Answer:
The measure of angle LMW is
![m\angle LMW=67\°](https://tex.z-dn.net/?f=m%5Cangle%20LMW%3D67%5C%C2%B0)
Step-by-step explanation:
see the attached figure to better understand the problem
step 1
Find the measure of arc MW
we know that
The inscribed angle measures half that of the arc comprising
so
![m\angle MLK=\frac{1}{2}[arc\ MW+arc\ WK]](https://tex.z-dn.net/?f=m%5Cangle%20MLK%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20MW%2Barc%5C%20WK%5D)
substitute the given values
![65\°=\frac{1}{2}[arc\ MW+68\°]\\ 130\°=[arc\ MW+68\°]\\ arc\ MW=130\°-68\°=62\°](https://tex.z-dn.net/?f=65%5C%C2%B0%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20MW%2B68%5C%C2%B0%5D%5C%5C%20130%5C%C2%B0%3D%5Barc%5C%20MW%2B68%5C%C2%B0%5D%5C%5C%20arc%5C%20MW%3D130%5C%C2%B0-68%5C%C2%B0%3D62%5C%C2%B0)
step 2
Find the measure of arc LK
we know that
-----> by complete circle
substitute the given values
![164\°+62\°+68\°+arc\ LK=360\°\\ 294\°+arc\ LK=360\°\\ arc\ LK=360\°-294\°=66\°](https://tex.z-dn.net/?f=164%5C%C2%B0%2B62%5C%C2%B0%2B68%5C%C2%B0%2Barc%5C%20LK%3D360%5C%C2%B0%5C%5C%20294%5C%C2%B0%2Barc%5C%20LK%3D360%5C%C2%B0%5C%5C%20arc%5C%20LK%3D360%5C%C2%B0-294%5C%C2%B0%3D66%5C%C2%B0)
step 3
Find the measure of angle LMW
we know that
The inscribed angle measures half that of the arc comprising
so
![m\angle LMW=\frac{1}{2}[arc\ LK+arc\ WK]](https://tex.z-dn.net/?f=m%5Cangle%20LMW%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20LK%2Barc%5C%20WK%5D)
substitute the given values
![m\angle LMW=\frac{1}{2}[66\°+68\°]=67\°](https://tex.z-dn.net/?f=m%5Cangle%20LMW%3D%5Cfrac%7B1%7D%7B2%7D%5B66%5C%C2%B0%2B68%5C%C2%B0%5D%3D67%5C%C2%B0)