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Question

An isosceles triangle is marked clockwise from the lower left vertex as upper A upper B upper C. The sides upper A upper B and upper B upper C are marked with single congruent tick marks.

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

PROOF: Given isosceles △ABC
with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯
, I can construct BD←→
, the angle bisector of ∠B
. _____________________. I also know that line segments are congruent to themselves, so BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯
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
by the SAS Congruence Theorem. Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C
.

(1 point)
Responses

 Then, by the definition of a midpoint, I know that AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯
.
 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¯¯¯¯¯¯¯¯
.
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
.
Then, by the definition of an angle bisector, I know that angle upper A upper B upper D congruent to angle upper C upper B upper D .

Then, by the definition of an angle bisector, I know that ∠BAC≅∠BCA
.
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 .
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Answer 1

The statement that should fill in the blank in the proof is:

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

This statement logically follows from the construction of line segment \( BD \) as the angle bisector of \( \angle B \), which means it divides \( \angle B \) into two equal angles \( \angle ABD \) and \( \angle CBD \).