, but rather the segments.

Correction:

To fix the mistake in Reason 3:

• Reason 3 can remain as is: “$BD\cong DC$” as it correctly reflects the definition of a midpoint.
• However, the critique that should address the angle statements is targeted towards Reason 6. It does not have a mistake, but the confusion might arise if one assumes we’re proving different angles that aren't required in this scenario.

Conclusion:

The confusion appears to be in how the statements were interpreted. The correct assessment is that all statements are mostly valid. The critique about statement 3 being about angle congruence doesn’t apply as it concerns segment lengths and respects the definition set by midpoint.