## Monday, February 13, 2012

### crowdsourcing a geometry poser

One problem that I encountered while tutoring my goddaughter stumped me, and I'm still working on it. It was one of a set of problems for which the instructions were these:

Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer.

The accompanying image shows this: For problem #2 of the aforementioned set, we're given only this information:

Angle 10 is congruent to angle 16.

I'm pretty sure there's some way to establish that lines C and D are parallel, but because we're given no information about lines M and N, I don't think we can prove they're parallel.

Normally, to prove that two lines are parallel, we look for certain data:

1. If corresponding angles are congruent, the lines are parallel. In the above illustration, the following pairs of corresponding angles would, if congruent, prove lines C and D are parallel: 9 & 13, 10 & 14, 11 & 15, 12, & 16.

2. If alternate interior angles are congruent, the lines are parallel. The following angle pairs would, if congruent, prove lines C and D are parallel: 11 & 14, 12 & 13.

3. If alternate exterior angles are congruent, the lines are parallel. The following angle pairs would, if congruent, prove lines C and D are parallel: 9 & 16, 10 & 15.

4. If consecutive interior angles are supplementary, the lines are parallel. The following angle pairs, if added together for a sum of 180 degrees, would prove lines C and D are parallel: 11 & 13, 12 & 14.

Intuitively, I know that, if angles 10 and 16 are congruent, the only possible configuration for these non-consecutive exterior angles lines is for them to be 90 degrees. But that's working the problem backwards: I need to show how they're 90 degrees, and that's where I'm stuck.

Here's what I can deduce (in an effort to establish that lines C and D are parallel):

1. ∠10 ≅ ∠16 (Given.)
2. ∠16 ≅ ∠13; ∠10 ≅ ∠11 (Vertical angles.)
3. ∠10 ≅ ∠13 (Transitive: ∠10≅∠16≅∠13.)
4. ∠13 ≅ ∠11 (Transitive: ∠13≅∠10≅∠11.)
5. ∠11 ≅ ∠16 (Transitive: ∠16≅∠10≅∠11.)
6. m∠11 + m∠12 = m∠13 + m∠14 = 180 (Steps 1-5, definition of supplemental angles.)
7. ∠12 ≅ ∠14 (Algebra. (a + b = a + c, ∴ b = c))
8. m∠10 + m∠12 = m∠15 + m∠16 = 180 (Steps 1-5, def. of supplementary angles.)
9. ∠12 ≅ ∠15 (Algebra. (a + b = a + c, ∴ b = c))

Feel free to write in with comments. If you have a short, sweet way to establish that lines C and D are parallel, I'm all ears. Meanwhile, I'll keep working at this on my own.

UPDATE, 3/19/12: Be sure to read this subsequent post. That the lines are parallel cannot be proven.

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