Okay, let's complete the statements step-by-step to prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].
To prove this, let's complete each statement with the appropriate reason:
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: Corresponding angles created by a transversal with parallel lines are congruent
4. Statement: [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex]
Reason: Definition of congruent angles
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
So, the completed statements are as follows:
[tex]\[
\begin{array}{|l|l|}
\hline
\text{Statement} & \text{Reason} \\
\hline
\text{Points } A, B, \text{ and } C \text{ form a triangle.} & \text{given} \\
\hline
\text{Let } \overline{DE} \text{ be a line passing through } B \text{ and parallel to } \overline{AC} & \text{definition of parallel lines} \\
\hline
\angle 3 \cong \angle 5 \text{ and } \angle 1 \cong \angle 4 & \text{corresponding angles created by a transversal with parallel lines are congruent} \\
\hline
m \angle 1 = m \angle 4 \text{ and } m \angle 3 = m \angle 5 & \text{definition of congruent angles} \\
\hline
m \angle 4 + m \angle 2 + m \angle 5 = 180^{\circ} & \text{angle addition and definition of a straight line} \\
\hline
m \angle 1 + m \angle 2 + m \angle 3 = 180^{\circ} & \text{substitution} \\
\hline
\end{array}
\][/tex]
This completes the proof that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].
