Question

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Triangle Proofs Quick Check
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Use the image to answer the 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.
▼Click for Long Desc
A horizontal line upper A upper D is drawn from vertex A on the right to midpoint upper D on the left side base upper B upper C. Newly formed line segments upper B upper D and upper D upper C are both marked with double congruent tick marks.


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


Identify Mistake in Steps to Prove Congruency Between Angles
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 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 ."

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

Answer 1

Reason 5 is wrong. It should say "SSS Congruence Theorem."

Justification: Triangles ABD and ACD have three pairs of congruent sides (AB ≅ AC, BD ≅ DC, AD ≅ AD), so SSS, not SAS, is the correct congruence reason.