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



Answer :

GinnyAnswer
To complete the statements to prove that the sum of the interior angles of [tex]$\triangle ABC$[/tex] is [tex]$180^\circ$[/tex], follow these steps:

1. Points [tex]$A, B$[/tex], and [tex]$C$[/tex] form a triangle.
- Reason: Given.

2. 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. [tex]$\angle 3 \simeq \angle 5$[/tex] and [tex]$\angle 1 \approx \angle 4$[/tex]
- Reason: Alternate interior angles formed by a transversal with parallel lines.

4. [tex]$m \angle 1 = m \angle 4$[/tex] and [tex]$m \angle 3 = m \angle 5$[/tex]
- Reason: Definition of congruent angles.

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

6. [tex]$m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ$[/tex]
- Reason: Substitution.

Therefore, by substituting the angles, we have shown that the sum of the interior angles of [tex]$\triangle ABC$[/tex] is [tex]$180^\circ$[/tex].

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