Question 630504
{{{drawing(400,400,-1.5,1.5,-1.5,1.5,
line(cos(pi/2-0*2pi/7),sin(pi/2-0*2pi/7),cos(pi/2-1*2pi/7),sin(pi/2-1*2pi/7)),
line(cos(pi/2-1*2pi/7),sin(pi/2-1*2pi/7),cos(pi/2-2*2pi/7),sin(pi/2-2*2pi/7)),
line(cos(pi/2-2*2pi/7),sin(pi/2-2*2pi/7),cos(pi/2-3*2pi/7),sin(pi/2-3*2pi/7)),
line(cos(pi/2-3*2pi/7),sin(pi/2-3*2pi/7),cos(pi/2-4*2pi/7),sin(pi/2-4*2pi/7)),
line(cos(pi/2-4*2pi/7),sin(pi/2-4*2pi/7),cos(pi/2-5*2pi/7),sin(pi/2-5*2pi/7)),
line(cos(pi/2-5*2pi/7),sin(pi/2-5*2pi/7),cos(pi/2-6*2pi/7),sin(pi/2-6*2pi/7)),
line(cos(pi/2-6*2pi/7),sin(pi/2-6*2pi/7),cos(pi/2-7*2pi/7),sin(pi/2-7*2pi/7)),
line(cos(pi/2-7*2pi/7),sin(pi/2-7*2pi/7),cos(pi/2-8*2pi/7),sin(pi/2-8*2pi/7)),

locate(cos(pi/2-5*2pi/7)-.05,sin(pi/2-5*2pi/7),A),
locate(cos(pi/2-4*2pi/7),sin(pi/2-4*2pi/7),B),
locate(cos(pi/2-3*2pi/7),sin(pi/2-3*2pi/7),C),
locate(cos(pi/2-2*2pi/7),sin(pi/2-2*2pi/7),D),
locate(0-.05,0+.1,O),


green(line(cos(pi/2-3*2pi/7),sin(pi/2-3*2pi/7),0,0),
line(cos(pi/2-4*2pi/7),sin(pi/2-4*2pi/7),0,0))




)}}}
<pre>
Given: OB bisects &#8736;ABC
       OC bisects &#8736;BCD

To find: the measure of &#8736;BOC

[Note: if you had proved that OB and OC intersect at the center, you
could just observe that &#8736;BOC is a central angle and has measure {{{"360°"/7}}}.  
But we will assume you haven't proved that.]  

The sum of the measures of the interior angles of an n-sided polygon
is given by the formula (n-2)×180°

Since this is a heptagon, n=7 and the sum of the measures of the 
interior angles is (7-2)×180° = (5)×180° = 900°

Since this heptagon is regular, all the interior angles are congruent
and have equal measure.  Therefore

m&#8736;ABC = m&#8736;BCD = {{{"900°"/7}}}

Since OB and OC bisect those interior angles,

m&#8736;OBC = m&#8736;OCB = {{{1/2}}}×{{{"900°"/7}}} = {{{"450°"/7}}}

Since the sum of the measures of the interior angles of &#5123;BOC is 180°,

m&#8736;OBC + m&#8736;OCB + m&#8736;BOC = 180°
{{{"450°"/7}}} + {{{"450°"/7}}} + m&#8736;BOC = 180°

                 {{{"900°"/7}}} + m&#8736;BOC = 180°

                                  m&#8736;BOC = 180° - {{{"900°"/7}}}

                                  m&#8736;BOC = {{{"1260°"/7}}} - {{{"900°"/7}}}
                                  
                                  m&#8736;BOC = {{{"360°"/7}}} = {{{51&3/7}}}°

Edwin</pre>