Let's complete the statements to prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
\begin{tabular}{|l|l|}
\hline Statement & Reason \\
\hline Points [tex]$A, B$[/tex], and [tex]$C$[/tex] form a triangle. & given \\
\hline Let [tex]$\overline{D E}$[/tex] be a line passing through [tex]$B$[/tex] and parallel to [tex]$\overline{A C}$[/tex] & definition of parallel lines \\
\hline [tex]$\angle 3 \cong \angle 5$[/tex] and [tex]$\angle 1 \cong \angle 4$[/tex] & alternate interior angles \\
\hline [tex]$m \angle 1= m \angle 4$[/tex] and [tex]$m \angle 3= m \angle 5$[/tex] & congruent angles have equal measures \\
\hline [tex]$m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}$[/tex] & angle addition and definition of a straight line \\
\hline [tex]$m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}$[/tex] & substitution \\
\hline
\end{tabular}
Here's a detailed, step-by-step explanation of the proof process:
1. Given: Points [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex] form a triangle.
- Reason: This is the starting information provided in the problem.
2. Definition of parallel lines: Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
- Reason: According to the definition, we establish a line [tex]\(\overline{DE}\)[/tex] parallel to [tex]\(\overline{AC}\)[/tex] through point [tex]\(B\)[/tex].
3. Alternate interior angles: [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex].
- Reason: For parallel lines, alternate interior angles are congruent. Since [tex]\(\overline{DE} \parallel \overline{AC}\)[/tex], the angles formed adhere to this property.
4. Congruent angles have equal measures: [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex].
- Reason: When angles are congruent, their measures are equal. Thus, from the congruences in the previous step, we equate their measures.
5. Angle addition and definition of a straight line: [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ\)[/tex].
- Reason: The angles forming a straight line sum to [tex]\(180^\circ\)[/tex]. In this case, the angles along [tex]\(\overline{DE}\)[/tex] (specifically, [tex]\(\angle 4\)[/tex], [tex]\(\angle 2\)[/tex], and [tex]\(\angle 5\)[/tex]) sum up to form a straight line, thus totaling [tex]\(180^\circ\)[/tex].
6. Substitution: [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ\)[/tex].
- Reason: By substituting [tex]\(m \angle 1\)[/tex] for [tex]\(m \angle 4\)[/tex] and [tex]\(m \angle 3\)[/tex] for [tex]\(m \angle 5\)[/tex] from the previously stated equal measures, we conclude that the interior angles of the triangle ([tex]\(\angle 1\)[/tex], [tex]\(\angle 2\)[/tex], and [tex]\(\angle 3\)[/tex]) sum to [tex]\(180^\circ\)[/tex].
Thus, we have proven that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
