Question

Question

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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 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.”
Reason 3 has a mistake. It should say “Definition of angle bisector.”

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.”
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Answer 1

In analyzing Fox's proof of the Base Angles Theorem, let's review the proof step by step and identify any mistakes:

1. The Given statement correctly states that triangle ABC is isosceles with \( AB \cong AC \).
2. In the Construction step, Fox correctly identifies D as the midpoint of \( BC \) and states he has constructed the line segment \( AD \).
3. Statement 3 asserts \( BD \cong DC \) which is correct because D is the midpoint of \( BC \).
4. Statement 4 claims \( AD \cong AD \), which is true due to the reflexive property of congruence and does not contain a mistake.
5. Statement 5 invokes the SAS Congruence Theorem. Here, Fox states that the two triangles \( \triangle ABD \) and \( \triangle ACD \) are congruent based on:
   • \( AB \cong AC \) (Given)
   • \( AD \cong AD \) (Reflexive property)
   • \( BD \cong DC \) (From the midpoint) This indeed shows that the two triangles are congruent, hence SAS is appropriate.
6. Finally, statement 6 concludes that \( \angle B \cong \angle C \), which correctly follows from the corresponding parts of congruent triangles (CPCTC).

However, the critique focuses on:

Statement 3 has a mistake. It should say “∠BAD≅∠CAD.”

The correct relation involves angles formed at point A when AD is drawn as an angle bisector. Therefore, this statement should indeed clarify that angles \( \angle BAD \) and \( \angle CAD \) are congruent due to AD being the angle bisector.

To fix the mistake, Fox should state in statement 3 that \( \angle BAD \cong \angle CAD \) and justify this with the definition of an angle bisector. The step statements should directly communicate the correct identification of angles rather than segments.

So the improvements to the proof should clarify the angle relationships rather than segment relationships at certain steps. The corrections are needed for clarity and accuracy in the geometric relationships involved in the proof.