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 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 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
.
1 answer
The correct statement to fill in the blank in the proof is:
"Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD."
This statement connects the fact that \( BD \) is the angle bisector of \( \angle ABC \) and directly supports the reasoning that leads to the conclusion that triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent by the SAS Congruence Theorem.
