It is given that the angle measures of a triangle are each a multiple of 12. We will find the probability that the triangle is equiangular. Recall that an equiangular triangle is a triangle such that all three interior angles are 60^(∘).
Let m∠1, m∠2, and m∠3 be the angle measures of the triangles. Let's list all of the possible triangles that satisfy the given requirement. We will start by listing all the triangles with the smallest angle measure of 12.
Triangles
m∠1
m∠2
m∠3
Sum of the Angles
Triangle I
12
12
156
180
Triangle II
12
24
144
180
Triangle III
12
36
132
180
Triangle IV
12
48
120
180
Triangle V
12
60
108
180
Triangle VI
12
72
96
180
Triangle VII
12
84
84
180
As we can see, there are seven triangles with the smallest angle measure of 12. The next multiple of 12 is 24, so let's list all of the triangles with the smallest angle measure of 24.
Triangles
m∠1
m∠2
m∠3
Sum of the Angles
Triangle VIII
24
24
132
180
Triangle IX
24
36
120
180
Triangle X
24
48
108
180
Triangle XI
24
60
96
180
Triangle XII
24
72
84
180
So far we have listed 12 triangles, 5 of which have the smallest angle measure being 24. We should repeat this procedure for all other multiples of 12, increasing the smallest angle measure by 12 every step.
Smallest angle measure
No. of triangles
12
7
24
5
36
4
48
2
60
1
Overall, there are nineteen possible triangles and one is equiangular. We can find the probability by dividing the number of favorable outcomes by the number of possible outcomes.
Number of favorable outcomes/Number of possible outcomes=1/19
