Given:
The figure of circle E. [tex]m\angle ABD=(11x-3)^\circ,m\angle ACD=(8x+15)^\circ[/tex].
To find:
The measure of arc AD.
Solution:
We know that the inscribed angles on the same arc are congruent and their measures are equal.
[tex]\angle ABD[/tex] and [tex]\angle ACD[/tex] are inscribed angles on the same arc AD. So,
[tex]m\angle ABD=m\angle ACD[/tex]
[tex]11x-3=8x+15[/tex]
[tex]11x-8x=3+15[/tex]
[tex]3x=18[/tex]
[tex]x=6[/tex]
Now,
[tex]m\angle ABD=(11x-3)^\circ[/tex]
[tex]m\angle ABD=(11(6)-3)^\circ[/tex]
[tex]m\angle ABD=(66-3)^\circ[/tex]
[tex]m\angle ABD=63^\circ[/tex]
We know that the intercepted arc is always twice of the inscribed angle.
[tex]m(Arc(AD))=2\times m\angle ABD[/tex]
[tex]m(Arc(AD))=2\times 63^\circ[/tex]
[tex]m(Arc(AD))=126^\circ[/tex]
Therefore, the measure of arc AD is 126 degrees.
