text: PROBLEM    10 – 0165:    Show that the lines containing the three altitudes are
  
  concurrent.
  
  Solution:                 We have shown that the perpendicular bisectors are concurrent.
  
  Here we are asked to show that another set of lines are concurrent. If we
  
  show that the lines that contain the altitudes of a given triangle also
  
  con­tain the perpendicular bisectors of a second triangle, then the
  
  concurrency follows immediately.
  
  Given: The altitudes of ∆ABC, L ,m, and n .
  
  Prove: ℓ, m, n are concurrent.
  
   STATEMENTS REASONS 1. The altitudes of ∆ABC, ℓ ,m, and n. 1. Given. 2. Construct DF ∥ AB through C  BE ∥ BC through A, and EF ∥ AC through B 2. Through a given external point only one line can be drawn  parallel to a given line. 3. Quadrilateral ABFC is a paral­lelogram. 3. If opposite sides of a quad­rilateral (AC and BF, CF and AB ) are parallel, then the quad­rilateral is a paral­lelogram. 4. Quadrilateral ACBE is a parallel­ogram. Quadrilateral ADCB is a parallel­ogram 4. Same reason as Step 3 5. (a) DC ≅ AB, CF ≅AB     (b) BE ≅ CA, FB ≅ CA     (c)  DA ≅CB, AE ≅CB 5. Opposite sides of a parallel­ogram are congruent 6. (a) DC ≅ CF     (b)  BE ≅ FB      (c) DA ≅ AE 6. Segments congruent to the same segments are congruent. 7.(a) C is the midpoint of DF   (b) B is the midpoint of FE   (c) A is the midpoint of DE 7. Definition of midpoint. 8. (a) m ⊥ AB 8. Definition of the altitude of a triangle. 9. (a) m ⊥ DF     (b) n ⊥ FE        (c) l ⊥ DE 9. In a plane, if a line is per­pendicular to one of two parallel lines, then it is per­pendicular to the other para­llel line. 10. (a) m is the perpendicular bisector  of DF (b) n is the perpendicular bisector  of FE. (c) ℓ is the perpendicular bisector of DE 10. A line perpendicular to a segment at its midpoint is the perpendicular bisector of the segment. 11. DF, FE, DE are sides of ∆DFE 11. In a plane, three non-collinear points determine a triangle. 12. m, n and ℓ are concurrent. 12. The perpendicular bisectors the sides of a triangle are concurrent
explanation: The document contains a detailed geometric proof that demonstrates the concurrency of the lines containing the altitudes of a triangle. It includes a problem statement, a solution, and a step-by-step proof with reasons for each step. The document does not contain a separate question and answer format but rather presents a complete proof.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: PROBLEM    10 – 0165:    Show that the lines containing the three altitudes are
  
  concurrent.
  
  Solution:                 We have shown that the perpendicular bisectors are concurrent.
  
  Here we are asked to show that another set of lines are concurrent. If we
  
  show that the lines that contain the altitudes of a given triangle also
  
  con­tain the perpendicular bisectors of a second triangle, then the
  
  concurrency follows immediately.
  
  Given: The altitudes of ∆ABC, L ,m, and n .
  
  Prove: ℓ, m, n are concurrent.
  
   STATEMENTS REASONS 1. The altitudes of ∆ABC, ℓ ,m, and n. 1. Given. 2. Construct DF ∥ AB through C  BE ∥ BC through A, and EF ∥ AC through B 2. Through a given external point only one line can be drawn  parallel to a given line. 3. Quadrilateral ABFC is a paral­lelogram. 3. If opposite sides of a quad­rilateral (AC and BF, CF and AB ) are parallel, then the quad­rilateral is a paral­lelogram. 4. Quadrilateral ACBE is a parallel­ogram. Quadrilateral ADCB is a parallel­ogram 4. Same reason as Step 3 5. (a) DC ≅ AB, CF ≅AB     (b) BE ≅ CA, FB ≅ CA     (c)  DA ≅CB, AE ≅CB 5. Opposite sides of a parallel­ogram are congruent 6. (a) DC ≅ CF     (b)  BE ≅ FB      (c) DA ≅ AE 6. Segments congruent to the same segments are congruent. 7.(a) C is the midpoint of DF   (b) B is the midpoint of FE   (c) A is the midpoint of DE 7. Definition of midpoint. 8. (a) m ⊥ AB 8. Definition of the altitude of a triangle. 9. (a) m ⊥ DF     (b) n ⊥ FE        (c) l ⊥ DE 9. In a plane, if a line is per­pendicular to one of two parallel lines, then it is per­pendicular to the other para­llel line. 10. (a) m is the perpendicular bisector  of DF (b) n is the perpendicular bisector  of FE. (c) ℓ is the perpendicular bisector of DE 10. A line perpendicular to a segment at its midpoint is the perpendicular bisector of the segment. 11. DF, FE, DE are sides of ∆DFE 11. In a plane, three non-collinear points determine a triangle. 12. m, n and ℓ are concurrent. 12. The perpendicular bisectors the sides of a triangle are concurrent
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a detailed geometric proof that demonstrates the concurrency of the lines containing the altitudes of a triangle. It includes a problem statement, a solution, and a step-by-step proof with reasons for each step. The document does not contain a separate question and answer format but rather presents a complete proof.
  
  Label: question-and-answer