Hello, AirForceOne!
A regular (equilateral) triangle has three \(\displaystyle 60^o\) angles.
A regular quadrilateral (square) has four \(\displaystyle 90^o\) angles.
A regular pentagon has five \(\displaystyle 108^o\) angles (<u>not</u> a multiple of \(\displaystyle 30^o\))
A regular hexabon has six \(\displaystyle 120^o\) angles.
There are 18 vertex angles in the drawer.
. . Thirteen of them are multiples of \(\displaystyle 30^o\).
Therefore, the probability is \(\displaystyle \frac{13}{18}\).
An exterior angle and an interior angle are supplementary; their sum is \(\displaystyle 180^o\).
. . So we have: \(\displaystyle \,x\,+\,\frac{1}{4}x\:=\:180\;\;\Rightarrow\;\;\frac{5}{4}x\,=\,180\;\;\Rightarrow\;\;x\,=\,144^o\)
Each interior angle is \(\displaystyle 144^o\).
.How many sides does it have?
In a polygon of \(\displaystyle n\) sides, the sum of the interior angles is: \(\displaystyle \,180(n\,-\,2)\) degrees.
If the polygon is equiangular, each interior angle has: \(\displaystyle \,\frac{180(n\,-\,2)}{n}\) degrees.
Our polygon has an interior angle of \(\displaystyle 144^o\).
. . So we have: \(\displaystyle \,\frac{180(n\,-\,2)}{n}\:=\:144\;\;\Rightarrow\;\;180n\,-\,360\:=\:144n\;\;\Rightarrow\;\;36n\,=\,360\)
Therefore: \(\displaystyle \,n\,=\,10\;\) . . . It is a decagon.
Edit: Ha! .... miscounted on #1.
.Thanks for catching it, jacket81.