To find the possible measurements of the angles of the three pieces of pizza, we can use the fact that the sum of the angles of any triangle is always 180 degrees. Since Kinser split the pizza into three pieces, we can imagine that the three pieces form a triangle, and the angles of the pieces are the angles of the triangle.
Let x, y, and z be the measures of the angles of the three pieces of pizza. Then we have:
x + y + z = 180 (the sum of the angles of a triangle is 180 degrees)
We know that one of the angles (angle KLM) measures 138 degrees. Let's assume that Kinser split the pizza into two smaller pieces (pieces A and B) with equal angles, and a third piece (piece C) with a smaller angle. Then we have:
x = y (pieces A and B have equal angles)
x + y + z = 180 (the sum of the angles of the triangle is 180 degrees)
x + y = 138 (angle KLM measures 138 degrees)
Substituting the second equation into the third equation, we get:
2x + z = 138
Since the three pieces of pizza are not equal in size, we can assume that piece C has the smallest angle. Let's assume that piece C has an angle of 18 degrees. Then we can solve for x and y as follows:
2x + 18 = 138
2x = 120
x = y = 60
Now we can find the measure of angle z:
z = 180 - x - y
z = 180 - 60 - 60
z = 60
Therefore, one possible way to split the pizza into three pieces with non-equal angles is to make two pieces (pieces A and B) with equal angles of 60 degrees each, and a third piece (piece C) with an angle of 18 degrees. Another possible arrangement is to make all three pieces unequal in size by varying the angles of pieces A and B. For example, if piece A has an angle of 50 degrees and piece B has an angle of 70 degrees, then piece C would have an angle of 60 degrees (to satisfy the equation x + y + z = 180).