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Triangle Sum Theorem Proof

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About This Proof

This step-by-step visualization demonstrates why the sum of the three interior angles of a triangle is always 180°. The proof uses parallel lines and alternate interior angles.

Proof Steps

  1. Start with triangle ABC with angles ∠1, ∠2, and ∠3
  2. Draw parallel line ℓ through B parallel to side AC
  3. Identify ∠4 ≅ ∠1 (alternate interior angles with transversal AB)
  4. Identify ∠5 ≅ ∠3 (alternate interior angles with transversal BC)
  5. Angles on a line ∠4 + ∠2 + ∠5 = 180° (straight angle)
  6. Substitute ∠1 + ∠2 + ∠3 = 180°

Key Concepts Used

  • Parallel lines and transversals
  • Alternate interior angles are congruent
  • Straight angle = 180°
  • Substitution property

Triangle Sum Theorem

The sum of the measures of the three interior angles of a triangle is always 180°.

\[m\angle A + m\angle B + m\angle C = 180°\]

Interaction

Click anywhere to advance through the proof steps. Click again to cycle back to the beginning.

Learning Objectives

  • Understand why triangle angles sum to 180°
  • Analyze the proof structure using parallel lines

Bloom's Taxonomy Level

Understanding and Analyzing

Iframe Embed Code

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