PROOF: Given isosceles ABC with AB # BC, I can construct BD, the angle bisector of L B. Then, by the definition of an angle bisector, I know that LABD ~ 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 AABD ~ ACBD by the SAS Congruence Theorem.

Question

PROOF: Given isosceles ABC with AB # BC, I can construct BD, the angle bisector of L B. Then, by the definition of an angle bisector, I know that LABD ~ 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 AABD ~ ACBD by the SAS Congruence Theorem.
What statement should be placed at the end of the proof?

GPT-4o mini · Answer

To conclude the proof, you should have a statement that encapsulates the main result or implication of your derivation. In this case, since you have shown that triangles ABD and CBD are similar (or congruent), you need to summarize this outcome effectively. A suitable statement could be:

"Thus, triangles ABD and CBD are similar (or congruent), which implies that the segments AB and BC are proportional, confirming that triangle ABC has the properties defined by its isosceles nature."

Alternatively, if your proof is intended to demonstrate the congruence of these triangles, you could conclude with:

"Therefore, triangles ABD and CBD are congruent by the SAS Congruence Theorem."

Make sure the final statement reflects the goal of your proof regarding the relationships within the triangles or any specific properties you aim to demonstrate.