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 D , the midpoint of AC¯¯¯¯¯¯¯¯ . By the definition of a midpoint, I know that AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ . I can also construct BD←→ through points B and D . Because any line segment is congruent to itself by the reflexive property of congruence, BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯ . __________________________. Corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C . (1 point) Responses I now have three pairs of sides congruent, so △ABD≅△CBD by the SAS Congruence Theorem I now have three pairs of sides congruent, so triangle upper A upper B upper D congruent to triangle upper C upper B upper D by the SAS Congruence Theorem I also know that △ABC≅△ABC by the reflexive property of congruence I also know that triangle upper A upper B upper C congruent to triangle upper A upper B upper C by the reflexive property of congruence I now have three pairs of sides congruent, so △ABD≅△CBD by the SSS Congruence Theorem I now have three pairs of sides congruent, so triangle upper A upper B upper D congruent to triangle upper C upper B upper D by the SSS Congruence Theorem I also know that by the definition of an angle bisector, ∠ABD≅∠CBD

Answer 1

The statement that should fill in the blank is:

"I now have three pairs of sides congruent, so triangle \( \triangle ABD \cong \triangle CBD \) by the SAS Congruence Theorem."

This statement appropriately follows from the earlier statements where it mentions the congruences of sides \( AB \), \( BD \), and \( AD \) for triangle \( ABD \) and sides \( BC \), \( BD \), and \( DC \) for triangle \( CBD \). Thus, it leads directly to the conclusion that the triangles are congruent, allowing us to use CPCTC to conclude that \( \angle A \cong \angle C \).