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.
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?
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Responses
Finally, ∠A≅∠C by the Base Angles Theorem.
Finally, angle upper A congruent to angle upper C by the Base Angles Theorem.
Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯.
Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so Modifying above upper A upper B with bar congruent to Modifying above upper B upper C with bar .
Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠B.
Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so angle upper A congruent to angle upper B .
Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C.
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The statement that should be placed at the end of the proof is:
Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C.
This follows logically from the earlier steps of the proof, which establish that triangles ABD and CBD are congruent. Since corresponding angles of congruent triangles are also congruent, we conclude that the angles at A and C are congruent as well.
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