Question

An isosceles triangle is marked clockwise from the lower left vertex as upper A upper B upper C. The sides upper A upper B and upper B upper C are marked with single congruent tick marks.
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 BD←→ , the angle bisector of ∠B . _____________________. 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. Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C .

(1 point)
Responses

Then, by the definition of an angle bisector, I know that ∠BAC≅∠BCA.
Then, by the definition of an angle bisector, I know that angle upper B upper A upper C congruent to angle upper B upper C upper A .

Then, by the definition of an isosceles triangle, I know that AB¯¯¯¯¯¯¯¯≅CA¯¯¯¯¯¯¯¯.
Then, by the definition of an isosceles triangle, I know that Modifying above upper A upper B with bar congruent to Modifying above upper C upper A with bar .

Answer 1

The best statement to fill in the blank in the proof is:

**Then, by the definition of an angle bisector, I know that ∠BAC≅∠BCA.**

This statement correctly connects the first part of the proof regarding the construction of the angle bisector and aligns with the overall goal of showing that the angles ∠A and ∠C are congruent.