To list the sides of triangle RST in ascending order based on their lengths, we first need to determine the angle measures for angles R, S, and T using the provided expressions.
The angles of a triangle sum up to \(180\) degrees. So we can set up the equation as follows:
\[ m<R + m<S + m<T = 180 \]
Substituting the expressions given for the angles:
\[ (x + 28) + (2x + 16) + (x + 12) = 180 \]
Combining like terms:
\[ x + 28 + 2x + 16 + x + 12 = 180 \]
This simplifies to:
\[ 4x + 56 = 180 \]
Next, we subtract \(56\) from both sides:
\[ 4x = 124 \]
Now, dividing by \(4\):
\[ x = 31 \]
Now we can find the measures of the angles:
\[ m<R = x + 28 = 31 + 28 = 59 \text{ degrees} \]
\[ m<S = 2x + 16 = 2(31) + 16 = 62 + 16 = 78 \text{ degrees} \]
\[ m<T = x + 12 = 31 + 12 = 43 \text{ degrees} \]
Now we have the measures of the angles:
- \(m<R = 59\) degrees
- \(m<S = 78\) degrees
- \(m<T = 43\) degrees
To list them in ascending order, we arrange the angles:
\[ m<T < m<R < m<S \]
Now we will associate these angles with the corresponding sides of the triangle. The side opposite the smallest angle is the shortest, and the side opposite the largest angle is the longest. Thus, we identify the sides:
- Side opposite \(m<T\) (43 degrees) is the shortest.
- Side opposite \(m<R\) (59 degrees) is in the middle.
- Side opposite \(m<S\) (78 degrees) is the longest.
Therefore, the sides of triangle RST in ascending order (from shortest to longest) are:
- Side opposite angle \(T\) (shortest)
- Side opposite angle \(R\) (middle)
- Side opposite angle \(S\) (longest)
In summary, the sides of triangle RST in ascending order are:
RST (side opposite T), RST (side opposite R), RST (side opposite S).