Let's go through the steps to prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
\begin{tabular}{|l|l|}
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Statement & Reason \\
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Points [tex]$A, B$[/tex], and [tex]$C$[/tex] form a triangle. & Given \\
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Let [tex]$\overline{DE}$[/tex] be a line passing through [tex]$B$[/tex] and parallel to [tex]$\overline{AC}$[/tex]. & Definition of parallel lines \\
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[tex]$\angle 3 \cong \angle 5$[/tex] and [tex]$\angle 1 \cong \angle 4$[/tex]. & Alternate interior angles are congruent when a transversal intersects parallel lines \\
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[tex]$m \angle 1 = m \angle 4$[/tex] and [tex]$m \angle 3 = m \angle 5$[/tex]. & Measure of congruent angles are equal \\
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[tex]$m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ$[/tex]. & Angle addition and definition of a straight line \\
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[tex]$m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ$[/tex]. & Substitution \\
\hline
\end{tabular}
Here’s an explanation of each step:
1. Given: Points [tex]\(A, B\)[/tex], and [tex]\(C\)[/tex] form a triangle. This establishes our starting figure.
2. Definition of Parallel Lines: We draw a line [tex]\(\overline{DE}\)[/tex] through point [tex]\(B\)[/tex] such that [tex]\(\overline{DE}\)[/tex] is parallel to [tex]\(\overline{AC}\)[/tex]. This construction is essential for using properties of parallel lines and transversals.
3. Alternate Interior Angles: Given [tex]\(\overline{DE} \parallel \overline{AC}\)[/tex] and that [tex]\(B\)[/tex] is the point of intersection of the transversal [tex]\(\overline{AB}\)[/tex]:
- [tex]\(\angle 3\)[/tex] is congruent to [tex]\(\angle 5\)[/tex] due to the Alternate Interior Angles Theorem.
- [tex]\(\angle 1\)[/tex] is congruent to [tex]\(\angle 4\)[/tex] once again due to the Alternate Interior Angles Theorem.
4. Measure of Congruent Angles: Since [tex]\(\angle 1 \cong \angle 4\)[/tex] and [tex]\(\angle 3 \cong \angle 5\)[/tex], their measures are equal. Therefore:
- [tex]\(m \angle 1 = m \angle 4\)[/tex]
- [tex]\(m \angle 3 = m \angle 5\)[/tex]
5. Angle Addition and Definition of a Straight Line: Along the straight line formed by [tex]\(\overline{DE}\)[/tex], the sum of the angles formed at [tex]\(B\)[/tex] must be [tex]\(180^\circ\)[/tex]:
- [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ \)[/tex]
6. Substitution: Substituting [tex]\(m \angle 4 = m \angle 1\)[/tex] and [tex]\(m \angle 5 = m \angle 3\)[/tex] from the previous statement into the equation from step 5, we get:
- [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ \)[/tex]
Therefore, we have proven that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
