To solve this problem, we start by determining the side lengths of triangle \( ABC \) given the perimeter and the expressions for the side lengths.
The perimeter of triangle \( ABC \) is given as 37 units, and we have:
- \( AB = 10 \)
- \( CA = x + 6 \)
- \( BC = 2x \)
Using the perimeter formula:
\[ AB + CA + BC = 37 \]
Substituting the side lengths:
\[ 10 + (x + 6) + (2x) = 37 \]
Combining like terms:
\[ 10 + x + 6 + 2x = 37 \]
\[ 3x + 16 = 37 \]
Now, we solve for \( x \):
\[ 3x = 37 - 16 \] \[ 3x = 21 \] \[ x = 7 \]
Next, we find the lengths of \( CA \) and \( BC \):
\[ CA = x + 6 = 7 + 6 = 13 \] \[ BC = 2x = 2(7) = 14 \]
Now we have the side lengths:
- \( AB = 10 \)
- \( CA = 13 \)
- \( BC = 14 \)
To determine the order of the angle measures, we can use the relationships between the sides and angles in a triangle: the larger the side, the larger the opposite angle.
Given the side lengths:
- \( AB = 10 \) (opposite \( \angle C \))
- \( CA = 13 \) (opposite \( \angle B \))
- \( BC = 14 \) (opposite \( \angle A \))
We can order the sides from smallest to largest: \[ AB < CA < BC \quad \text{(10 < 13 < 14)} \]
Now we can determine the order of the angles:
- Since \( AB \) is the smallest side, \( \angle C \) is the smallest angle.
- Since \( CA \) is the middle side, \( \angle B \) is the middle angle.
- Since \( BC \) is the largest side, \( \angle A \) is the largest angle.
Thus, we have:
\[ m\angle C < m\angle B < m\angle A \]
The correct order of the angles from least to greatest is:
\(\boxed{m\angle C < m\angle B < m\angle A}\)
This matches the response option:
m∠C < m∠B < m∠A.
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