What is the smallest integer (in degrees) angle, which can be constructed with straightedge and compass?

Answer 1

With a straightedge and compass, you can always bisect any angle...
.. you can also create 2pi/5 = 72 degrees

thus you can make 36 degrees... and since you can make 30 degrees,

... you produce 6 degrees... bisecting...
... i guess, it will be 3 degrees. Unless there is a way to trisect an angle ... and this is impossible... §

so my answer is 3 degrees.
(unless you are using a marked straightedge, then 1 degree is possible.)

Answer 2

Yes, DC's guess about 3° is right, and here is the proof. Why 1° and 2° can not be constructed with straightedge and compass follows not from the impossibility of the famous ancient problem of angle trisection /there can be eventually another construction, not involving trisection/, but from the following: the prominent German scientist Gauss proved some 200 years ago the possibility to construct a regular N-gon /N=2,3,.../ ONLY when N has a form:
N = 2^k*p1*p2*....pm, here k=0,1,2,3,....,
p1, p2, ..., pm are "m" DISTINCT prime numbers /m=0,1,2,.../, each of the form
2^(2^q) + 1 /so called Fermat primes, q=0,1,2,.../.
The construction of angle 3° is equivalent to the construction of a regular 120-gon and the latter is possible because
120 = 2^3 * 3 * 5 /3 = 2^(2^0) + 1 and 5 = 2^(2^1) + 1 are the first Fermat primes/. Regular 9-gon is IMPOSSIBLE to construct because 9 = 3 * 3 and the prime factors are not distinct. The constructions of angles 2° and 1° is equivalent to the construction of a regular 180-gon and 360-gon, but
180 = 2^2*3*3*5, 360 = 2^3*3*3*5 and the prime factorizations contain two 3's and these polygons are not constructible as well as the 9-gon.

Answer 3

build a perpendicular to a line... that makes ninety tiers.. you like yet another 15. Now build a 60 degree perspective... placed the ingredient of the compass on the element the place the perpendicular intersects with the line, and build 1 / 4 circle. the place the circumference of the circle intersects with the perpendicular, place the ingredient of your compass and lay off the radius alongside that circle. That perspective would be 60 tiers. (total one hundred fifty tiers) Bisect the perspective between the perpendicular and the perspective purely formed. that provide you with with 30 tiers. (total a hundred and twenty tiers.... getting nearer). back, bisect the perspective between the perpendicular and the final bisection. that provide you with with 15 tiers (total one 0 five). Draw a line from the element the place the perpendicular lines intersect to the final built element... the bisection. the better perspective so formed would be one 0 five tiers (by using shape) performed.

Answer 4

1º angle try dividing 90º again and again