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

 Then, by the definition of a midpoint, I know that AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯.
 Then, by the definition of a midpoint, I know that Modifying above upper A upper D with bar congruent to Modifying above upper D upper C with bar .

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 .

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

Answer 1

The correct statement to fill in the blank is:

"Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD."

This statement is appropriate because it establishes that the angle bisector (line segment BD in this case) creates two angles (∠ABD and ∠CBD) that are congruent, which is crucial for applying the SAS Congruence Theorem later in the proof.