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  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)ResponsesI 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 TheoremI 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 also know that by the definition of an angle bisector, ∠ABD≅∠CBDI also know that by the definition of an angle bisector,  angle upper A upper B upper D congruent to angle upper C upper B upper DI now have three pairs of sides congruent, so △ABD≅△CBD by the SSS Congruence Theorem

Answer 1

The appropriate statement to fill in the blank is:

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

This correctly follows the earlier statements, which indicate that two sides are congruent from the isosceles triangle and one side (BD) is being referenced. This proves that the two triangles \(ABD\) and \(CBD\) are congruent.