Angles of a Quadrilateral
Simple Question
a) What is the minimum number of acute angles in a convex quadrilateral? Prove.
That may be too easy. So, I'll give a point to (or split point between) the most creative and entertaining answer(s).
b) What is the minimum number of acute angles in a concave quadrilateral? Prove.
Clarification:
By acute, I mean angles that are 90 deg or lower. Acute = Non-obtuse.
a) What is the minimum number of acute angles in a convex quadrilateral? Prove.
That may be too easy. So, I'll give a point to (or split point between) the most creative and entertaining answer(s).
b) What is the minimum number of acute angles in a concave quadrilateral? Prove.
Clarification:
By acute, I mean angles that are 90 deg or lower. Acute = Non-obtuse.
posted by Sisyphus at 7/18/2006 01:11:00 PM
5 Comments:
1) Convex - minimum 0. All four angles can be 90 deg and so none of the angles needs to be obtuse.
2) Concave - 2. Atleast one angle is more than 180 deg and sum of other three angles is less than 180. So even if one of the other three angles is obtuse (>90), the remaining two have to be acute.
So atleast 2 angles of a convex quad. should be acute.
w+x+y+z=360,
I.
Let us assume w,x,y are all in (90,180)
=> (w+x+y) is in (270,540)
=> z can lie in (0,90)
Thus minimum no of acute angles in a convex quad. is ONE.
II.
Let w be the reflex angle.
=>w lies in (180,360)
=>x+y+z lies in (0,180)
Thus,there cannot be two obtuse angles in x,y or z simultaneously.
Thus in a concave quad., the minimum no. of acute angles is TWO.
BTW,long time between puzzles,eh?
Raju and Sai did not read my clarification (which I posted along with the question). So, no points for you.
Grenade has invented a better problem and provided a satisfying answer.
Oh Blogmaster, the great littlecow, could you give a point to Grenade?!
Thanks.
Enjoyed a lot! »
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