Use the image to answer the question. 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)ResponsesThen, 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 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 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 .
1 answer
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 shows that the angle bisector BD creates two congruent angles (∠ABD and ∠CBD) from the given isosceles triangle △ABC, which is essential for applying the SAS (Side-Angle-Side) congruence theorem in the subsequent steps of the proof.
