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

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



Answer :

GinnyAnswer
Sure! Let's complete the statements with the correct reasons for each step.

[tex]\[ \begin{tabular}{|l|l|} \hline Statement & Reason \\ \hline Points $A, B$, and C form a triangle. & given \\ \hline Let $\overline{D E}$ be a line passing through $B$ and parallel to $\overline{A C}$ & definition of parallel lines \\ \hline $\angle 3 \cong \angle 5$ and $\angle 1 \cong \angle 4$ & Alternate Interior Angles Theorem \\ \hline $m \angle 1= m \angle 4$ and $m \angle 3= m \angle 5$ & Congruent Angles Have Equal Measures \\ \hline $m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}$ & angle addition and definition of a straight line \\ \hline $m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}$ & substitution \\ \hline \end{tabular} \][/tex]

This step-by-step proof demonstrates that the sum of the interior angles of [tex]$\triangle ABC$[/tex] is [tex]$180^{\circ}$[/tex], using the properties of parallel lines and congruent angles.

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