An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords. Using the following formula
[tex]Inscribed\:Angle=\frac{1}{2}Intercepted\:Arc[/tex]
We have that
[tex]m∠ADB=\frac{1}{2}mAB[/tex]
We have the measure of the angle ∠ADB, which is 43º. The arc AB is:
[tex]mAB=2\cdot43=86[/tex]
∠AOB is a central angle, therefore, the measure of the intercepted arc(which is AB) is equal to the measure of ∠AOB.
[tex]mAB=m∠AOB=86^o[/tex]
The measure of ∠AOB is equal to the measure of m∠BOC, therefore
[tex]m∠BOC=86^o[/tex]
The angles ∠AOB and ∠BOC together create the central angle ∠AOC, that intercepts the arc AC, therefore
[tex]86^o+86^o=mAC\implies mAC=172^o[/tex]
The arc AC is also intercepted by the chords that form the angle ∠ADC, therefore
[tex]m∠ADC=\frac{1}{2}mAC\implies m∠ADC=86^o[/tex]
The angle ∠ADC is formed by the sum of the angles ∠ADB and ∠BDC, therefore, we have
[tex]\begin{gathered} m∠ADC=m∠ADB+m∠BDC \\ \implies86^o=43^o+m∠BDC \\ \implies m∠BDC=43^o \end{gathered}[/tex]
The measure of the angle ∠BDC is 43º.
