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 .
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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 .
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¯¯¯¯¯¯¯¯.
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The statement that should fill in the blank is:
Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD.
This makes sense in the context of the proof because the angle bisector BD will create two angles ∠ABD and ∠CBD that are congruent, which is an important step in proving the congruence of triangles ABD and CBD using the SAS Congruence Theorem.
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