To determine the relationship between the angles in triangle \( \triangle ABC \) based on the side lengths, we first need to express the measures of the sides in terms of \( x \) and then solve for \( x \).
The side lengths are:
- \( AB = 10 \)
- \( CA = x + 6 \)
- \( BC = 2x \)
The perimeter of triangle \( ABC \) is given as 37 units: \[ AB + CA + BC = 37 \] Substituting the lengths into the equation: \[ 10 + (x + 6) + (2x) = 37 \] Simplifying this gives: \[ 10 + x + 6 + 2x = 37 \] \[ 3x + 16 = 37 \] Subtracting 16 from both sides: \[ 3x = 21 \] Dividing by 3: \[ x = 7 \]
Now we can find the lengths of each side:
- \( AB = 10 \)
- \( CA = x + 6 = 7 + 6 = 13 \)
- \( BC = 2x = 2(7) = 14 \)
Now we have the side lengths:
- \( AB = 10 \)
- \( CA = 13 \)
- \( BC = 14 \)
Next, we use the properties of triangles, which state that the angle opposite the longer side is larger than the angle opposite the shorter side.
Ordering the side lengths from least to greatest:
- \( AB < CA < BC \)
- \( 10 < 13 < 14 \)
This corresponds to the angles as follows:
- Angle opposite to \( AB \) (which is \( \angle C \)) is the smallest,
- Angle opposite to \( CA \) (which is \( \angle B \)) is in the middle,
- Angle opposite to \( BC \) (which is \( \angle A \)) is the largest.
Therefore, the order of the angles from least to greatest is: \[ m\angle C < m\angle B < m\angle A \]
Thus, the correct response is: m∠C < m∠B < m∠A.