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
The text passage contains a geometry problem and its solution, which involves proving that the lines containing the altitudes of a triangle are concurrent. The solution is structured as a series of statements and reasons.
no_question

the passage does not contain a mathematical question in the form of a word problem or calculation. instead, it presents a proof for a geometric property.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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
user
The text passage contains a geometry problem and its solution, which involves proving that the lines containing the altitudes of a triangle are concurrent. The solution is structured as a series of statements and reasons.

TYPE: NO_QUESTION

The passage does not contain a mathematical question in the form of a word problem or calculation. Instead, it presents a proof for a geometric property.