To prove that the sum of the interior angles of a triangle is 180 degrees, we can use a simple geometric approach involving parallel lines. Here’s a step-by-step proof:
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Consider a triangle: Let triangle \( ABC \) have interior angles \( \angle A \), \( \angle B \), and \( \angle C \).
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Extend one side: Extend the side \( BC \) of the triangle to a point \( D \) beyond \( C \).
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Draw a parallel line: Through point \( A \), draw a line parallel to line \( BC \). Let this line intersect line \( AD \) at some point.
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Identify angles: Since line \( AD \) is a transversal intersecting the two parallel lines \( BC \) and the line through \( A \), we can use the properties of alternate interior angles:
- \( \angle CAB \) (which is \( \angle A \)) is equal to the angle formed at \( A \) by the transversal and the parallel line, let's call it \( \angle EAD \).
- \( \angle ABC \) (which is \( \angle B \)) is equal to the angle \( \angle ABD \) formed at point \( B \) by the transversal \( AD \).
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Sum of angles: Now we observe that the angles on a straight line must sum to 180 degrees: \[ \angle EAD + \angle ABC + \angle ACB = 180^\circ \]
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Replace angles: Now substituting the equal angles we identified earlier into this equation gives: \[ \angle A + \angle B + \angle C = 180^\circ \]
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Conclusion: Hence, we have proved that the sum of the interior angles of triangle \( ABC \) is indeed 180 degrees.
\(\boxed{180^\circ}\)