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].
