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



Answer :

GinnyAnswer
To prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex], we can complete the statements with the appropriate reasons in the following manner:

1. Points [tex]\(A, B\)[/tex], and [tex]\(C\)[/tex] form a triangle. & [tex]\( \text{given} \)[/tex]

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

3. [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex]. & [tex]\( \text{corresponding angles are congruent} \)[/tex]

4. [tex]\(m\angle 1 = m\angle 4\)[/tex] and [tex]\(m\angle 3 = m\angle 5\)[/tex]. & [tex]\(\text{measures of congruent angles are equal}\)[/tex]

5. [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ\)[/tex]. & [tex]\(\text{sum of angles on a straight line}\)[/tex]

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

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