Question 148657
Triangle ABC is inscribed in a circle. given that AB is a 40 degree arc and ABC is a 50 degree angle, find the sizes of the other arcs and angles in the figure

<pre><font size = 4 color ="indigo"><b>
{{{drawing(400,400,-1.5,1.5,-1.5,1.5, circle(0,0,1),
triangle(-1,0,-.7660444431,.6427876097,.17364818,-.9848078),
locate(-1.1,.4,"40°"), locate(-.82,.45,"50°"), locate(-1.1,.1,"A"),
locate(-.9,.7,"B"),locate(.2,-1,"C") )}}}

Angle ACB is an inscribed angle, subtending a 40° arc.
An inscribed angle has {{{1/2}}} the measure of its inscribed arc.
Therefore angle ACB has measure 20°, so we write that in:

{{{drawing(400,400,-1.5,1.5,-1.5,1.5, circle(0,0,1),
triangle(-1,0,-.7660444431,.6427876097,.17364818,-.9848078),
locate(-1.1,.4,"40°"), locate(-.82,.45,"50°"), locate(-1.1,.1,"A"),
locate(-.9,.7,"B"),locate(.2,-1,"C"),locate(-.27,-.5,"20°") )}}}

Since the three angles of any triangle total 180°, we find the 
remaining angle BAC by adding 50°+20°, getting 70°, then subtracting
from 180° and getting 110°, so we write that in for angle BAC:

{{{drawing(400,400,-1.5,1.5,-1.5,1.5, circle(0,0,1),
triangle(-1,0,-.7660444431,.6427876097,.17364818,-.9848078),
locate(-1.1,.4,"40°"), locate(-.82,.45,"50°"), locate(-1.1,.1,"A"),
locate(-.9,.7,"B"),locate(.2,-1,"C"),locate(-.27,-.5,"20°"),
locate(-.95,.1,"110°")  )}}}

The inscribed angle at B is 50°. It subtends arc AC, and since it 
is {{{1/2}}} of the measure of its inscribed arc, the arc AC must be
2x50° or 100°, so we write in 100° for arc AC:

{{{drawing(400,400,-1.5,1.5,-1.5,1.5, circle(0,0,1),
triangle(-1,0,-.7660444431,.6427876097,.17364818,-.9848078),
locate(-1.1,.4,"40°"), locate(-.82,.45,"50°"), locate(-1.1,.1,"A"),
locate(-.9,.7,"B"),locate(.2,-1,"C"),locate(-.27,-.5,"20°"),
locate(-.95,.1,"110°"), locate(-.9,-.8,"100°")  )}}}

The big major arc going clockwise from B around to C is subtended by 
the 110° angle at A.  And since it is {{{1/2}}} of the measure of its 
inscribed arc, the large major arc BC must be 2x110° or 220°, so we 
write in 220° for major arc BC, going clockwise from B around to C:

{{{drawing(400,400,-1.5,1.5,-1.5,1.5, circle(0,0,1),
triangle(-1,0,-.7660444431,.6427876097,.17364818,-.9848078),
locate(-1.1,.4,"40°"), locate(-.82,.45,"50°"), locate(-1.1,.1,"A"),
locate(-.9,.7,"B"),locate(.2,-1,"C"),locate(-.27,-.5,"20°"),
locate(-.95,.1,"110°"), locate(-.9,-.8,"100°"), locate(.9,.5,"220°"))}}}

Notice as a partial check that the three arcs have sum 360°.

minor are AB =  40²
minor arc AC = 100°
major arc BC = 220°
-------------------
       total = 360°

Edwin</pre>