Question
An isosceles triangle with a sideways orientation is marked counter-clockwise from the right as upper A upper B upper C. Sides upper A upper B and upper A upper C are marked with a single congruent tick mark. A line bisects the triangle.
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.”
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.”
Answers
GPT-4o mini
The Base Angles Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent. In this proof, Fox is attempting to use the properties of an isosceles triangle to prove that the base angles (angles B and C) are congruent.
Let's critique the provided proof and identify the mistakes:
1. **Statement 6**: The error lies in the conclusion at this step. Statement 6 claims "∠B≅∠C". This is, in fact, the correct conclusion according to the Base Angles Theorem since angles B and C are opposite to the congruent sides AB and AC. Thus, there is no mistake here.
2. **Statement 3**: This statement is correct as stated. It says "BD≅DC" which accurately reflects the definition of a midpoint.
3. **Reason 5**: The reason given as "SAS Congruence Theorem" corresponds correctly to the triangle congruence criterion used. Since we have two sides (AD and AB or AC) that are congruent and the angle (∠BAD and ∠CAD) between them is equal, SAS is indeed applicable here.
4. **Reason 3**: This is also correct in citing the "Definition of midpoint" which states that a midpoint divides a segment into two equal parts.
Overall, Fox's proof is mostly correct, **but here is the primary critique**:
The confusion might arise if anyone misreads the triangles involved or mixes up angles A versus angles B and C. To clarify, when using CPCTC (Corresponding Parts of Congruent Triangles are Congruent), it's important to clearly establish what angles correspond to what sides.
In the end, the critique would be:
- **Statement 6 is actually correct**, as it properly states that angles B and C are congruent.
- There are no mistakes in the reasons as stated, they correctly follow the definitions and properties needed.
Thus, it appears the critique statements provided do not address an actual mistake in Fox's proof. If there’s confusion about angles, it’s worth reinforcing the importance of marking angles properly on the triangle diagram and clearly stating Corresponding Parts.
In summary, the reasoning given by Fox to conclude that ∠B ≅ ∠C is valid, and there doesn't appear to be a mistake in his proof as presented. If any confusion persists, it's primarily about clarification, not a fundamental error in logic or reasoning.
Let's critique the provided proof and identify the mistakes:
1. **Statement 6**: The error lies in the conclusion at this step. Statement 6 claims "∠B≅∠C". This is, in fact, the correct conclusion according to the Base Angles Theorem since angles B and C are opposite to the congruent sides AB and AC. Thus, there is no mistake here.
2. **Statement 3**: This statement is correct as stated. It says "BD≅DC" which accurately reflects the definition of a midpoint.
3. **Reason 5**: The reason given as "SAS Congruence Theorem" corresponds correctly to the triangle congruence criterion used. Since we have two sides (AD and AB or AC) that are congruent and the angle (∠BAD and ∠CAD) between them is equal, SAS is indeed applicable here.
4. **Reason 3**: This is also correct in citing the "Definition of midpoint" which states that a midpoint divides a segment into two equal parts.
Overall, Fox's proof is mostly correct, **but here is the primary critique**:
The confusion might arise if anyone misreads the triangles involved or mixes up angles A versus angles B and C. To clarify, when using CPCTC (Corresponding Parts of Congruent Triangles are Congruent), it's important to clearly establish what angles correspond to what sides.
In the end, the critique would be:
- **Statement 6 is actually correct**, as it properly states that angles B and C are congruent.
- There are no mistakes in the reasons as stated, they correctly follow the definitions and properties needed.
Thus, it appears the critique statements provided do not address an actual mistake in Fox's proof. If there’s confusion about angles, it’s worth reinforcing the importance of marking angles properly on the triangle diagram and clearly stating Corresponding Parts.
In summary, the reasoning given by Fox to conclude that ∠B ≅ ∠C is valid, and there doesn't appear to be a mistake in his proof as presented. If any confusion persists, it's primarily about clarification, not a fundamental error in logic or reasoning.