triangle proofs quick check
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consider the following proof of the base angles theorem. which statement should fill in the blank?
proof given isosceles △abc 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)
then, by the definition of an angle bisector, i know that $ \angle b a c \cong \angle b c a $
then, by the definition of an angle bisector, i know that $ \angle a b d \cong \angle c b d $
then, by the definition of a midpoint, i know that $ \overline { a d } \cong \overline { d c } $
then, by the definition of an isosceles triangle, i know that $ \overline { a b } \cong \overline { c a } $

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Accepted Answer

# Answer:
B. Then, by the definition of an angle bisector, I know that $ \angle ABD\cong\angle CBD $

# Explanation:
## Step1: Analyze the role of angle bisector
An angle bisector divides an angle into two congruent angles. Here, $ \overrightarrow{BD} $ is the angle bisector of $ \angle B$.
## Step2: Apply the definition of angle bisector
By the definition of an angle bisector, if $ \overrightarrow{BD} $ bisects $ \angle B$ in $ \triangle ABC$, then $ \angle ABD$ and $ \angle CBD$ are the two angles formed by the bisector. So, $ \angle ABD\cong\angle CBD$

Option A is wrong because $ \angle BAC$ and $ \angle BCA$ are not related to the angle - bisector $ \overrightarrow{BD}$ in terms of the angle - bisector definition. Option C is wrong because there is no information given about $ D$ being a mid - point (a mid - point would make $ \overline{AD}\cong\overline{DC}$, but we are dealing with an angle bisector here). Option D is wrong because $ \overline{AB}\cong\overline{BC}$ is given (the triangle is isosceles with $ \overline{AB}\cong\overline{BC}$), not $ \overline{AB}\cong\overline{CA}$

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze the role of angle bisector

Step2: Apply the definition of angle bisector

Answer: