with
the angle opposite side .
For an angle to be acute, . Therefore, an acute triangle satisfies , , and .
The smallest number of acute triangles into which an arbitrary obtuse triangle can be dissected is seven if , , and otherwise eight (Manheimer 1960,
Gardner 1981, Wells 1991). A square can be dissected into
as few as 9 acute triangles (Gardner 1981, Wells 1991).
Gardner, M. "Mathematical Games: A Fifth Collection of 'Brain-Teasers.' " Sci. Amer.202, 150-154, Feb. 1960.Gardner,
M. "Mathematical Games: The Games and Puzzles of Lewis Carroll and the Answers
to February's Problems." Sci. Amer.202, 172-182, Mar. 1960.Gardner,
M. "Mathematical Games: The Inspired Geometrical Symmetries of Scott Kim."
Sci. Amer.244, 22-31, Jun. 1981.Goldberg, G. "Problem
E1406." Amer. Math. Monthly67, 923, 1960.Hoggatt,
V. E. Jr. "Acute Isosceles Dissection of an Obtuse Triangle." Amer.
Math. Monthly68, 912-913, 1961.Johnson, R. S. "Problem
256 [1977: 155]." Crux Math.4, 53-54, 1978.Manheimer,
W. "Dissecting an Obtuse Triangle into Acute Triangles." Solution to Problem
E1406. Amer. Math. Monthly67, 923, 1960.Nelson, H. L.
"Solution to Problem 256." Crux Math.4, 102-104, 1978.Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 1-2, 1991.
,
, and 
,
the angle opposite side 
.
For an angle to be acute, 
. Therefore, an acute triangle satisfies 
, 
, and 
.
, 
, and otherwise eight (Manheimer 1960,
Gardner 1981, Wells 1991). A square can be dissected into
as few as 9 acute triangles (Gardner 1981, Wells 1991).
