Calling all Geniouses!?

Question

I have an interesting math teacher that is obsessed with homework, and I need some help!
What are these polygons, and can you use logic and describe them to me?

1.) It is a closed figure with straight sides.
2.) It has sides of two different lengths
3.) One of its angles measures 135 degrees.
4.) Another of its angles is congruent to the angle in clue 3
5.) The angles in clues 3 and 4 are not consecutive angles
6.) Its other angles are larger than acute but smaller than obtuse
7.) It has two sides that are parallel
8.) It has ten pairs of angles

Next.....
1.) It is a polygon
2.) Some of its angles measure 120 degrees
3.) Its other angles measure 150 degrees
4.) Its sides aren't congruent
5.) It has an even number of both measures of angles
6.) Its smaller angles are consecutive angles
7.) It ahs more of the larger angles than the smaller
8.) It has 45 pairs of angles.
Thank You SOOOO... Much!!!!~~~~~

Sorry,
7.) It has more of the larger angles than the smaller.

Answer 1

Let's start with the first one:

#1 tells us that it's a shape made of connecting segments and to end so that they don't overlap. So it's some kind of polygon. But #2 means it's not a regular polygon (a fact which others here seemed to have overlooked). Notice also that we don't know right away whether or not it's convex, where all the corners would point out.

#3 & #4 tell us that two of the angles measure 135 degrees. And #8 implies there are 20 angles in total. But #6 tells us that all of the other angles, and I'm assuming "interior angles", measure 90 degrees. This would mean that the shape is convex (no interior angles are greater than 180 degrees), but ultimately this doesn't make any sense. The sum of the interior angles would be 2*135 + 18*90 = 1890 degrees in total, but for a convex polygon of 20 angles (and thus 20 sides), the sum should be 180(n-2) = 180*18 = 3240. For #6, did your teacher perhaps mean larger than acute but smaller than REFLEX, meaning between 90 and 180 degrees?

As for the second one:

It's a polygon, and none of the angles are greater than 180 degrees, so it's convex too. Let A be the number of angles that measure 120 degrees, and B the number of angles that measure 150 degrees. #8 tells us that there are 90 angles total, so A + B = 90. #8 also tells us that there are 90 sides to this convex polygon. So the sum of the interior angles is 180(90 - 2) = 15,840. We know that this must be the sum of all the 120 and 150-degree angles, so 120A + 150B = 15840. Using this with A+B=90 and solving two equations for two unknowns, you get A= -78 and B=168, which again doesn't make any sense.

Either there's some big assumption I keep making, or your teacher has to word these problems a little better.

Answer 2

Comment on the 2nd problem.

If there are 45 pairs of angles then there are 90 sides. If there are 90 sides, then the sum of the angles is (90-2)(180)= 15,840 degrees. Even if all 90 angles were 150 degrees, that would only add up to 90*150 = 13,500 degrees. Therefore the polygon your teacher suggests is impossible.

Comment on 2nd question:
Statement 6 implies that all othe angles are right angles. Therefore there is one pair of angles that ar each 135 degrees and 9 pairs that are each 90 degrees for a total of 10 pairs as indicated in Statement 8.

Therefore, the total number of degrees in the polygon is
9(180)+2*135 = 1890.

But this is a 20-sided figure which should have a total of 18*180 = 3240 degrees. So something is screwy. Perhaps She is using permutaions or combinations when she/he is talking about pairs of angles. Perhaps your teacher is including exterior angles. Who knows. This has got to be a trick question or your teacher is wrong.

Answer 3

I am a little confused - if a polygon has 45 pairs of angles, that's 90 angles which means it would have 90 sides. But then the angles would be much larger than 120 and 150 degrees.

Answer 4

Your closed figure with straight sides has 2 angles of 135° and 18 90° angles. I don't see how that is possible.

Your polygon with 45 pairs of angles would have 90 angles and therefore 90 sides. A polygon with that many sides would have angles greater than 120° or 150°.

Answer 5

In the first one, all information other than 8. is extraneous.
A polygon with 10 pairs of angles has 20 angles, and therefore 20 sides. It is called a dodecagon.

2*45 = 90 angles ==> 90 sides. I'm not sure, but I think you would call it a novodecagon. (Latin for 9 is novum, Latin for 10 is decem.)

Answer 6

While I don't think ALL credible theorists hold these 'great captains' in this same severe light, those that do are doing what we all call "Monday Morning Quarterbacking" . Its easy to point fingers & claim 'well they should've known!' but these types don't usually put things into perspective, namely how the commander on the ground saw what variables he had at the time & how he then came to particular conclusion. In some cases the captains are trapped either by tradition (i.e this is how we've always done it & its brought success so why change it?), or reputation (i.e. Napoleon's unbeatable streak). What's more, say in the popular case of commanders in the American Civil War utlizing Napoleonic tactics at a time when the armament had advanced greatly, and a similar tactical problem in WW1. In either case, the military has generally never been a bastion of inventive, progressive thinking, choosing instead to go with what's always been done. New ideas are generally frowned upon, regardless of the results it brings. Witness Billy Mitchell's example of how planes could take out capital ships, & the military's disregard of such. So to damn ACW or WW1 officers for operating as they did is wholly unfair b/c to do otherwise would've run contradictory to their training. Then again, some armchair generals would beg to differ.

Answer 7

It doesn't take a genies to figure those out. It takes time, I just want the points. Bye.