Answer:
[tex]m\angle MEJ=25^{\circ}.[/tex]
Step-by-step explanation:
The arcs LY, KM, KL and MJ together form the full revolution angle, thus
[tex]4x+50^{\circ}+6x+x+10^{\circ}+4x=360^{\circ},\\ \\15x=300^{\circ},\\ \\x=20^{\circ}.[/tex]
Note that
[tex]m\angle MOJ=4x=80^{\circ},[/tex]
then
[tex]m\angle MLJ=\dfrac{1}{2}\cdot 80^{\circ}=40^{\circ}.[/tex]
So,
[tex]m\angle ELM=180^{\circ}-40^{\circ}=140^{\circ}.[/tex]
Also
[tex]m\angle LOK=30^{\circ},[/tex]
so
[tex]m\angle KML=\dfrac{1}{2}\cdot 30^{\circ}=15^{\circ}.[/tex]
In triangle EML,
[tex]m\angle MEL+m\angle EML+m\angle ELM=180^{\circ},\\ \\m\angle MEL=180^{\circ}-15^{\circ}-140^{\circ}=25^{\circ}.[/tex]
Thus, [tex]m\angle MEJ=25^{\circ}.[/tex]
