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=\frac{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}`