The following is the proof of the Base Angles Theorem. Fill in the blank with the correct statement. PROOF: 1. Given isosceles ΔABC with AB¯¯¯¯¯¯¯¯≅BC,¯¯¯¯¯¯¯¯¯ I can construct BD←→ , the angle bisector of ∠B. 2. ________________________________ 3. I also know that line segments are congruent to themselves, so BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯ by the reflexive property of congruence. 4. I now have two pairs of sides and an included angle that are congruent, so I know that ΔABD ≅ΔCBD by the SAS congruence theorem. 5. 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 ∠BAC ≅∠BCA . Then, by the definition of an isosceles triangle, I know that AB¯¯¯¯¯¯¯¯≅CA¯¯¯¯¯¯¯¯ . Then, by the definition of an isosceles triangle, I know that line segment cap A cap b is congruent to line segment cap c cap A. Then, by the definition of a midpoint, I know that AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ . Then, by the definition of a midpoint, I know that line segment cap A cap d is congruent to line segment cap d cap c. Then, by the definition of an angle bisector, I know that ∠ABD ≅∠CBD .

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 statement aligns with the steps in the proof, as it establishes that the angle bisector divides angle B into two congruent angles, which is crucial for applying the SAS (Side-Angle-Side) Congruence Theorem in the subsequent steps.