consider the following proof of the base angles theorem. which statement should fill in the blank?

proof: given isosceles $ \triangle a b c $ with $ \overline{a b} \cong \overline{b c} $, i can construct $ \overleftrightarrow{b d} $, the angle bisector of $ \angle b $. _______________. i also know that line segments are congruent to themselves, so $ \overline{b d} \cong \overline{b d} $ by the reflexive property of congruence. i now have two pairs of sides and an included angle that are congruent, so i know that $ \triangle a b d \cong \triangle c b d $ by the sas congruence theorem. finally, corresponding parts of congruent triangles are congruent by the cpctc theorem, so $ \angle a \cong \angle c $.

(1 point)

$ \bigcirc $ then, by the definition of an isosceles triangle, i know that $ \overline{a b} \cong \overline{c a} $

$ \bigcirc $ then, by the definition of a midpoint, i know that $ \overline{a d} \cong \overline{d c} $

$ \bigcirc $ then, by the definition of an angle bisector, i know that $ \angle a b d \cong \angle c b d $

$ \bigcirc $ then, by the definition of an angle bisector, i know that $ \angle b a c \cong \angle b c a $

Question

consider the following proof of the base angles theorem. which statement should fill in the blank?

proof: given isosceles $ \triangle a b c $ with $ \overline{a b} \cong \overline{b c} $, i can construct $ \overleftrightarrow{b d} $, the angle bisector of $ \angle b $. _______________. i also know that line segments are congruent to themselves, so $ \overline{b d} \cong \overline{b d} $ by the reflexive property of congruence. i now have two pairs of sides and an included angle that are congruent, so i know that $ \triangle a b d \cong \triangle c b d $ by the sas congruence theorem. finally, corresponding parts of congruent triangles are congruent by the cpctc theorem, so $ \angle a \cong \angle c $.

(1 point)

$ \bigcirc $ then, by the definition of an isosceles triangle, i know that $ \overline{a b} \cong \overline{c a} $

$ \bigcirc $ then, by the definition of a midpoint, i know that $ \overline{a d} \cong \overline{d c} $

$ \bigcirc $ then, by the definition of an angle bisector, i know that $ \angle a b d \cong \angle c b d $

$ \bigcirc $ then, by the definition of an angle bisector, i know that $ \angle b a c \cong \angle b c a $

Accepted Answer

# Brief Explanations:
The problem is about proving the Base Angles Theorem in a triangle. The key is using the properties related to the angle - bisector. Since $ \overrightarrow{BD}$ is the angle bisector of $ \angle B$, by the definition of an angle bisector, it divides $ \angle B$ into two congruent angles. That is $ \angle ABD\cong\angle CBD$. This is necessary for the $SAS$ (Side - Angle - Side) congruence criterion as we already have $ \overline{AB}\cong\overline{BC}$ (given) and $ \overline{BD}\cong\overline{BD}$ (reflexive property).
# Answer:
Then, by the definition of an angle bisector, I know that $ \angle ABD\cong\angle CBD$.

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