The Triangle Angle Sum Theorem indeed states that the sum of the interior angles in any triangle is always 180 degrees. This theorem is fundamental in various geometric proofs, including those involving triangle congruences, such as AAS (Angle-Angle-Side) congruence.
In the context of AAS congruence, if you have two angles and the side that is not between them, you can determine that the triangles formed are congruent. Here's how the Triangle Angle Sum Theorem aids in this:
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Identify Known Angles: If you have triangle \( ABC \) and you know the measures of angles \( A \) and \( B \), you can use the Triangle Angle Sum Theorem to find angle \( C \): \[ C = 180^\circ - (A + B). \]
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Setup Another Triangle: Suppose you have another triangle \( DEF \), where you know angles \( D \) and \( E \) (which correspond to angles \( A \) and \( B \) from the first triangle) and a side \( DE \) (which corresponds to side \( AB \) in triangle \( ABC \)).
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Angle Matching: Since \( D \) and \( E \) correspond to \( A \) and \( B \), and if you have determined from the theorem that angle \( C \) is equal to angle \( F \) (since both angles share the same calculation based on their corresponding pairs), you can conclude that angle \( F = C \).
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Applying AAS: With two pairs of angles and the included side being equal, it meets the AAS criteria:
- \( \angle A = \angle D \)
- \( \angle B = \angle E \)
- Side \( AB = DE \)
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Conclusion: By the AAS theorem, triangle \( ABC \) is congruent to triangle \( DEF \).
Overall, the Triangle Angle Sum Theorem is crucial, as it provides the necessary angle measures to establish congruency through the AAS theorem, thereby reinforcing the interconnectedness of triangle properties in geometry.
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