Question

PROOF: Given isosceles △ABC with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ , I can construct BD←→ , the angle bisector of ∠B . Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD . 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. What statement should be placed at the end of the proof? (1 point) Responses Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ . Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so Modifying above upper A upper B with bar congruent to Modifying above upper B upper C with bar . Finally, ∠A≅∠C by the Base Angles Theorem. Finally, angle upper A congruent to angle upper C by the Base Angles Theorem. Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C . Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so angle upper A congruent to angle upper C . Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠B .

Answer 1

To conclude your proof, the appropriate statement to place at the end is:

"Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A ≅ ∠C."

This statement aligns with the conclusion drawn from the congruence of triangles $△ABD$ and $△CBD$ established through the SAS Congruence Theorem. The Base Angles Theorem states that in an isosceles triangle, the angles opposite the equal sides are also equal. Thus, since you have established that the triangles are congruent, the angles $\angle A$ and $\angle C$ must be congruent as well.