To prove that the sum of the measures of the interior angles of a triangle is always 180 degrees, we typically use the concept of corresponding angles and alternate interior angles that arise when a line is drawn parallel to one side of the triangle.
Here is a common method to demonstrate this:
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Draw a triangle \( ABC \).
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Extend one of the sides of the triangle, for example, line segment \( BC \), and draw a line parallel to \( AC \) through point \( B \). Let's call this line line \( l \).
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By the parallel line cut by a transversal \( AB \), the angle \( CAB \) (angle A) corresponds to a new angle formed at point \( B \) (we can call it angle \( 1 \)). Thus, \( \angle CAB = \angle 1 \).
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Additionally, angle \( ABC \) (angle B) and angle \( ACB \) (angle C) are alternate interior angles with respect to line \( l \) and the transversal \( AC \). Thus, \( \angle ACB = \angle 2 \) and these two angles correspond to each other.
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Since lines are parallel, the sum of angles formed along the straight line (which represents the sum of angles at point B and the angle at point A) is:
- \( \angle A + \angle B + \angle C + \angle 2 = 180^\circ \)
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Rearranging gives:
- \( \angle A + \angle B + \angle C = 180^\circ \)
Therefore, through the understanding of corresponding angles and alternate interior angles generated by a transversal cutting through parallel lines, we can conclude that the sum of the interior angles of a triangle is always \( 180^\circ \).
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