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.”
Answers
GPT-4o mini
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.
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.