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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
\textbf{Statement} & \textbf{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
\(\angle 3 \simeq \angle 5\) and \(\angle 1 \approx \angle 4\) & Alternate interior angles are congruent \\
\hline
\(m \angle 1 = m \angle 4\) and \(m \angle 3 = m \angle 5\) & Measures of congruent angles are equal \\
\hline
\(m \angle 4 + m \angle 2 + m \angle 5 = 180^{\circ}\) & Angle addition postulate \\
\hline
\(m \angle 1 + m \angle 2 + m \angle 3 = 180^{\circ}\) & Substitution and definition of a straight line \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnswer
Sure! Let's fill in the missing reasons step-by-step:

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

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

3. [tex]\(\angle 3 \simeq \angle 5\)[/tex] and [tex]\(\angle 1 \approx \angle 4\)[/tex]
- Statement: [tex]\(\angle 3 \simeq \angle 5\)[/tex] and [tex]\(\angle 1 \approx \angle 4\)[/tex]
- Reason: corresponding angles are equal when a transversal intersects parallel lines

4. [tex]\(m \angle 1= m \angle 4\)[/tex] and [tex]\(m \angle 3= m \angle 5\)[/tex]
- Statement: [tex]\(m \angle 1= m \angle 4\)[/tex] and [tex]\(m \angle 3= m \angle 5\)[/tex]
- Reason: because they are corresponding angles

5. [tex]\(m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}\)[/tex]
- Statement: [tex]\(m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}\)[/tex]
- Reason: sum of angles on a straight line is 180 degrees

6. [tex]\(m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}\)[/tex]
- Statement: [tex]\(m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}\)[/tex]
- Reason: angle addition and definition of a straight line

Thus, completing the steps proves that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].
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