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

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

Answer 1

The mistake in Fox's proof is found in Reason 5. The reasoning states that "△ABD≅△ACD" is proven by the "SAS Congruence Theorem," but the correct application here is actually based on the "SSS Congruence Theorem."

To fix this mistake, the statement should correctly indicate that the triangles are congruent by the SSS Congruence Theorem because:

1. AB and AC are congruent (given),
2. AD is common to both triangles (from the reflexive property),
3. BD and CD are congruent since D is the midpoint of BC.

Thus, the correct reasoning in Step 5 should read:

"5. △ABD≅△ACD by SSS Congruence Theorem."

The other statements and reasons in Fox's proof appear to be correct given the isosceles triangle conditions and the construction of the midpoint and angle bisector.