To find the value of \( x \) and the measures of the two angles, we start with the given equations for the angles:
- \( m \angle 1 = 3x \)
- \( m \angle 2 = 2x + 10 \)
We know that the sum of the measures of the two angles is 160 degrees, so we can set up the equation:
\[ m \angle 1 + m \angle 2 = 160 \]
Substituting the expressions for the angles into the equation gives us:
\[ 3x + (2x + 10) = 160 \]
Now, combine like terms:
\[ 3x + 2x + 10 = 160 \] \[ 5x + 10 = 160 \]
Next, subtract 10 from both sides:
\[ 5x = 150 \]
Now, divide by 5:
\[ x = 30 \]
Now that we have \( x \), we can find the measures of the two angles.
For \( m \angle 1 \):
\[ m \angle 1 = 3x = 3(30) = 90 \]
For \( m \angle 2 \):
\[ m \angle 2 = 2x + 10 = 2(30) + 10 = 60 + 10 = 70 \]
Thus, the measures of the two angles are:
\[ m \angle 1 = 90^\circ, \quad m \angle 2 = 70^\circ \]
Final answer:
- \( m \angle 1 = 90^\circ \)
- \( m \angle 2 = 70^\circ \)