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

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



Answer :

GinnyAnswer
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]

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