Triangle ABC
In triangle ABC, let's assume the lengths of the sides are a, b, and c respectively.

Given Ratio
According to the given ratio, a:b:c = 4:5:6.

Let's Find the Angle Measures
To find the angle measures in triangle ABC, we can use the Law of Cosines.

The Law of Cosines states that in a triangle with sides of lengths a, b, and c, the cosine of one of the angles, let's say angle A, can be found using the formula:
cos(A) = (b² + c² - a²) / (2bc)

Similarly, we can find the cosines of angles B and C using the same formula.

Calculating the Cosine of Angle A
Using the given ratio, we can express the lengths of the sides as follows:
a = 4x, b = 5x, and c = 6x.

Now, substituting these values into the Law of Cosines formula, we have:
cos(A) = (5x)² + (6x)² - (4x)² / (2 * 5x * 6x)

Simplifying the equation, we get:
cos(A) = (25x² + 36x² - 16x²) / (60x²)
cos(A) = 45x² / 60x²
cos(A) = 3/4

Finding Angle A
To find angle A, we can use the inverse cosine function (arccos) to find the angle whose cosine is 3/4.
A = arccos(3/4)

Using a calculator, we find A ≈ 41.41 degrees.

Calculating Angle B
Since we know the ratios of the sides, we can find the ratio of angle measures in triangle ABC.

A:B:C = a:b:c = 4:5:6

Therefore, angle A:B:C = 4:5:6

Using this ratio, we can find angle B as follows:
B = (180° / (4+5+6)) * 5
B = (180° / 15) * 5
B = 12° * 5
B = 60°

Calculating Angle C
To find angle C, we can use the fact that the sum of the angles in a triangle is 180 degrees.
C = 180° - A - B
C = 180° - 41.41° - 60°
C ≈ 78.59°

Conclusion
In triangle ABC, the angle measures are approximately A ≈ 41.41°, B = 60°, and C ≈ 78.59°.