Question

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

Answer 1

The proof is essentially correct, but the final step is stated too quickly and needs one small clarification.

After showing △ABD ≅ △ACD by SAS, CPCTC gives
∠ABD ≅ ∠ACD.
Because D is on BC (D is the midpoint), ray BD is the same as ray BC and ray CD is the same as ray CB, so
∠ABD = ∠ABC and ∠ACD = ∠ACB.
Therefore ∠ABC ≅ ∠ACB, i.e. ∠B ≅ ∠C.

So replace the terse Step 6 with these two lines (or add them as an extra step): (1) ∠ABD ≅ ∠ACD by CPCTC, and (2) since B, D, C are collinear, ∠ABD = ∠ABC and ∠ACD = ∠ACB, hence ∠B ≅ ∠C.