Question 387197
{{{drawing(400,400,-1.3,1.3,-1.3, 1.3,
locate(-.2,1.1,M), locate(1.05,0,P), locate(-.5,-.93,O), locate(-1.09,-.05,N),
red(arc(0,0,2,-2,0,100)),
green(arc(0,0,2,-2,100,184)),
red(arc(0,0,2,-2,184,244)),
green(arc(0,0,2,-2,244,360)), locate(.6,.9,"100°"),
locate(-.25,.07,Q),
line(1,0,cos(100*pi/180),sin(100*pi/180)),
locate(.64,.15,"42°"),

line(cos(244*pi/180),sin(244*pi/180),cos(184*pi/180),sin(184*pi/180)),

line(cos(100*pi/180),sin(100*pi/180),cos(244*pi/180),sin(244*pi/180)),
line(1,0,cos(184*pi/180),sin(184*pi/180))
 )}}}
<pre><font size = 4 color = "indigo"><b>
You don't even need to know the measure of arc PM.  All you need to notice
 
is that inscribed angle MPN and inscribed angle MON subtend the same arc MN.

Since the measure of an inscribed angle is 1/2 of its subtended arc,

both angles MPN and MON have measure of 1/2 of the measure or arc

MN which they subtend.  So they have equal measure.  Therefore the measure 

of angle MON is also 42°.  It didn't require any calculations at all. 

{{{drawing(400,400,-1.3,1.3,-1.3, 1.3,
locate(-.2,1.1,M), locate(1.05,0,P), locate(-.5,-.93,O), locate(-1.09,-.05,N),
red(arc(0,0,2,-2,0,100)),
green(arc(0,0,2,-2,100,184)),
red(arc(0,0,2,-2,184,244)),
green(arc(0,0,2,-2,244,360)), locate(.6,.9,"100°"),
locate(-.25,.07,Q),
line(1,0,cos(100*pi/180),sin(100*pi/180)),
locate(.64,.15,"42°"),

line(cos(244*pi/180),sin(244*pi/180),cos(184*pi/180),sin(184*pi/180)),

line(cos(100*pi/180),sin(100*pi/180),cos(244*pi/180),sin(244*pi/180)),
line(1,0,cos(184*pi/180),sin(184*pi/180)), locate(-.57,-.6,"42°") 
 )}}}

Edwin</pre>