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

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 ."

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 ."

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.”

Answer 1

In Fox's proof, the mistake is primarily found in **Statement 3** and its reasoning.

**Critique:**

- **Statement 3:** It states that "BD≅DC" which is indeed correct based on the definition of the midpoint. However, in the context of the Base Angles Theorem, Fox should focus on the angles formed with the segments - specifically, he should be asserting something related to angles rather than focusing solely on the lengths of those segments. It's unclear if he had identified the angles formed.

- **Statement 6:** There is no mistake in this statement regarding the angles ∠B and ∠C, as it accurately states what is being proven. The use of CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is also correctly applied.

- **Reason 5:** There is no mistake here. The SAS (Side-Angle-Side) Congruence Theorem is appropriately applied once it has been established that two sides and the included angle are congruent.

### Suggested Fix:

To fix the mistake in Statement 3 and keep the logic consistent, Fox should:

1. **Correct Statement 3**: Instead of stating the congruence of segments, he could either assert about the relevant angles formed by the construction of the midpoint or confirm the congruence of the segments while ensuring to include the angles.
2. For example, he might say: "∠BAD ≅ ∠CAD" and then use the definition of a ray or an angle bisector instead of just stating segment congruence.

To summarize, an ideal corrected version of that part of the proof would be:

**Statement 3:** ∠BAD ≅ ∠CAD (by the construction of AD as the angle bisector).

This keeps the logical structure intact and directly supports the proof of the Base Angles Theorem (∠B ≅ ∠C).