Answer :
ANSWER
[tex]\text{mPL}=62\degree[/tex]EXPLANATION
We want to find the measure of the angle of the arc PL.
To do this, we first have to find the value of x.
We have that:
[tex]\begin{gathered} <\text{LMP}=(5x-19)\degree \\ <\text{LNP}=(2x+11)\degree \end{gathered}[/tex]According to circle theorem, the angles subtended at the circumference by the same arc are equal. This means that:
[tex]\begin{gathered} <\text{LMP}=<\text{LNP} \\ \Rightarrow(5x-19)\degree=(2x+11)\degree \end{gathered}[/tex]Collect like terms and simplify:
[tex]\begin{gathered} 5x-19=2x+11 \\ 5x-2x=11+19 \\ 3x=30 \end{gathered}[/tex]Divide both sides by 3:
[tex]\begin{gathered} \frac{3x}{3}=\frac{30}{3} \\ x=10 \end{gathered}[/tex]According to circle theorems, we have that the angle of an arc is equal to twice the angle subtended at the circumference by that arc.
This means that:
[tex]\begin{gathered} mPL=2(5x-19) \\ \Rightarrow\text{mPL}=10x-38 \\ \text{mPL}=10(10)-38=100-38 \\ \text{mPL}=62\degree \end{gathered}[/tex]