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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."
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."
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GPT-4o mini
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 \( \triangle ABD \) and \( \triangle ACD \) 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 \( \angle B \cong \angle C \), 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.**
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 \( \triangle ABD \) and \( \triangle ACD \) 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 \( \angle B \cong \angle C \), 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.**
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