Question

onsider the following proof of the Base Angles Theorem. Which statement should fill in the blank?

PROOF: Given isosceles △ABC
△
𝐴
𝐵
𝐶
with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯
𝐴
𝐵
¯
≅
𝐵
𝐶
¯
, I can construct BD←→
𝐵
𝐷
↔
, the angle bisector of ∠B
∠
𝐵
. _____________________. I also know that line segments are congruent to themselves, so BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯
𝐵
𝐷
¯
≅
𝐵
𝐷
¯
by the reflexive property of congruence. I now have two pairs of sides and an included angle that are congruent, so I know that △ABD≅△CBD
△
𝐴
𝐵
𝐷
≅
△
𝐶
𝐵
𝐷
by the SAS Congruence Theorem. Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C
∠
𝐴
≅
∠
𝐶
.

(1 point)
Responses

Then, by the definition of an angle bisector, I know that ∠BAC≅∠BCA
∠
𝐵
𝐴
𝐶
≅
∠
𝐵
𝐶
𝐴
.
Then, by the definition of an angle bisector, I know that angle upper B upper A upper C congruent to angle upper B upper C upper A .

 Then, by the definition of a midpoint, I know that AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯
𝐴
𝐷
¯
≅
𝐷
𝐶
¯
.
 Then, by the definition of a midpoint, I know that Modifying above upper A upper D with bar congruent to Modifying above upper D upper C with bar .

Then, by the definition of an isosceles triangle, I know that AB¯¯¯¯¯¯¯¯≅CA¯¯¯¯¯¯¯¯
𝐴
𝐵
¯
≅
𝐶
𝐴
¯
.
Then, by the definition of an isosceles triangle, I know that Modifying above upper A upper B with bar congruent to Modifying above upper C upper A with bar .

Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD
∠
𝐴
𝐵
𝐷
≅
∠
𝐶
𝐵
𝐷
.

Answer 1

The appropriate statement to fill in the blank is:

**Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD.**

This choice confirms that angle BD bisects angle ABC, establishing that the two angles (∠ABD and ∠CBD) formed by the bisector are congruent before proceeding with the proof. It aligns with the given information and the logical progression of the proof that follows.