Explanation:
When two angles form a linear pair, they are supplementary, meaning their measures sum up to \(180^\circ\).
Let's denote the measures of the two angles as \(m<1\) and \(m<2\).
Given that \(m<1 = 5x + 9\) and \(m<2 = 3x + 11\), we can set up the equation for the sum of the angles:
\[m<1 + m<2 = (5x + 9) + (3x + 11) = 180^\circ.\]
Now, solve for \(x\) and then find the measures of each angle using the value of \(x\).
\[5x + 9 + 3x + 11 = 180\]
Combine like terms:
\[8x + 20 = 180\]
Subtract 20 from both sides:
\[8x = 160\]
Divide both sides by 8:
\[x = 20\]
Now that we have \(x = 20\), we can find the measures of \(<1\) and \(<2\) by substituting this value back into their expressions:
\[m<1 = 5x + 9 = 5(20) + 9 = 100 + 9 = 109^\circ,\]
and
\[m<2 = 3x + 11 = 3(20) + 11 = 60 + 11 = 71^\circ.\]
So, the measure of \(<1\) is \(109^\circ\) and the measure of \(<2\) is \(71^\circ\).