Given that ZPMO and KLMN form a linear pair, we have:
[tex]\text{ZPMO + KLMN = 180}[/tex]Since ZPMO=X+30 and ZLMN=2X, then:
[tex]\begin{gathered} (x+30)+(2x)=180 \\ \Rightarrow3x=180-30 \\ \Rightarrow3x=150 \\ \Rightarrow x=\frac{150}{3}=50 \\ x=50 \end{gathered}[/tex]Therefore, x = 50.
Finally, to find the measure of both angles, we just substitute x=50:
[tex]\begin{gathered} x=50 \\ \text{ZPMO}=X+30 \\ \Rightarrow ZPMO=50+30=80 \\ \text{ZLMN}=2x \\ \Rightarrow ZLMN=2\cdot50=100 \end{gathered}[/tex]Therefore, ZPMO=80 AND ZLMN=100
