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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

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

Reason 5 has a mistake. It should say “SSS Congruence Theorem.”
Reason 5 has a mistake. It should say “SSS Congruence Theorem.”

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 ."
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Answer 1

In Fox's proof, it seems the primary mistake lies within the reasoning given for Statement 3. The correct interpretation of Statement 3 should be related to the segments BD and DC, which arise from point D being the midpoint of segment BC. Therefore, rather than referencing an angle bisector, the reasoning should clarify that the segments are congruent due to the definition of a midpoint.

Here's a critique of the relevant statements and their errors:

Mistakes:

• Reason 3: The statement correctly identifies that D is the midpoint, but the reasoning given is confusing because it incorrectly implies that it involves angles. It should clearly express that because D is the midpoint, segments BD and DC are congruent, not angles.

Suggested Correction:

• Reason 3 should be: "Definition of midpoint," and it should explicitly state that BD ≅ DC due to D being the midpoint of BC.

Clarified Proof:

Here's how the proof can be modified for clarity:

• 3. BD ≅ DC (statement) Reason: "Definition of midpoint."

By correcting the language in Statement 3 and explicitly referring to segment congruence, the argument would more clearly establish the basis necessary for applying the SAS Congruence Theorem in Statement 5. The other statements in this proof align correctly with their corresponding reasons.

It’s worth noting that the conclusions reached by the proof regarding angles B and C congruence (in Statement 6) are ultimately correct based on the established congruency of triangles ABD and ACD, which follow from the corrected statements.