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 .

Question

Possible Answers:
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 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 ∠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 . Then, by the definition of a midpoint, I know that AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯.

Answer 1

The correct statement to fill in the blank is:

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

This fits logically into the proof as it establishes that since BD is the angle bisector of ∠B, it divides the angle into two equal parts, which is essential for concluding that the triangles ABD and CBD are congruent.