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.

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?

GPT-4o mini · Answer

At the end of the proof, you should write a statement that concludes what you have proven about triangle \( ABC \). Since you have established that \( \triangle ABD \cong \triangle CBD \), you can conclude that the base angles \( \angle A \) and \( \angle C \) are congruent because corresponding parts of congruent triangles are congruent (CPCTC).

Thus, the statement to place at the end of the proof could be:

"Therefore, \( \angle A \cong \angle C \), which proves that the base angles of isosceles triangle \( ABC \) are congruent."