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 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¯¯¯¯¯¯¯¯. 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.
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The most suitable statement to fill in the blank in the proof of the Base Angles Theorem is:
"Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD."
This statement logically follows the construction of the angle bisector BD and pertains to the property of angle bisectors, which is essential to demonstrate that the triangles ABD and CBD have two angles congruent, leading to the application of the SAS Congruence Theorem.
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