Adam wants to prove the Base Angles Theorem. On which statement or reason did he make a mistake? Also, how can fix his mistake?

Question

Given: Isosceles ∆ABC with AB ≅ AC
Prove: ∠B ≅ ∠C

Statements

1. Isosceles ∆ABC with AB ≅ AC
2. D, midpoint of BC, AD through points A and D
3. BD ≅ DC
4. AD ≅ AD
5. ∆ABD ≅ ∆ACD
6. ∠B ≅ ∠C

Reasons
1. Given
2. Construction
3. Definition of midpoint
4. Reflexive property of congruence
5. SAS Congruence Theorem
6. CPCTC Theorem

a. Statement 3 has a mistake. It should say ∠BAD ≅ ∠CAD
b. Reason 3 is incorrect. It should say "Definition of angle bisector."
c. Statement 6 is incorrect. It should say "∠A ≅ ∠C."
d. Reason 5 is incorrect. It should say "SSS Congruence Theorem"

Answer 1

To analyze the proof step by step, we will evaluate the statements and reasons provided.

1. Statement 1 is correct: It states the given information about triangle ∆ABC being isosceles with \( AB ≅ AC \).

2. In Statement 2, the construction of point D, the midpoint of BC, is appropriate.

3. Statement 3 asserts \( BD ≅ DC \) which is correct according to the definition of a midpoint.

4. Statement 4 asserts \( AD ≅ AD \), which is also correct by the reflexive property.

5. Statement 5 states that \( ∆ABD ≅ ∆ACD \) and uses the SAS Congruence Theorem. For SAS, we need two sides and the included angle of one triangle to be congruent to two sides and the included angle of the other triangle. Since \( BD ≅ DC \) (Statement 3), \( AB ≅ AC \) (from Statement 1), and \( AD ≅ AD \) (Statement 4), the angle at A (∠BAD and ∠CAD) is the included angle for both triangles.

   Therefore, the application of the SAS Congruence Theorem is valid given the sides and the included angle are congruent.

6. In Statement 6, it follows that \( ∠B ≅ ∠C \) due to the previously established congruence of the two triangles (ABD and ACD), which is justified by the corresponding parts of congruent triangles (CPCTC).

Reviewing the options for mistakes:

• a. Statement 3 is correct as it is, \( BD ≅ DC \), related to the midpoint, not angles.
• b. Reason 3 is correct as it states the definition of the midpoint as \( BD ≅ DC \).
• c. Statement 6 is correct as it is, asserting \( ∠B ≅ ∠C \) based on the triangle congruence.
• d. Reason 5 is also correct. The triangles are congruent by SAS, not SSS.

After analyzing, none of the provided options (a, b, c, d) actually highlight a real mistake in Adam's reasoning. Therefore, Adam has made no mistakes based on the information given, and the proof is valid as it stands.

If you need clarification on any specific part of this proof or have additional questions, please let me know!