Answer :
Let's complete the statements and the reasons for each.
1. Points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] form a triangle. (given)
2. Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex]. (definition of parallel lines)
3. [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex]. The reason is alternate interior angles and corresponding angles.
4. [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex]. The reason is measures of corresponding and alternate interior angles.
5. [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ\)[/tex]. The reason is angle addition and definition of a straight line.
6. [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ\)[/tex]. The reason is substitution.
So here is the completed table:
[tex]\[ \begin{tabular}{|l|l|} \hline Statement & Reason \\ \hline Points $A, B$, and $C$ form a triangle. & given \\ \hline Let $\overline{DE}$ be a line passing through $B$ and parallel to $\overline{AC}$ & definition of parallel lines \\ \hline $\angle 3 \cong \angle 5$ and $\angle 1 \cong \angle 4$ & alternate interior angles and corresponding angles \\ \hline $m \angle 1= m \angle 4$ and $m \angle 3= m \angle 5$ & measures of corresponding and alternate interior angles \\ \hline $m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}$ & angle addition and definition of a straight line \\ \hline $m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}$ & substitution \\ \hline \end{tabular} \][/tex]
1. Points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] form a triangle. (given)
2. Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex]. (definition of parallel lines)
3. [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex]. The reason is alternate interior angles and corresponding angles.
4. [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex]. The reason is measures of corresponding and alternate interior angles.
5. [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ\)[/tex]. The reason is angle addition and definition of a straight line.
6. [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ\)[/tex]. The reason is substitution.
So here is the completed table:
[tex]\[ \begin{tabular}{|l|l|} \hline Statement & Reason \\ \hline Points $A, B$, and $C$ form a triangle. & given \\ \hline Let $\overline{DE}$ be a line passing through $B$ and parallel to $\overline{AC}$ & definition of parallel lines \\ \hline $\angle 3 \cong \angle 5$ and $\angle 1 \cong \angle 4$ & alternate interior angles and corresponding angles \\ \hline $m \angle 1= m \angle 4$ and $m \angle 3= m \angle 5$ & measures of corresponding and alternate interior angles \\ \hline $m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}$ & angle addition and definition of a straight line \\ \hline $m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}$ & substitution \\ \hline \end{tabular} \][/tex]
