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

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

So the complete reasoning to prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex] follows these detailed steps:

1. Statement: Points [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex] form a triangle.
Reason: given

2. Statement: Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
Reason: definition of parallel lines

3. Statement: [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex].
Reason: alternate interior angles are congruent

4. Statement: [tex]\(m\angle 1 = m\angle 4\)[/tex] and [tex]\(m\angle 3 = m\angle 5\)[/tex].
Reason: angles with equal measures are equal

5. Statement: [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ\)[/tex].
Reason: angle addition and definition of a straight line

6. Statement: [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ\)[/tex].
Reason: substitution

By following these logical steps, we prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is indeed [tex]\(180^\circ\)[/tex].
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