We are given a two-column proof with several blanks and asked to fill in those blank spaces. Let's begin by looking at the given information and the desired outcome of the proof.
Given:& AB ≅ AC
Prove:& ∠ B ≅ ∠ C
Now, let's take a look at the statements that need to be completed one at a time.
Missing information 2
The first blank reason justifies congruence of ∠ CAD and ∠ BAD. We know this is true because of the first reason which tells us that AD is an angle bisector of ∠ CAB. By definition, it divides the angle in two congruent halves.
\begin{gathered}
\underline\textbf{Reason}\\
m\angle CAD \cong m \angle BAD\\
\bm{2.} \underline{ \, \text{by the definition of an angle bisector,} \, } \\
\end{gathered}
Missing information 3
The third statement is the given information.
\begin{gathered}
\underline\textbf{Reason}\\
\overline{AB} \cong \overline{AC}\\
\bm{3.} \underline{ \, \text{because this is given information} \, } \\
\end{gathered}
Missing information 4
The fourth statement says that DA is congruent with itself. This is known as the Reflexive Property of Congruence.
\begin{gathered}
\underline\textbf{Reason}\\
\overline{DA} \cong \overline{DA}\\
\bm{4.} \underline{ \, \text{ by the Reflexive Property of Congruence} \, } \\
\end{gathered}
Missing information 5
Since we know there are two congruent sides and the included angles are congruent in △ ADB and △ ADC, we are able to prove that these triangles are congruent by the SAS Congruence Theorem.
\begin{gathered}
\underline\textbf{Reason}\\
\triangle ADB \cong \triangle ADC \\
\bm{5.} \underline{ \, \text{ by the SAS Congruence Theorem} \, } \\
\end{gathered}
Missing information 6
Having proved that △ ADB≅ △ ADC, we can proceed to identify ∠ B and ∠ C as corresponding angles which means they are congruent.
\begin{gathered}
\underline\textbf{Reason}\\
\angle B \cong \angle C \\
\bm{6.} \underline{ \, \text{Corresponding parts are congruent} \, } \\
\end{gathered}
Final proof
Having justified all of the statements, we can go ahead and complete the two-column proof.
