Points $A, B,$ and $C$ form a triangle. Complete the statements to prove that the sum of the interior angles of $\triangle ABC$ is $180^{\circ}$.

[tex]\[
\begin{array}{|l|l|}
\hline \text{Statement} & \text{Reason} \\
\hline \text{Points } A, B,\text{ and } C \text{ form a triangle.} & \text{Given} \\
\hline \text{Let } \overline{DE} \text{ be a line passing through } B \text{ and parallel to } \overline{AC}. & \text{Definition of parallel lines} \\
\hline \angle 3 \approx \angle 5 \text{ and } \angle 1 \approx \angle 4 & \text{Alternate interior angles} \\
\hline m \angle 1 = m \angle 4 \text{ and } m \angle 3 = m \angle 5 & \text{Corresponding angle measures} \\
\hline m \angle 4 + m \angle 2 + m \angle 5 = 180^{\circ} & \text{Angle addition and definition of a straight line} \\
\hline m \angle 1 + m \angle 2 + m \angle 3 = 180^{\circ} & \text{Substitution} \\
\hline
\end{array}
\][/tex]

To prove that the sum of the interior angles of $\triangle ABC$ is $180^\circ$, we can proceed with the following detailed steps:

\begin{tabular}{|l|l|}
\hline
Statement & Reason \\
\hline
Points $A, B$, and $C$ form a triangle. & given \\
\hline
Let $\overline{DE}$ be a line passing through $B$ and parallel to $\overline{AC}$. & definition of parallel lines \\
\hline
$\angle 3 \approx \angle 5$ and $\angle 1 \approx \angle 4$ & alternate interior angles theorem \\
\hline
$m\angle 1 = m\angle 4$ and $m\angle 3 = m\angle 5$ & corresponding angles in parallel lines are equal \\
\hline
$m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ$ & angle addition and definition of a straight line \\
\hline
$m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$ & substitution \\
\hline
\end{tabular}

### Explanation:
1. Given Statement: Points $A, B$, and $C$ form a triangle.

2. Parallel Line Construction: We draw a line $\overline{DE}$ through point $B$ which is parallel to $\overline{AC}$.

3. Alternate Interior Angles Theorem: Because $\overline{DE}$ is parallel to $\overline{AC}$, by the alternate interior angles theorem:
- $\angle 3 \approx \angle 5$
- $\angle 1 \approx \angle 4$

4. Equal Angles: These congruences translate into equal angle measures.
- Hence, $m\angle 1 = m\angle 4$
- And $m\angle 3 = m\angle 5$

5. Straight Line Angle Sum: Considering the angles on line $\overline{DE}$:
- The angles on a straight line sum to $180^\circ$.
- Therefore, $m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ$

6. Substitution: Using $m\angle 1 = m\angle 4$ and $m\angle 3 = m\angle 5$:
- Substitute $m\angle 4$ and $m\angle 5$ with $m\angle 1$ and $m\angle 3$ respectively.
- It follows that $m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$.

This completes the proof that the sum of the interior angles of [tex]$\triangle ABC$[/tex] is [tex]$180^\circ$[/tex].