Question

Given: circle k(O), m LM = 164° m WK = 68°, m∠MLK = 65° Find: m∠LMW

Answer 1

Let P be a point outside the circle such that triangle LMP has legs coincident with chords MW and LK (i.e. M, W, and P are colinear, and L, K, and P are colinear). By the intersecting secants theorem,

$m\angle LPM=\dfrac{m\widehat{LM}-m\widehat{WK}}2\impliesm\angle LPM=48^\circ$

The angles in any triangle add to 180 degrees in measure, and $\angle MLK\congruent\angle MLP$ and $m\angle LMW=m\angle LMP$, so that

$m\angle MLK+m\angle LPM+m\angle LMP=180^\circ$

$\implies\boxed{m\angle LMW=67^\circ}$