Final answer:
To find the smallest angle of the triangle formed by dividing a circle in a 4:5:6 ratio, calculate the central angles for each division, find their supplementary angles, and the smallest internal angle of the triangle is the supplement of the largest central angle, which is 36 degrees.
Step-by-step explanation:
The question involves dividing a circle into parts in a ratio and then forming a triangle by connecting the dividing points. Given the ratio of 4:5:6, this represents the division of the circle into angles proportional to these numbers. To find the total degrees in the circle, which is 360 degrees, we multiply the sum of the ratio parts (4+5+6=15) by the number of degrees per part. We then find that each part represents 24 degrees (360/15). Multiplying each part of the ratio by 24 gives us the angles at the center of the circle corresponding to the triangle's vertices: 96 degrees (4 parts), 120 degrees (5 parts), and 144 degrees (6 parts).
The next step is to find the smallest angle of the triangle formed by the dividing points. For this, we need to calculate the external angles at the center, which are supplementary to the internal angles of the triangle. Subtract each angle from 180 degrees (180-96=84, 180-120=60, 180-144=36 degrees), then the smallest internal angle of the triangle will be the supplement of the largest angle at the center, which is 36 degrees.
Therefore, the measure of the smallest angle of the obtained triangle is 36 degrees.