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Given: [tex]\overline{A B}=\overline{B C}[/tex]

Prove: The base angles of an isosceles triangle are congruent.

The two-column proof with missing statement proves the base angles of an isosceles triangle are congruent:

\begin{tabular}{|l|l|}
\hline
\multicolumn{1}{|c|}{ Statement } & \multicolumn{1}{c|}{ Reason } \\
\hline
1. [tex]\overline{B D}[/tex] is an angle bisector of [tex]\angle A B C[/tex] & 1. By construction \\
\hline
2. [tex]\angle A B D = \angle C B D[/tex] & 2. Definition of an angle bisector \\
\hline
3. [tex]\overline{B D} = \overline{B D}[/tex] & 3. Reflexive property \\
\hline
4. [tex]\triangle A B D = \triangle C B D[/tex] & 4. Side-Angle-Side (SAS) postulate \\
\hline
5. [tex]\angle B A C = \angle B C A[/tex] & 5. CPCTC \\
\hline
\end{tabular}

Which statement can be used to fill in the numbered blank space?

A. [tex]\triangle D A B = \triangle D B C[/tex]
B. [tex]\triangle A B D = \triangle A B C[/tex]
C. [tex]\triangle A B C = \triangle C B D[/tex]
D. [tex]\triangle A B D = \triangle C B D[/tex]



Answer :

GinnyAnswer
Let's complete the proof step by step:

Given:
- [tex]\( \overline{AB} = \overline{BC} \)[/tex]

We are to prove that the base angles of an isosceles triangle are congruent, specifically for [tex]\( \triangle ABC \)[/tex] where [tex]\( AB = BC \)[/tex].

1. [tex]\( \overline{BD} \)[/tex] is an angle bisector of [tex]\( \angle ABC \)[/tex].
Reason: by Construction

2. [tex]\( \angle ABD = \angle CBD \)[/tex].
Reason: Definition of an Angle Bisector

3. [tex]\( \overline{BD} = \overline{BD} \)[/tex].
Reason: Reflexive Property

At this stage, we need to show that [tex]\( \triangle ABD \)[/tex] is congruent to [tex]\( \triangle CBD \)[/tex] using the given information.

- We have [tex]\( \overline{AB} = \overline{BC} \)[/tex] (Given).
- We have [tex]\( \angle ABD = \angle CBD \)[/tex] (Definition of an Angle Bisector).
- We have [tex]\( \overline{BD} = \overline{BD} \)[/tex] (Reflexive Property).

Putting it all together, we have:
- [tex]\( \overline{AB} = \overline{BC} \)[/tex]
- [tex]\( \angle ABD = \angle CBD \)[/tex]
- [tex]\( \overline{BD} = \overline{BD} \)[/tex]

According to the Side-Angle-Side (SAS) Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

4. [tex]\( \triangle ABD = \triangle CBD \)[/tex].
Reason: Side-Angle-Side (SAS) Postulate.

Once we have established the congruence of [tex]\( \triangle ABD \)[/tex] and [tex]\( \triangle CBD \)[/tex], we can use the property of Corresponding Parts of Congruent Triangles are Congruent (CPCTC) to show the base angles are congruent.

5. [tex]\( \angle BAC = \angle BCA \)[/tex].
Reason: CPCTC

Thus, the statement that completes the proof is:

[tex]\( \triangle ABD = \triangle CBD \)[/tex].

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