text: # Prove Triangle Theorems
  
  Videos, solutions, and lessons to help High School students learn how to prove theorems about triangles.
  
  Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
  
  Common Core: HSG-SRT.B.4
  
  ## Triangle Proportionality Theorem
  
  The Triangle Proportionality Theorem
  If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.
  Demonstration of the Triangle Proportionality Theorem
  In the triangle ABC let PQ be parallel to BC, with P on AB and Q on AC. Then |AP|/|PB| = |AQ|/|QC|.
  
  Using the Properties of the Triangle Proportionality Theorem to Solve for Unknown Values.
  Proof: Converse of the Triangle Proportionality Theorem
  Proving -- Converse of the Triangle Proportionality Theorem: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
  
  ## Pythagorean Theorem
  
  Prove the Pythagorean Theorem using similar triangles
  In this lesson, you will learn how to prove the Pythagorean Theorem using similar triangles.
  Pythagorean Theorem Proof Using Similarity
  Proof of the Pythagorean Theorem using similarity.
  Similar Triangles: Ratio of Areas.
explanation: The document provides a series of educational descriptions and proofs related to triangle theorems, including the Triangle Proportionality Theorem and the Pythagorean Theorem. It does not contain a specific question or a question-and-answer format but rather presents theorems and their proofs.
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distilabel_metadata.content: # Prove Triangle Theorems
  
  Videos, solutions, and lessons to help High School students learn how to prove theorems about triangles.
  
  Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
  
  Common Core: HSG-SRT.B.4
  
  ## Triangle Proportionality Theorem
  
  The Triangle Proportionality Theorem
  If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.
  Demonstration of the Triangle Proportionality Theorem
  In the triangle ABC let PQ be parallel to BC, with P on AB and Q on AC. Then |AP|/|PB| = |AQ|/|QC|.
  
  Using the Properties of the Triangle Proportionality Theorem to Solve for Unknown Values.
  Proof: Converse of the Triangle Proportionality Theorem
  Proving -- Converse of the Triangle Proportionality Theorem: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
  
  ## Pythagorean Theorem
  
  Prove the Pythagorean Theorem using similar triangles
  In this lesson, you will learn how to prove the Pythagorean Theorem using similar triangles.
  Pythagorean Theorem Proof Using Similarity
  Proof of the Pythagorean Theorem using similarity.
  Similar Triangles: Ratio of Areas.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides a series of educational descriptions and proofs related to triangle theorems, including the Triangle Proportionality Theorem and the Pythagorean Theorem. It does not contain a specific question or a question-and-answer format but rather presents theorems and their proofs.
  
  Label: no-question